Reader
Elektrotechniek,
Wiskunde en Informatica
Amplifiers
Dr. J. Hoekstra
Reader for the course Amplifiers and Instrumentation (EE1C31)
Supplementary material for the course Transistor Circuits (EE2C1)
(version december-2014).
Delft
University of
Technology
Artikelnummer 06926880024
Contents
1 Introduction
1.1 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Information-carrying signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Information loss, dissipation, and noise . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Amplification of signal power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Top-down approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Analysis and synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Circuit Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Simulation codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
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2 Electrical Signals, Components, and Circuit Theory
2.1 Electrical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Continuous or discrete signals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Sine waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Average power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Time domain and frequency domain . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Phasor diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.7 Bandwidth and transmission channels . . . . . . . . . . . . . . . . . . . . . . . .
2.1.8 Pulse waveforms and duty cycle . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Components and elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Nonlinear resistor and diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Small-signal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Nonlinear resistors can introduce harmonics . . . . . . . . . . . . . . . . . . . . .
2.2.5 Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.7 * Memristor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.8 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.9 * Gyrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.10 Ideal voltage source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.11 Ideal current source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.12 Nullator, norator, and nullor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.13 The dependent voltage source . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.14 The dependent current source . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
2.3
Circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Kirchhoff’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Linear circuits and superposition . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 * Tellegen’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Thévenin and Norton equivalents . . . . . . . . . . . . . . . . . . . . . . . . . .
Real sources and equivalent circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Real sources, sensors and actuators . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
38
39
39
42
44
45
45
48
50
3 Processing of Voltage and Current Signals
3.1 Addition and subtraction of current and voltage signals . . . . . . . . . . . . . . . . . . .
3.1.1 Addition and subtraction of ideal current and voltage signal sources . . . . . . . .
3.1.2 Addition and subtraction of real current signal sources . . . . . . . . . . . . . . .
3.1.3 Addition and subtraction of real voltage sources . . . . . . . . . . . . . . . . . . .
3.2 Division of voltage signals and division of current signals . . . . . . . . . . . . . . . . . .
3.2.1 Division of voltage signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Division of current signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Multiplication of voltage and current signals . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Passive multiplication of voltage and current signals . . . . . . . . . . . . . . . .
3.3.2 Active multiplication of voltage signals . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Active multiplication of current signals . . . . . . . . . . . . . . . . . . . . . . .
3.4 Conversion from current signals to voltage signals, and conversion from voltage signals to
current signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Conversion from current signal to voltage signal . . . . . . . . . . . . . . . . . .
3.4.2 Conversion from voltage signal to current signal . . . . . . . . . . . . . . . . . .
3.5 Filtering signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Impedances of series and parallel reactive circuits . . . . . . . . . . . . . . . . . .
3.5.2 Transfer functions and passive filters . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Simulation codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
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52
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60
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61
4 Amplifier Models, Two-ports, and Nullor
4.1 Modeling of amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Sources and loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Equivalent circuits for the amplifier . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Open-circuit voltage gain and loading . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Ideal voltage amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Two-ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The two-port admittance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 The two-port impedance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 The two-port transmission or chain matrix . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Input and output resistances using the chain matrix . . . . . . . . . . . . . . . . .
4.2.5 Combining two-ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Transactors and nullor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Chain matrix of the nullor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Transactors: ideal amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 *Example: proof of vi = 0 and ii = 0 for the voltage-to-voltage transactor with
μ→∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Amplifier configurations with the nullor, revised . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 The voltage-voltage converter or voltage amplifier . . . . . . . . . . . . . . . . .
4.4.2 The voltage-current converter or transadmittance amplifier . . . . . . . . . . . . .
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97
2.4
2.5
2.6
3
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66
70
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79
101
102
103
104
4.5
4.4.3 The current-voltage converter or transimpedance amplifier . . . . . . . . . . . . . 105
4.4.4 The current-current converter or current amplifier . . . . . . . . . . . . . . . . . . 106
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Amplifier configurations I: The op amp nullor
5.1 Circuits with the ideal operational amplifier . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Voltage amplifier based circuits . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Transresistance amplifier based circuits . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Additional configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Differential amplifier and operational amplifier with finite gain . . . . . . . . . . . . . . .
5.3 Non-inverting voltage amplifier circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Open-loop gain and closed-loop gain . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Output resistance and input resistance . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Special case: the voltage follower . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Dynamic model for the op amp: saturation, stability, and feedback . . . . . . . . .
5.3.5 Gain-Bandwith product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Inverting voltage amplifier circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Open-loop gain and closed-loop gain . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Input resistance and output resistance . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Gain-bandwidth product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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135
6 Amplifier configurations II: The single bipolar junction transistor nullor
6.1 npn transistor characteristics in CE configuration . . . . . . . . . . . . . . . . . . . . . .
6.2 Small-signal equivalent circuits in CE configuration . . . . . . . . . . . . . . . . . . . . .
6.2.1 Approaching the nullor with BJT in CE configuration . . . . . . . . . . . . . . . .
6.3 Simple voltage amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Amplifiers based on a bipolar transistor nullor implementation . . . . . . . . . . . . . . .
6.4.1 Transconductance based voltage amplifier . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Voltage follower or emitter follower . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Common-base amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4 Transresistance based amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.5 Basic building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.6 Differential amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 pnp transistor characteristics in CE configuration . . . . . . . . . . . . . . . . . . . . . .
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152
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155
7 *Amplifier configurations III: The field-effect transistor nullor
7.1 FET transistor characteristics in CE configuration . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Junction field-effect transistor (JFET) . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Metal oxide field-effect transistor (MOSFET) . . . . . . . . . . . . . . . . . . . .
7.2 Basic building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4
Chapter
1
Introduction
This chapter introduces the basic ideas concerning amplification with electronic circuits. In electronic
circuits, charges—currents of electrons—are manipulated. The goal of this manipulation is the processing
or transfer of signals. These signals may carry information; the electronic circuit can then be seen as an
information processing tool. In electronic circuits, signals carrying information can be in time varying
voltages and in time varying currents.
Information can be lost by interference from the environment or to a phenomenon called noise. Besides
this, the strength of the information carrying signals may decrease due to dissipation during transmission or
processing. In all cases amplification of signals is necessary. In case of noise or interference the signals must
be amplified before and in case of dissipation the signals must, in general, be amplified during transmission
or processing.
In the electronics’ domain this amplification is done using transistors, often integrated with resistors,
small capacitors, and very simple inductors in integrated circuits (IC’s), popularly called chips.
This course deals with amplification by means of electronic components. It is a first introduction to
devices, circuits, and concepts, which make amplification in the electronics’ domain so successful in our
current information en communication oriented society.
1.1 Circuits
A physical circuit is made up of physical elements like: resistors, capacitors, inductors, transformers,
diodes, and transistors—connected to energy sources that give rise to currents and voltages in the circuits.
In circuits, currents can be maintained when a closed circuit (sometimes also called complete circuit) exists
and energy is delivered.
Circuit theory and the theoretical description of electronics are based on the concept of modeling. We
describe the physical circuit in terms of an idealized model that is an interconnection of idealized elements.
The idealized elements are simple models of the physical elements or physical phenomena.
Although physical elements and phenomena may be described only approximately, idealized elements
are by definition characterized precisely. Of the physical elements we can model their resistances by ideal
resistors, their capacitances by ideal capacitors, their (self-) inductances by ideal inductors. Electric fields
and magnetic fields are modeled by adding ideal capacitors and ideal inductors, respectively.
The ideal circuit elements are called lumped elements, that is, their size is assumed to be small compared to the wavelength corresponding to their highest frequency of normal operation. Lumped elements
may have two terminals, as in a resistor, or more than two terminals, as in a transformer or a transistor.
For two-terminal lumped circuits, the current through the element and the voltage across it are well-defined
quantities. For lumped elements with more than two terminals, the current entering any terminal and the
voltage across any pair of terminals are well defined at all times.
It is customary to choose a reference direction for the current through any two-terminal lumped element
5
A
i
+
v
B
Figure 1.1: Definition of reference voltage and current in a two-terminal lumped element.
with terminals A and B as shown in Figure 1.1. For example, it may be a resistor, capacitor, or diode. It is
of fundamental importance to be very precise concerning the reference direction of the voltage across the
element and the direction of the current through it. The reference voltage is indicated by the plus and minus
symbols located near the terminals A and B. The reference direction of the current is indicated by an arrow.
By convention the current i is positive at time t [that is, i(t) > 0] whenever a net flow of positive charges
enter at terminal with the positive voltage sign A and leaves the element at terminal with the negative voltage
sign B at the same time t; this convention is generally called: the passive sign convention.
1.2 Information-carrying signals
Information is coded in signals. Signals are functions of time, or place, or color, or some other quantity that
can have different values. In electronics the information carrying signals are in time varying voltages and
in time varying currents; the amplitude of the physical quantity representing the signal is always a function
of time. Often, the signals have to be processed in order to have good information transfer.
In any point in an electronic circuit the current and the voltage represent the signal power p = p(t)
according to
p(t) = i(t)v(t).
(1.1)
Power is the rate of change of electrical energy w = w(t):
p(t) =
dw
.
dt
(1.2)
Sometimes an electronic circuit does not influence the signal power, but often part of the signal power is
converted to heat during transport or processing of the signal. Consequently the signal power decreases and
so does the reliability of the information processing. To prevent this, the possibility of addition of signal
power must be available. The dimension of power is energy divided by time, so the law of conservation of
energy requires that for this addition additional energy sources must be available.
Fortunately, circuit components are available that are able to increase signal power when an additional
energy source is added. Often the additional energy source is a battery or accu. The components that can
control the addition of energy are transistors and are known as active components.
1.3 Signal processing
Amongst others, electronics circuits are used in various signals processing tasks such as:
• transporting information, whether it is by wiring or through the air;
6
v
v
t
t
a) analogue signal
b) analogue signal plus noise
Figure 1.2: Noisy signals.
• storing information;
• extracting information from sensory inputs;
• calculating, like in computers or mobile phones.
There are certain elements in these processing tasks that are common to many electronic systems. These
include amplification, the addition and subtraction of signals, division, integration, differentiation, and
filtering (that is, changing the relative magnitude of signals at different frequencies). All these processing
tasks will be treated in this course.
1.4 Information loss, dissipation, and noise
Information might be lost due to dissipation in the processing circuitry or due to noise. Noise is caused in
the circuitry itself. Both dissipation and noise are energetic phenomena in electronic circuits.
Transporting signals by wiring or processing signals in (sub)circuits causes dissipation. Dissipation
is loss of (signal) energy. Part of the energy supplied to the circuits is converted to heat; this is why
processors in computers have to be cooled. The losses can be due to finite resistance of the conductors in
the circuits or to imperfections of isolating materials (dielectric losses). In electronics circuits this loss of
energy is modeled by (non)linear passive resistors, in which the power entering these elements is always
positive. Due to dissipation the amplitude of the signals decreases and eventually the information will be
lost due to noise. Compensating dissipation needs the addition of energy in such a way so as to maintain
the information in the signal. Accordingly the signal power has to be amplified.
In electronic circuits noise is also an energetic phenomenon. It comes to circuits as thermal energy, or
other mechanisms in which electrons in different energy levels are involved. Being a random phenomenon
noise is not confined to a single frequency.
All electronic components that possess resistance generate what is called thermal noise, also called
Johnson noise, as a result of random, thermally induced motion of electrons. The random motions of electrons cause random variations in current through and voltage across elements having resistances. Thermal
noise can be reduced by reducing the temperature of the circuit. Thermal noise has components at all frequencies with equal noise power in all parts of the spectrum. For this reason it is often described as white
noise. Figure 1.2 shows a signal without and with noise.
Although theoretically the noise has an infinite bandwidth—that is, an infinite frequency range—for
a given application only the noise within the bandwidth B = B (Δf ) of the operation of the circuit is
important. This gives us the opportunity to define the noise power Pn = P (T, B) related to the absolute
temperature T and the bandwidth B:
Pn = 4kT B.
(1.3)
7
The term 4kT is the thermal energy related to average kinetic energy of the electrons, k is Boltzmann’s
constant (k = 1.3805 x 10−23 J/K). The definition of the bandwidth with dimension of frequency (time−1 )
completes this definition of noise power.
For a given resistor with resistance of value R, the thermal power dissipated in it, is related to the root
mean square (rms) value of the voltage V across the resistor, V (rms). As we will see in section 2.1.4:
1√
V2
=
2V
(1.4)
V (rms) =
2
2
Using
P =
V2
,
R
(1.5)
we obtain
1
Vn (rms) = (4kT BR) 2 .
(1.6)
Thus although the noise power generated by all resistors is equal and dependent on temperature; the noise
voltage increases with the value R of the resistor. Larger resistors have larger random voltage fluctuations
than smaller resistors.
For the current fluctuations due to temperature in a resistor holds a similar criterium. For a given resistor
with resistance of value R, the thermal power dissipated in it, is related to the root mean square (rms) value
of the current I through the resistor:
P = I 2 R,
(1.7)
which results in
In (rms) =
4kT B
R
21
(1.8)
.
The noise current decreases with the value R of the resistor. Larger resistors have smaller random current
fluctuations than smaller resistors.
The noise phenomenon limits the minimum signal level a circuit can usefully respond to, because there
will always be a small but significant amount of thermal noise in the circuit.
Due to semiconductor devices there are several other sources of noise in electronic circuits such as shot
noise, seen in very low-level signals where the finite number of electrons in a current becomes significant—
often because of a quantum mechanical phenomenon called tunneling—or flicker noise (1/f noise), which
has often to do with effects at the surface of semiconductor devices . Shot noise power is directly proportional to current; its effect is greatest when a semiconductor junction has a high internal resistance. Flicker
noise power is inverse proportional to the frequency and is a dominant noise source only at low-frequencies.
Dealing with noise needs the addition of energy in such a way so as to maintain the information in the
signal before the information is processed. Again the signal power has to be amplified.
1.5 Signal-to-noise ratio
The quantity that measures the relationship between the strength of an information-carrying signal in an
electrical system and the random fluctuations in amplitude, phase, and frequency superimposed on that
signal and collectively referred to as noise is called the signal-to-noise ratio. For analog signals, the ratio,
denoted S/N , is usually stated in terms of the relative amounts of power contained in the signal and noise.
The signal-to-noise ratio is commonly expressed as the ratio of the signal power Ps , to the noise power Pn :
S/N ratio =
Ps
.
Pn
(1.9)
8
IE
c
b
VBE
+
+
e
VCE
IE
VBE
0.7
(a)
(b)
Figure 1.3: (a) Bipolar junction transistor in a so-called common-emitter configuration (oversimplified); (b)
relation between emitter current and base-emitter voltage (BJT-characteristic).
This result is commonly expressed in decibels:
S/N ratio = 10 log10
Ps
Pn
dB.
(1.10)
As the signal and the noise are present together at the same point in the circuit, they see the same impedance.
We can now express the power ratio in terms of the more convenient ratio of root mean square (rms) signal
voltage Vs to root mean square noise voltage Vn :
S/N ratio = 20 log10
Vs
Vn
dB.
(1.11)
1.6 Amplification of signal power
Now, it is clear that amplification of signal power is a major issue in analog electronic design. It is also
clear that for amplification of signal power one or more external energy sources must be available. Often
the external energy source can be a battery. The energy supplied by the battery, then, has to be converted to
additional signal power in a controlled manner. Controlled in a way that the to-be-amplified signal power
appears at the output of an electronic device such that both the voltage signal and the current signal are
multiplied by constants with absolute values larger than or equal to 1. Fortunately, transistors can perform
this conversion.
For example, signal power can be amplified using a bipolar junction transistor (BJT). It is common
to refer to bipolar transistors simply as ’transistors’. The bipolar transistor has three terminals—called
emitter, base, and collector. Figure 1.3(a) shows a BJT in a configuration called common-emitter amplifier;
the figure shows a somewhat oversimplified version of this amplifier. The ideal batteries VBE and VCE
adjust—in electronics often called bias—the transistor such that a small current IB enters the transistor at
the base, a large current IC enters at the collector, and a large current (the sum IB + IC ) leaves the emitter.
The construction of the transistor is such that for normal operation the current entering the collector is
proportional to the current leaving the emitter, that is,
IC = αIE .
(1.12)
In a good transistor the value of α is between 0.990 and 0.995. Because IE = IB + IC the above result
leads to:
IB = IC
1−α
IC
=
α
hF E
or
I C = hF E I B
9
(1.13)
IC+ic
Rl
IB+ib
VBE
IE
Vl+vl
+
+
IE
VCC
+
vbe ~
Q
ie
vbe
IE+ie
VBE
(a)
VBE
(b)
Figure 1.4: (a) Principle of signal amplification with a npn BJT; (b) VBE IE -characteristic of the BJT. The
operating point is Q; the dashed line is the tangent in Q.
with
hF E =
α
.
1−α
(1.14)
hF E is commonly called the dc current gain of the transistor. For good transistors, a typical value for hF E
is 200. However, typical general-purpose transistors of a given type might have a spread of current gain in
the range of 80 - 350.
When biased correctly the current at the emitter can be measured as a function of the voltage across
the base and the emitter, the base-emitter voltage. The result is a graph that shows a transfer characteristic
of the npn transistor, see Figure 1.3(b). Of course there are many characteristics because we have three
terminals, for example we could also measure the current at the collector as a function of the base-emitter
voltage, or the current at the collector as a function of the collector-emitter voltage, et cetera; they are all
called transistor characteristics.
It is also possible to calculate a characteristic using the theory of semiconductors. The advantage of
this is, that the formula obtained in this way gives us the opportunity to understand the influence of many
physical quantities related to the construction of the transistor, and makes it possible to design for required
specifications. The theory of semiconductors shows that there is a well defined relation between the baseemitter voltage and the current at the emitter. For an active silicon npn BJT holds:
VBE
,
(1.15)
IE = Is exp
VT
where VT is the so-called thermal voltage and has a value VT ≈ 25 mV at room temperature. Is is a
very small constant determined by the technology to make the base, and is typically 10−12 A in a silicon
BJT.
We are now going to consider the amplification of signal power in the common-emitter amplifier circuit.
In series with the dc voltage source VBE a signal voltage source vbe is placed. A load resistor is added;
the voltage source VCC provides the energy for biasing and amplification, see Figure 1.4(a). For example,
this signal voltage source could be a microphone. As a consequence of a varying base current, the collector
current will vary and thus the voltage over the load resistor. As an aside, please notice that in reality the
biasing of the transistor is never done with two dc voltage sources.
Figure 1.4(b) shows how the signal voltage vbe causes a variation in the emitter current ie . To obtain
an equation for the collector current ic as a function of vbe we replace the nonlinear characteristic with its
tangent in point Q, called operating point. From equation 1.15 we derive
IE
dIE
=
.
dVBE
VT
(1.16)
10
Using IC = αIE we obtain
dIC
= α.
dIE
(1.17)
Using both equations we find:
gm =
dIC
dIC dIE
IE
IC
=
=α
=
.
dVBE
dIE dVBE
VT
VT
(1.18)
gm is called the transconductance and depends on the value of IC .
For the variation of the collector current ic holds
ic = gm vbe .
(1.19)
So, for the variation of the voltage across the load resistor Rl we find:
vl = ic Rl = gm Rl vbe .
(1.20)
We find the absolute value of the voltage amplification of the amplifier |Av | = |vl /vbe | to be gm Rl , the
absolute value of the current amplification of the amplifier |Ai | = |ic /ib | to be hF E , and the power amplification |Ap | to be:
|Ap | =
|ic vl |
= g m R l hF E .
|ib vbe |
(1.21)
Of course, the reasoning is only valid if |vbe | VBE , and is called a ’small-signal approach’. DC
supply voltages were used to bias the nonlinear transistor at an operating point and a small ac signal is
injected into the circuit. To analyze the circuit the analysis is split into two parts. First, we analyze the
dc circuit to find the operating point. In this analysis of bias conditions, we must deal with the nonlinear
aspects of the transistor. In the second part of the analysis, we consider the small ac signal. Because the
transistor characteristics are approximately linear if we consider sufficiently small regions of operation, we
can deal with a linear small-signal model to use in the ac analysis.
If we look carefully at what we have been doing, we notice that the energy source VCC provides the
power for the amplification. The voltage and current signals at the input in fact control the voltage and
the current at the output of the amplifier, and thus the power delivered. To actually see how the amplifier
function is controlled it is worth seeing how a single control device may be used to form an amplifier.
Consider a voltage divider circuit made of a resistor and a variable resistor, see figure 1.5(a). The current
Io and the output voltage Vo are determined by
Io =
V
R1 + R2
Vo = I o R 2 =
(1.22)
V
R2
R2 =
V.
R1 + R2
R1 + R2
(1.23)
The variable resistance R2 controls both current and output voltage; if R2 is decreased the current is
increased and the output voltage is decreased, if R2 is increased the current is decreased and the output
voltage is increased. If we now replace R2 with some control device that has an input Vi that controls its
resistance, a varying Vi will vary the output Vo . In fact, the control device does not have to be a voltage
controlled resistance at all; any device in which the current is controlled by an external input can be used.
This is what we we saw when we were considering the amplifier transistor circuit.
This opens the way to have a control device in our amplifier circuit, in which only a voltage is present
at the input that determines the current through the device. These devices exist and are called field effect
transistors (FETs); they are commonly referred to as FET’s (examples are the so-called MOSFETs you
can find in any computer chip).
We remark finally that having a small input voltage and a large output voltage in our amplification
circuit, we can cascade (put behind each other) various amplifier stages to obtain a large amplification. The
overall gain is then the product of the gains of all the stages.
11
V
V
R1
R1
Vo
R2
Io
Vo
Vi
control
device
Io
0V
0V
(b)
(a)
Figure 1.5: (a) simple voltage divider circuit; (b) voltage divider circuit using a control device.
1.7 Top-down approach
In the design process various levels exist. Designs can be made on a single transistor level, or even at the
level of technology (design based on topological issues on a chip), or at a level of combining several stages,
or at a real system level, or on the level of the application.
There are at least two reasons to approach general amplifier design starting from application-level working downwards to the implementation-level—that is, top-down. First on the application level sensory inputs have specific electronic properties and often have a nonlinear iv-characteristic; second on the single
transistor-level or implementation-level the properties of the transistors and FETs vary, often called device
variability. A suitable system design concept is necessary to cope with this variability.
For this reason this course starts with circuit and system concepts before the physics of the semiconductor devices is dealt with.
1.8 Analysis and synthesis
Another reason for a top-down approach is the challenge of amplifier circuit synthesis, in plain words: how
do we make good amplifier circuits. Is good knowledge of semiconductor device physics and basic electronics often enough to analyse existing circuits, for designing amplifier circuits this is generally speaking
not enough. Unless we make or select various different designs for a specific application to find the best
one, we have no structural guidelines only using analysis tools.
1.9 Circuit Simulation
Simulation of circuits is an important tool for analyzing circuits. Examples are the comprehensive Pspice,
or the simple circuit simulator applet at www.falstad.com/circuit/ . At the end of most chapters simulation
codes of circuits can be found that can be used to simulate them with the falstad simulation tool.
1.10 Problems
1. For the load in Figure 1.1 , v(t) = u(t) V; u(t) is the step function—that is, the voltage is switched
on at t = 0—and i(t) = 10e−2t u(t) mA. Find the power absorbed by the load at t = 0.5 s. Find
12
and sketch w(t) in mJ. What is the total energy generated by the load? (Ans. p(0.5) = 3, 68mW;
w(t) = 5(1 − e−2t )u(t); w(t = ∞) = −5mJ)
2. Estimate the rms noise voltage and rms noise current produced by thermal noise in a 10 kΩ resistor
at room temperature when measured over the typical audio-frequency bandwidth of 20 kHz. (Ans.
1.82 μV; 0.18 nA)
3. At a particular point in the circuit, a signal of 5 volts rms is corrupted by 10 mV of noise. What is the
signal-to-noise ratio (in decibels) at this point? (Ans. 54 dB)
4. Calculate the absolute voltage amplification, the absolute current amplification, and the signal power
amplification in the simplified common-emitter amplifier circuit of fig. 1.4, when given that the load
resistor has a resistance of 2.7 kΩ. The transistor is characterized by hF E = 200. The voltage sources
VBE and VCC have values 0.6 V and 10 V, respectively; with this bias the collector current IC in the
operating point Q (VBE = 0.6 V) is IC = 2.6 mA. (Ans. |Av | = 280; |Ai | = 200; |Ap | = 56, 160)
5. When we consider the transistor amplifier as a voltage divider subcircuit in which one of the resistors is replaced by a voltage or current controlled nonlinear resistor (the transistor), is the variable
nonlinear resistor (the transistor) then generating or absorbing power?
6. Do a computer simulation of the circuit of problem 4. Discuss the results.
1.11 Simulation codes
Simulation of the circuit in figure 1.3:
$ 1 5.0E-6 10.2 71 5.0 50
t 448 320 496 320 0 1 -9.4 0.6 100.0
w 448 320 416 320 0
w 496 304 496 272 0
w 496 272 528 272 0
w 496 336 496 368 0
w 496 368 528 368 0
w 496 368 416 368 0
v 416 368 416 320 0 0 40.0 0.6 0.0 0.0 0.5
v 528 368 528 272 0 0 40.0 10.0 0.0 0.0 0.5
Simulation of the circuit in figure 1.4:
$ 1 5.0E-6 10.2 71 5.0 50
t 480 320 528 320 0 1 -3.6 0.595 100.0
w 480 320 448 320 0
w 528 304 528 272 0
w 528 272 560 272 0
w 528 336 528 368 0
w 528 464 560 464 0
w 528 464 448 464 0
v 448 368 448 320 0 0 40.0 0.6 0.0 0.0 0.5
v 560 464 560 368 0 0 40.0 10.0 0.0 0.0 0.5
r 560 272 560 368 0 2700.0
w 528 368 528 464 0
v 448 464 448 368 0 1 40.0 0.0050 0.0 0.0 0.5
g 528 464 528 480 0
o 11 64 0 35 0.02 3.9E-4 0 -1
o 9 64 0 35 10.0 0.00625 1 -1
13
Intentionally left blank.
14
Chapter
2
Electrical Signals, Components, and
Circuit Theory
In this chapter elementary properties of signals, components, and circuit analysis are introduced.
2.1 Electrical signals
In electronic circuits the important signals are in time varying currents and voltages. The signals come from
the outside world by sensors, and are produced to control the outside world through actuators. Sensors
sense external quantities and actuators control external quantities. Together they are called transducers.
In the outside world the signals are often not electrical. The sensors transform those external signals into
electrical signals—that is, into time varying currents and voltages.
2.1.1
Continuous or discrete signals
Most physical quantities from the outside world vary in a continuous manner leading to continuous signals.
Continuous signals are generally referred to as analogue signals. Sometimes physical quantities change
discontinuous, abrupt, or step wise. If quantities change abruptly between a number of fixed values, they
are referred to as discrete signals.
Analogue signals
Analogue signals have a continuously changing or constant amplitude during its existence. An example, in
this case a time varying voltage, is shown in Figure 2.1(a).
Sampled signals and sample-and-hold
Sometimes we are interested in the magnitude of the signal at specific points in time. In this case we take
a sample of the signal at these points in time. If we give the new signal the value zero in between the
measuring points, the process is called sampling and the resulting signal is a sampled signal (Figure 2.1(b).
And, the circuit that samples a signal is called a sampling circuit. As an example, Figure 2.2(a) shows a
simple circuit performing the action of sampling a voltage signal. The switch is closed only when we want
to sample the circuit. When the switch is closed the sampled voltage appears over the resistor R.
If we want to remember the value of the last sampled voltage, we can use a so-called sample-and-hold
circuit see Figure 2.1(c). In essence, a sample-and-hold circuit can be simply a capacitor and a switch
providing that the circuitry at the input delivers the required current to charge the capacitor. As shown in
figure 2.2(b) when the switch is closed, the capacitor quickly charges or discharges (a negative current from
15
v
v
t
t
a) analogue signal
b) sampled signal
v
v
t
t
c) sample-and-hold
d) binary signal
Figure 2.1: Various signals.
the input circuitry) so that the voltage across the capacitor—and hence the voltage at the output—equals the
input voltage. If the switch is now opened, the capacitor holds it charge and the output voltage is constant.
In practice this simple circuit requires a low impedance of the source circuitry to have a fast charging or
discharging of the capacitor. However, having a low impedance the input circuitry is heavily loaded (must
be able to provide a large current if necessary, and in both directions), possibly distorting the input value.
Also the capacitor should be connected to a load that will tend to discharge the capacitor due to its finite
resistance, thus a load with a very high resistance is preferred. As we will see later these problems can be
overcome by using buffer amplifiers (amplifiers with an amplification of one and having a very high input
and a very low output resistance); Figure 2.3 shows the architecture and a possible implementation. The
application of these buffers is a fine example of a top-down designing process.
Digital and binary
Digital signals are signals that are discrete in both magnitude and time. If a sample-and-hold circuit only
samples at unit time steps and produces sampled values that are multiples of a unit value, the signal is both
discrete in time and discrete in magnitude. The signal that results is called a multi-valued digital signal.
Discrete signals may take a number of forms dependent on the number of discrete levels. Is the number
of discrete levels only two—for example 0 and 1—the signal is called a binary signal, see Figure 2.1(d).
Binary signals are the signals typically used in electronic logic circuitry and computers.
vi
vo
vo
vi
R
C
(b) sample-and-hold circuit
(a) sampling circuit
Figure 2.2: Simplified circuits for sampling and sample-and-hold.
16
buffer
amplifier
input
buffer
amplifier
output
(a)
v2(t)
+
+
+
_
v1(t)
+
output
(b)
Figure 2.3: (a) Using buffers in the simple sample-and-hold circuit for sample-and-hold; (b) a simplified
practical circuit for sampling-and-holding a voltage signal, in which the buffers are implemented by socalled unity gain buffer amplifiers (these will be discussed in chapter 5), and the switch by an field-effecttransistor pulsed by a voltage source (the field-effect-transistor will be discussed in chapter 7)
2.1.2
Sine waves
The sine wave is a common type of alternating current (ac) and alternating voltage. It is also referred to as
a sinusoidal wave or just sinusoid. The general expression for a sinusoidal wave describing a voltage signal
or a current signal is, in the time domain:
v(t) = V sin(ωt + φ)
and
(2.1)
i(t) = I sin(ωt + φ).
(2.2)
Note that time independent voltages and currents are in capitals, while time dependent currents and voltages
are in lower case letters. In these formulas V and I are the peak values in absolute values or amplitude of
the wave, ω the angular frequency
ω = 2πf =
2π
,
T
(2.3)
and φ the phase. The phase φ determines the value of the wave at time t = 0. The dimension of ω is radians
per second.
2.1.3
Periodicity
A signal is periodic when it repeats itself in identical cycles after a time T , called its period. The frequency
f of a signal is the number of cycles that a signal completes in one second:
f=
1
.
T
(2.4)
In electronics periodic signals are used for certain signal processing tasks and as signals for measuring
electronic properties of devices and circuits. Besides measuring circuits the analysis of a circuit’s response
17
to periodic excitations can be understood by realizing that every periodic signal x(t) can be decomposed in
sinusoidal waves of different frequencies:
x(t) =
∞
Ak cos(ωk t + φk )
with
ωk = kω,
(2.5)
k=0
with ω = 2π/T . The above equation comes from a branch in mathematics called Fourier analysis, it tells
that a repetitive waveform is always composed of the sum of a constant, representing the average value of
the signal (sometimes called the zero harmonics) and sine waves of various frequencies that are multiples of
the so-called first harmonic or fundamental frequency ω0 = 2π/T (the term for k = 1). Figure 2.4 shows
how a periodic square wave is related to its Fourier components.
k=0
∑
k=1
k=0
k=1
∑
k=16
k=0
k=3
k=2
Figure 2.4: A periodic waveform as a sum of fourier components
2.1.4
Average power
When we excite a current in a resistor with a sinusoidal voltage source v = V sin(ωt) the current will be a
sinusoidal too and is equal to
i = I sin(ωt) =
V
sin(ωt).
R
(2.6)
The average power Pave delivered to the resistor and provided by the source is
1 T
1 T v2
dt =
vidt =
Pave =
T 0
T 0 R
1 1 T
1 V2 T 1
2
=
(V sin(ωt)) dt =
(1 − cos(2ωt))dt =
TR 0
T R 0 2
18
2
1
1 V2 T
V
2 1
= √
= Veff
(2.7)
T R 2
R
R
2
√
The expression V / 2 is called the effective value or root mean square value of the sinusoidal voltage.
We see it is useful to define
1
2
·
Pave = Veff
(2.8)
R
=
Also we can define
2
·R
Pave = Ieff
(2.9)
Note that to obtain the average power in equation 2.7 the current i and the voltage v must have the same
phase.
2.1.5
Time domain and frequency domain
A sine wave can be represented in the time domain, but also in the frequency domain. In the frequency
domain the sine wave is characterized by its frequency and its effective value. However, in the frequency
domain phase information is lost.
Figure 2.5 shows the sine wave in the time domain (a) and frequency domain (b).
V
Veff
t
f
(a)
(b)
Figure 2.5: representation of a sine wave in (a) time domain and (b) frequency domain
2.1.6
Phasor diagrams
Sinusoidal signals are characterized by their amplitude A, their angular frequency ω, and their phase ϕ. In
linear circuits, that are, circuits only consisting of linear elements, voltages and currents at different points
in a circuit excited by a single energy source all have a common frequency. However, the amplitude of the
signals and the phase relationship between these signals may differ.
A common tool to analyze circuits with sinusoidal signals and sources is the phasor diagram, which
allows us to represent both amplitude and phase of a signal. For example, the two sine waves in Figure 2.6(a)
can be represented by a phasor diagram in figure 2.6(b). As can be seen, sine wave B leads sine wave A by
30◦ and has lower amplitude. Mathematically phasor quantities can be expressed as complex numbers.
For any element the sinusoidal steady-state voltage across it and the sinusoidal steady-state current
through it, we can write
v(t) = V cos(ωt + ϕv )
(2.10)
i(t) = I cos(ωt + ϕi ).
(2.11)
and
19
Im
V
A
B
φ
B
φ
t
A
Re
(b)
(a)
Figure 2.6: (a) two sine waves and (b) phasor diagram
We consider both i(t) and v(t) to be phasors defined by:
V = V ejϕv
(2.12)
I = Iejϕi .
(2.13)
and
Conversely if we know the frequency, the phasors V and I specify uniquely the sinusoid by taking the
real parts of the phasors multiplied by ejωt :
Re(Vejωt ) = Re(V ej(ωt+ϕv ) ) = V cos(ωt + ϕv )
(2.14)
Re(Iejωt ) = Re(Iej(ωt+ϕi ) ) = I cos(ωt + ϕi ).
(2.15)
and
2.1.7
Bandwidth and transmission channels
Knowledge of the frequency components of a signal is very important for designing electronic circuits and
systems. Circuits have the properties not to treat the different frequencies in the same way. Often a lowest
frequency and an upper frequency are known that can be handled sufficiently in a circuit. The frequency
range between this upper frequency and lower frequency is called the bandwidth of the circuit. More
generally we can define a frequency range in which information can be processed reliably. This frequency
range is called transmission channel.
2.1.8
Pulse waveforms and duty cycle
Sinusoidal waves are important in electronics, but not the only ones. Another major type is the pulse
waveform. The pulse waveform is described as a very rapid transition (leading egde) from one voltage or
current level to another voltage or current level, and then, after an interval of time, a very rapid transition
(trailing edge) back to the original level. Although real pulses have a certain rise time and fall time, analysis
is often simplified by treating all pulses as ideal in many applications.
Often periodic pulse waveforms, see Fig. 2.7 are used, for example in switching circuits. An important
characteristic of a repetitive pulse waveform is the duty cycle: the ratio of the pulse width (tw ) to the period
(T ) and is expressed as a percentage:
tw
100%
(2.16)
percent duty cycle T
A square wave is a pulse train with a duty cycle of 50%.
20
tw
T
Figure 2.7: Repetitive pulse waveform, and the definition of the pulse width and period.
2.2 Components and elements
A circuit is characterized by one or more sources of electrical energy interconnected with one or more
receivers, or sinks of electrical energy. In fact, in the circuit energy is supplied by sources, transferred
from one place in the circuit to another, and temporarily stored or dissipated in circuit components. In
general, for the circuit holds the principle of conservation of energy. The sum of the energy delivered by
the independent sources in the circuit is equal to the energy absorbed by all other elements of the circuit.
To obtain a subcircuit we separate conceptually some part of the circuit from its surroundings. In this way
an entity is obtained between elements and the complete circuit; and an element is simply a subcircuit that
cannot be decomposed any further. Since the total current entering an isolated circuit is zero, a subcircuit
has at least two terminals, and the current entering the subcircuit through one terminal then leaves it through
the other one. Since a voltage can only be defined, or measured, between two terminals, the description of
a two-terminal subcircuit involves only one current—through the subcircuit—and one voltage—across the
subcircuit. A terminal pair is called a port whenever it is characterized by one current and one voltage. A
subcircuit having only such terminal pairs is called an n-port. All one-port elements will be represented
schematically as shown in Figure 2.8; that is, both current and power are always directed towards the
element. For the analysis of the (equivalent) circuits, the following one-port elements are generally defined:
+
_
i
ik
k
vk
j
pk= vkik
Figure 2.8: Schematic representation of a one-port network element k. The arrows follow the passive sign
convention, so the current and the power are directed towards the element.
linear and nonlinear resistors (R), linear and nonlinear capacitors (C), and linear and nonlinear inductors (L).
The ideal energy generators voltage source (vs ) and current source (is ) can be seen as nonlinear resistors.
The elements are idealizations of the actual physical devices—the components—and relate the physical
quantities current (i), voltage (v), charge (q), and flux (φ). In circuit theory elements as: resistors, capacitors,
and inductors, are abstractly defined. The various electronic components are introduced. Discussed are the
properties linear, non-linear, time-variant, time-invariant, as well as, energetic properties, and impedances.
Active and passive components
Electronic devices that control the addition of energy during amplification are generally known as active
components, they are capable of raising the energy level of signals. Also energy sources are grouped under
active components. Passive components cannot raise energy levels of signals. Often they even loose energy
as heat, dissipation, or radiation. Examples are resistors, capacitors, and inductors.
21
Resistance and impedance
Currents in electronic circuits may be constant or varying in time. When the current remains constant with
time, it is called a direct current (dc). When the current varies sinusoidally with time, it is referred to as
alternating current (ac). A common form of alternating current is the sinusoidal waveform.
The resistance of a component or circuit is the opposition to the flow of dc electrical current. The
impedance of a component or circuit is the opposition to the flow of a sinusoidal ac current. A more formal
definition is: the impedance Z of a one-port at the angular frequency ω is the ratio of the output voltage
phasor V and the input current phasor I.
Switching and steady-state
Switching on an excitation often causes transient currents and transient voltages in the circuit. Once the
voltages and currents in a dc circuit reach a stable value, the stationary state is reached. Once the voltages
and the currents in an ac circuit reach an average end value, the steady-state condition is reached that
continues as along as the periodic excitation continues.
2.2.1
Resistor
A one-port element is called a resistor if at any instant of time t its voltage v(t) and its current i(t) satisfy
a relation FR defined by a curve fR in the iv-plane (or vi-plane), the curve may be time dependent:
FR = {(i, v) : fR (i, v, t) = 0, with i = i(t) and v = v(t)}
(2.17)
The curve is called the characteristic of the resistor at time t. It specifies the set of all possible values that
the pair of variables i(t) and v(t) may take at time t. Usual physical resistors are time-invariant resistors;
that is, its characteristic does not vary with time. A two-terminal subcircuit can behave like a time-varying
resistor, if its characteristic in the iv-plane varies with time. For example, a switch can be considered a
linear time-varying resistor. Any resistor can be classified in four ways depending upon whether it is linear
or nonlinear and whether it is time-varying or time-invariant.
A resistor is called a linear resistor, often called just resistor R, if its characteristic is at all times a
straight line through the origin. Accordingly, a resistor that has a nonlinear resistance is called a nonlinear
resistor, see figure 2.9. A linear time-invariant resistor, by definition, has a characteristic that does not vary
with time and is a straight line through the origin.
v(t)
v(t)
0
0
i(t)
i(t)
(b)
(a)
Figure 2.9: Characteristics of: (a) (passive) time-invariant linear resistive element, (b) (passive) timeinvariant nonlinear resistive element.
For a linear resistor with resistance R the relation between its instantaneous voltage v(t) and current
i(t) is expressed by Ohm’s law:
v(t) = Ri(t)
(2.18)
22
as:
FR = {(i(t), v(t)) : v(t) − Ri(t) = 0},
(2.19)
FR = {(i(t), v(t)) : i(t) − Gv(t) = 0},
(2.20)
or
R and G are constants independent of i,v, and t. R is called the resistance and G is called the conductance.
The two two-terminal elements short circuit and open circuit can be considered to be linear timeinvariant resistors with resistance zero and conductance zero, respectively; see figure 2.10. These elements
are sometimes called non-energetic elements, because the power absorbed or generated by these elements
is always zero.
v(t)
i(t)
characteristic of a
short circuit (R=0)
0
characteristic of an
open circuit (G=0)
0
i(t)
(a)
v(t)
(b)
Figure 2.10: Characteristics of: (a) short circuit, (b) open circuit.
The impedance ZR of a linear resistor is found by dividing the voltage phasor V by the current phasor
I according to Ohm’s law. Both are complex numbers. The impedance of the resistor is:
ZR =
v(t)
Re(Vejωt )
V
=
= ,
i(t)
Re(Iejωt )
I
(2.21)
and is a real number. And, thus, equals ZR = R. The complex numbers V and I must have the same angle,
see Figure 2.11
The admittance YR of a linear resistor is found by dividing the current phasor I by the voltage phasor
V according to Ohm’s law. Both are complex numbers. The admittance of the resistor is:
YR =
Re(Iejωt )
I
i(t)
=
= ,
jωt
v(t)
Re(Ve )
V
(2.22)
and equals YR = 1/R.
The power p dissipated in a resistor is:
p(t) = v(t)i(t) = Ri2 (t),
(2.23)
and the amount of energy w(t) dissipated in the resistor:
t
w(t) =
∞
p dτ.
(2.24)
23
j
i(t) = Re(Iejωt)
linear
circuit in
steadystate
v(t) =
Re(Vejωt)
V=RI
R
I
0
t
0
i(t) = v(t)/R
v(t) = V cos(ωt + φv)
Figure 2.11: Sinusoidal steady-state characterization of a resistor.
2.2.2
Nonlinear resistor and diode
In electronics, nonlinear devices that can be modeled by nonlinear resistors have become increasingly important. It is necessary to define the concept of a nonlinear resistor in a general way. Because the tangent
of the curved vi-characteristic of a nonlinear resistor changes, the resistance R—as in the case of a linear
resistor—cannot be defined. The best we can do is to define a differential resistance in a point of operation. An important problem in electronics is the determination of points of operation. We will discuss
this problem. Associated is small-signal analysis. Its relation to the points of operation and the derivation
of small-signal equivalent circuits is illustrated. Finally, the property that nonlinear resistors can produce
harmonics, a property that is completely unknown for linear resistors is treated.
A nonlinear resistor is characterized by a nonlinear curve in the vi-plane. An example is the diode, see
Figure 2.12. At room temperature the diode is described approximately by the equation
V
−1 ,
(2.25)
I = Is exp
VT
in which I is the current through the diode and V the voltage across it, VT is the thermal voltage and has a
value VT ≈ 25 mV at room temperature.
It is clear that we cannot define a resistance for the diode, the ratio V /I varies over the curve (there is no
straight line through the origin), however, in any point P on the curve (Ip , Vp ) we can define a differential
conductance gd and a differential resistance rd by:
dI
gd = (rd )−1 .
dV
(2.26)
For any non-zero current through the diode, we see that the diode dissipates energy: p = iv is always
positive.
In contrast to linear resistors, a nonlinear resistor in general has a vi-chacteristic which is not symmetric
with respect to the origin, a symmetry called bilateral. For example, for the diode it is clear that f (v, i) =
f (−v, −i). As a consequence the absolute value of the current through the diode is different for (+v) or
(−v), and the amount of energy absorbed is different too. For these reasons, it is important that the symbol
for a nonlinear resistor indicates its orientation. If the nonlinear resistor is not bilateral we will use the
24
ID
(A)
ID
(nA)
4
60
3
40
2
20
0
1
0
50
100
150
0
200
0
0.2
0.4
0.6
0.8
1
VD (mV)
1.2
VD(V)
Figure 2.12: Characteristic of a diode
i
+
v
Figure 2.13: Symbol for a nonlinear resistor (unless the nonlinear resistor is bilateral).
symbol as in Fig. 2.13 and will call it a nonlinear resistor. A special case of a nonbilateral resistor is the
piecewise linear resistor of Fig. 2.14.
The solutions of a nonlinear circuit to dc inputs are called operating points. This subject is of major
importance in electronics. Due to the nonlinear vi-characteristic, an analytical solution to a dc input cannot always be found. For example, consider the circuit of Fig. 2.15. For the diode we approximate the
diode equation (assuming that for the voltage across the diode in the point of operation, VD , holds that
exp(VD /VT ) 1), by:
VD
.
(2.27)
I = Is exp
VT
Using Kirchhoff’s current law (KCL) we find the current through the load resistor to be equal to the
V
R1
1
I
1
R2
Figure 2.14: Piecewise linear but not bilateral resistor.
25
D
I
E
+
R
Figure 2.15: Simple circuit with a diode and load resistor.
current through the diode:
1
(E − VD ) = Is exp
R
VD
VT
(2.28)
This equation in VD cannot be solved analytically. However we can solve the equation graphically by
letting VD run over all possible values (along the x-axis), constructing a “load” line (a line of all possible
solutions of the current through the resistor in the circuit), and selecting that value where the currents are
equal. The solution gives us the point of operation, the operating point. Figure 2.16 shows for the current
through the resistor
I=
1
(E − VD ),
R
the load line, and for the current through the diode
VD
,
I = Is exp
VT
and the operating point as the crossing of both lines. This model with the nonlinear diode characteristic is
called the large-signal model of the diode.
I
operating point
E/R
ID
(actual ID)
VD
E
VD
(possible VD)
(actual VD)
Figure 2.16: Finding the operation point by crossing the load line and the diode curve
2.2.3
Small-signal models
Now, the same circuit is considered with a relatively small ac voltage signal, Δvs superposed on the dc
voltage E, see Fig. 2.17. How will this small voltage signal change the current the circuit? And what
will be the ac component Δvd of the voltage VD + Δvd across the diode? The necessary equations can be
obtained by the following substitutions:
E → E + Δvs
26
I+∆i
D
+
∆vs
E
R
+
Figure 2.17: Circuit in which the diode and the resistor are excited by an additional small ac voltage signal
in the operating point.
VD → VD + Δvd
I → I + Δi.
When we consider the addition of a small voltage signal to the circuit—biased in the operation point—we
notice that the nonlinear diode characteristic can be approximated by the tangent in the operating point, see
Fig. 2.18. When biased in the operating point, we see that for small ac signals the diode can be represented
by a resistor.
I
operating point
E/R
∆id
ID
(actual ID)
E
VD
(actual VD)
VD
(possible VD)
∆vd
Figure 2.18: Finding Δi (= Δid ) in the operation point of the diode curve
This statement can be made clear by translating the curve of Fig. 2.18 such that the coordinates of point
P are translated to the origin, see Fig. 2.19.
The effect of the translation is that the original circuit is changed if we only consider the influence of
the small ac voltage signal. The new circuit is called the small-signal circuit, and is a circuit with only the
small-signal voltage and small-signal current, which must be superposed on the large-signal voltage VD and
the large-signal current ID .
To really find the small-signal circuit, we notice that we can replace the nonlinear diode characteristic
by a linear resistor with the value of the differential resistance, rd , in the operating point. To do so we
translate the original vi-characteristic of the diode such that the operating point is moved to the origin by
(1) subtracting the actual value of the dc voltage across the diode VD from the possible VD (the x-axis) and
(2) subtracting the actual dc current through the diode ID from the current through the circuit. This can be
done by making the dc voltage source zero, while keeping the value of the differential resistance in P (the
dc voltage source, with value E, caused the voltage VD and ID in the circuit).
That is, in the small-signal circuit the dc voltage source is replaced by a short-circuit element (V = 0).
In addition, if we would have considered a dc current source, then in the small-signal circuit this dc current
27
∆id
(rd) -1
∆vd
Figure 2.19: Shifting the operation point to the origin, to obtain the resistive behavior of the diode for small
ac signals
.
source is replaced by a open-circuit element (I = 0). This all means that for the small-signal circuit we
obtain the following circuit, see Fig. 2.20:
∆i
∆vs
rd
+
R
Figure 2.20: Small-signal model of the circuit (also known as small-signal equivalent circuit).
The model with the approximation of the nonlinear diode curve by the tangent in the operation point
(its slope representing the differential resistance rd ), in case of small ac signals, is called the small-signal
model of the diode.
For Δi holds, see Fig. 2.20:
Δi =
Δvd
,
rd
(2.29)
where rd is the differential resistance.
Using equation 2.27:
rd =
dI
dV
−1
=
I
VT
−1
(2.30)
,
As VT = 25 mV, at room temperature, we obtain:
rd =
25
Ω.
I(in mA)
(2.31)
The value of the differential resistance depends on the actual value of the current in the operation point.
Based on the small-signal circuit we can find the ac component of the voltage across the diode:
Δvd =
rd
Δvs .
rd + R
(2.32)
28
Of course the total voltage across the diode D is VD + Δvd .
A last question is: How small must the small signal be? To answer this question we look into the effect
of the nonlinear resistor on incoming signals. Based on whether you find the effect acceptable or not, you
can decide on the magnitude of the small voltage signal.
2.2.4
Nonlinear resistors can introduce harmonics
In an operating point, where g is the differential conductance, the relation between the small-signal current
i(t) through a nonlinear resistor as a function of the small-signal voltage v(t) across it can be described
generally as:
i(t) = gv(t) + bv 2 (t) + cv 3 (t) + ...
(2.33)
The values of the constants g, b, c, ... all depend on the actual curvature of the characteristic. For a relatively
gradually changing characteristic the power series can be truncated after a few terms.
What happens if the small-signal voltage is a sinusoidal signal v(t) = V sin ωt? Application in equation 2.33 gives:
i(t) = gV sin ωt + b(V sin ωt)2 + c(V sin ωt)3 + ...
(2.34)
The first term is the linear approximation. After some mathematics the second term gives:
b(V sin ωt)2 =
1 2 1 2
bV − bV cos 2ωt.
2
2
(2.35)
An additional signal component is appearing with a double frequency (2ω) shifted in time (90 degrees phase
difference). Evaluating the third term, we can find
c(V sin ωt)3 =
3 3
1
cV sin ωt − cV 3 sin3ωt.
4
4
(2.36)
Again an additional signal component appears, now with an angular frequency of 3ω. In general, we can
see that as a consequence of the nonlinearity higher harmonics appear in the signal.
2.2.5
Capacitor
A capacitor consists of two conducting surfaces separated by an insulating layer. The two conducting
surfaces can store opposite charges creating an electric field between the plates. In general, a two-terminal
element is called a capacitor if at any time t its stored charge q(t) and its voltage v(t) satisfy a relation
defined by a curve in the vq-plane:
FC = {(v, q) : fC (v, q, t) = 0, with v = v(t) and q = q(t)}
(2.37)
This curve is called the characteristic of the capacitor at time t. It specifies the set of all possible values
that the pair of variables v(t) and q(t) may take at time t. As in the case of the resistor, a capacitor whose
characteristics is at all times a straight line through the origin of the vq-plane is called a linear capacitor.
Conversely, if at any time the characteristic is not a straight line through the origin of the vq-plane, the
capacitor is called a nonlinear capacitor. A capacitor whose characteristic does not change with time is
called a time-invariant capacitor. If the characteristic does change with time, the capacitor is called a timevarying capacitor.
A linear time-invariant capacitor, by definition, has a characteristic that does not vary with time and
is a straight line through the origin. The relation between its instantaneous voltage v(t) and stored charge
q(t) is expressed by:
FC = {(v(t), q(t)) : q(t) − Cv(t) = 0}
(2.38)
29
where C is a constant independent of q,v, and t. C is called the capacitance. The equation relating the
current through and the terminal voltage across the capacitor is:
i(t) =
dv
dq
=C
dt
dt
(2.39)
By integration of this equation the terminal voltage as a function of the current can be obtained within an
arbitrary integration constant:
1 t
1 t
i(α)dα = v(t0 ) +
i(α)dα
(2.40)
v(t) =
C −∞
C t0
and the supplementary datum v(t0 ), the initial voltage, must be specified in order to define the state of a
capacitance.
In a realistic circuit with a dc current a capacitor is charged until the voltage across it is stable. In this
stationary-state no current passes the one-port capacitor. The resistance is infinite. The dissipated power is
zero, however energy is stored in the electric field of the capacitor during charging.
To find the impedance of the capacitor we have to find the ratio of the phasor representing the voltage
and the phasor representing the current. Using equation 2.39:
V ej(ωt+ϕv )
dt
= jωCV ej(ωt+ϕv )
Iej(ωt+ϕi ) = C
(2.41)
(2.42)
and we find
I = jωCV
V=
or
1
I.
jωC
(2.43)
Because of the factor jω the current phasor I and the voltage phasor V differ by an angle of 90◦ when
plotted in the complex plane as shown in Figure 2.21. The current phasor leads the voltage phasor.
i(t) = Re(Iejωt)
linear
circuit in
steadystate
v(t) =
Re(Vejωt)
I
j
V
C
0
v(t) = V cos(ωt + φv)
t
0
i(t) = ωCV cos(ωt + φv + 90°)
Figure 2.21: Sinusoidal steady-state characterization of a capacitor.
The impedance and admittance of the capacitor are, thus,
ZC =
1
jωC
and
YC = jωC
(2.44)
30
and
|ZC | =
1
ω2 C 2
=
1
ωC
and
|YC | = ωC,
(2.45)
both depend on the angular frequency ω. Figure 2.22 shows the impedance characteristic of a capacitor with
a realistic value. Logarithmic scales are used to cover the large range of frequencies we often encounter in
log |ZC|
|ZC| = 1/C
Δϕ
C
+90o
0o
-90o
0
ω = 1/C
log ω
log ω
Figure 2.22: Impedance characteristic of a capacitor.
electronic circuits:
log |ZC | = − log ω − log C
(2.46)
For the phase difference Δϕ holds:
Δϕ = ϕv − ϕi = −90◦
(2.47)
As the frequency approaches zero, the admittance goes to zero. This resembles the dc case, the capacitor
acts like an open circuit (i = 0). As the frequency goes to infinity, the impedance approaches zero; meaning
that for high frequencies the behavior of the capacitor resembles a short circuit (v = 0).
The average power P dissipated in the capacitor is
1 T
P =
V sin(ωt)V ωC cos(ωt)dt = 0,
(2.48)
T 0
and the total energy W stored in the capacitor in stationary state is
WC =
2.2.6
1
CV 2
2
(2.49)
Inductor
A wire formed into a coil becomes an inductor. An inductor excited by a dc current source does not create
a voltage across it in steady-state. Its impedance is in this case zero. However, energy is stored in the
magnetic field surrounding the coil.
The following equations hold: the constitutional relation, relating the flux φ, the inductance L, and the
current I
φ = LI,
(2.50)
for the voltage across the inductor
V =L
dI
,
dt
(2.51)
31
and for the energy
WL =
1 2
LI .
2
(2.52)
For the relation between the phasors V and I and the impedance of the inductor we find (reasoning in a
similar way as with the capacitor) using equation 2.51:
V = jωLI
(2.53)
and
ZL = jωL
and
YL =
1
.
jωC
(2.54)
The amplitude of the sinusoidal voltage is obtained by multiplying the amplitude of the current by the
magnitude of the impedance (evaluated at the proper frequency), and the phase of the sinusoidal voltage
is obtained by adding 90◦ to the phase of the current. The steady-state characterization of the inductor is
shown in Figure 2.23.
j
i(t) = Re(Iejωt)
linear
circuit in
steadystate
v(t) =
Re(Vejωt)
V
L
0
I
v(t) = V cos(ωt + φv)
t
0
i(t) = V cos(ωt + φv - 90°)
ωL
Figure 2.23: Sinusoidal steady-state characterization of a inductor.
The current phasor lags the voltage phasor. The impedance characteristics of an inductor are shown in
Figure 2.24. When the frequency approaches zero, the impedance of the inductor goes to zero too; as in the
case of a dc current. When the frequency becomes very high, the admittance approaches zero; it resembles
a open circuit.
The average power P dissipated in the inductor is
P =
2.2.7
1
T
T
IωL cos(ωt)I sin(ωt)dt = 0.
(2.55)
0
* Memristor
The resistor is an element relating the voltage and the current, the capacitor is an element relating the
charge and the voltage, and the inductor is an element relating the flux and the current. Also an element
exist relating the charge and the flux, it is called a memristor. The memristor was predicted in the last
32
log |ZL|
Δϕ
L
+90o
0o
log ω
-90o
ω=1/L
log ω
Figure 2.24: Impedance characteristics of an inductor.
century, however the first device that claimed to be a memristor was manufactured recently. In 2008 HP
laboratories made a memristor.
The constitutional equation for a memristor can be written down as:
FM = {(φ, q) : fC (φ, q, t) = 0, with φ = φ(t) and q = q(t)}.
(2.56)
For a linear memristor holds:
φ(t) = M q(t),
(2.57)
M is called the memristance. Using the equations relating flux and voltage, and charge and current:
t
t
φ(t) =
v(τ )dτ
and q(t) =
i(τ )dτ
−∞
−∞
we find for the linear memristor
t
t
v(τ )dτ = M
i(τ )dτ,
−∞
(2.58)
−∞
which reduces by differentiation to
v(t) = M i(t)
or
i(t) = M −1 v(t).
(2.59)
And we see that at the terminals the memristor cannot be distinguished from a resistor. However, if we consider a nonlinear memristor a differential memristance can be defined only. This can be done in essentially
two ways:
−1
dq
dφ
or m(φ) =
.
(2.60)
m(q) =
dq
dφ
Using
v(t) = m(q(t))i(t)
or
i(t) = m−1 (φ(t))v(t)
the differential memristance can be used to express the vi-relation of the nonlinear memristor as:
t
v(t) = m
i(τ )dτ i(t)
−∞
or as
i(t) = m
−1
t
(2.61)
(2.62)
v(τ )dτ
(2.63)
v(t)
−∞
33
1 : n
ii
vi
io
vo
Figure 2.25: Transformer.
We see that the actual value of the differential memristance at any point t in time is determined by the
history of the current or the voltage. The nonlinear memristor behaves as a resistor or conductance with
memory. The name memristor is a contraction of memory resistor, because that is exactly its function: to
remember its history.
A memristor is a two-terminal device whose memristance depends on the magnitude and polarity of
the voltage or current applied to it and the length of time that the voltage or the current has been applied.
Ideally, when you turn off the voltage or the current, the memristor remembers its most recent resistance
until the next time you turn it on, whether that happens a day later or a year later.
2.2.8
Transformer
A transformer consists of two coils, a primary and a secondary. When a current goes through the coils,
the coils are coupled by a magnetic field. Because there is no electrical contact between the magnetically
coupled coils, energy is transferred in electrical isolation. An alternating voltage and current applied to the
primary will produce an alternating magnetic field, which causes an alternating voltage across and current
in the secondary coil, see Figure 2.25. The actual voltage or current is determined by the ratio of the number
of turns of the coils:
vo = nvi ,
(2.64)
1
ii ,
n
(2.65)
and
io =
The transformer works as an input voltage signal amplifier with a gain determined by the turns ratio,
however it is a passive amplifier: the current is divided by the same ratio (in the ideal case) because no other
energy source than the input signal is available. The fact that amplification of the voltage leads to division
of the current can successfully be used to transform a resistor. Therefore consider the circuit of figure 2.26.
The current through the resistor is
1
+
ii
: n
+
io
vi
R
-
-
=
+
vi
-
R
n2
Figure 2.26: Transforming the resistance.
io =
nvi
vo
=
.
R
R
(2.66)
34
With this current at the secondary coil the current at the primary coil is fixed:
ii = nio =
n2 vi
.
R
(2.67)
The impedance at the primary coil is now found as
Zi =
vi
vi R
R
= 2 = 2
ii
n vi
n
(2.68)
At the primary coil, the resistor with value R is seen as a resistor with value R/n2 . This technique of
making a load resistance to appear different is called impedance matching.
2.2.9
* Gyrator
You may think of a device that is ideally described by equations like the ones of the transformer but relating
voltages on one side to currents at the other side. The equations describing the device would be:
io = R g v i ,
(2.69)
1
ii ,
Rg
(2.70)
and
vo =
Rg is called the gyrator resistance. The gyrator, see Figure 2.27, which couples currents to voltages and
voltages to currents without energy loss was proposed by Tellegen.
Rg
+
+
io
ii
vo
vi
-
-
Figure 2.27: Gyrator.
2.2.10
Ideal voltage source
The ideal voltage source is an energy source defined in the iv-plane. As such it is, in fact, a nonlinear
resistor. If a positive current is leaving the ideal voltage source with the polarity indicated in Figure 2.28
the source is delivering power (energy). If a positive current is entering this ideal voltage source, the source
is absorbing energy just as in case of a rechargeable battery. In other words, an ideal voltage source delivers
to, or absorbs from, the circuit whatever current is necessary to maintain its terminal voltage.
2.2.11
Ideal current source
Just as the ideal voltage source, the ideal current source is an energy source defined in the iv-plane and
is a nonlinear resistor. If a positive current is leaving the source at the positive terminal the source is
providing energy, if a negative current is leaving at the positive terminal, the source is absorbing energy, see
Figure 2.29. The ideal current source provides the specified current completely independent of the voltage
across the source.
35
v
+
absorbing energy
i
delivering energy
V
} V is positve
+
V
i
0
-
} V is negative
absorbing energy
delivering energy
Figure 2.28: Ideal voltage source; Note the direction of the current is not according to the positive sign
convention inside the element.
i
+
absorbing energy
i
delivering energy
I
} I is positve
I
v
0
-
delivering energy
v
} I is negative
absorbing energy
Figure 2.29: Ideal current source; Note the direction of the current is not according to the positive sign
convention inside the element.
2.2.12
Nullator, norator, and nullor
In the iv-plane two other special elements can be defined that play an important role in the description of
amplifiers. The characteristics are shown in Figure 2.30. In (a) the element is called nullator; in (b) the
element is called norator. As you see the nullator is a nonenergetic element, while the norator is an element
that can provide any amount of energy. The nullator can be seen as an open circuit and closed circuit acting
simultaneously; or as a non-zero valued resistor through which the current is zero (in which case the voltage
must be zero too), or the voltage across it is zero (in which case the current through it must be zero).
The nullator and norator often occur in pairs. This is not surprising because the nullator cannot pass
any energy (i = 0 and v = 0), so the circuit behind the nullator often needs an additional energy source
with a specified current and voltage. This specific amount of energy can be provided by the norator. The
combination of the two elements, shown in Figure 2.31, has been given the name nullor.
2.2.13
The dependent voltage source
In this and the next subsection four additional active elements are introduced called dependent (or controlled) sources. Unlike the independent sources we treated until now, they must receive a stimulus from
somewhere else in the circuit. This stimulus can be either a voltage or a current, and the source itself may
be a voltage source or a current source. Four combinations are called the voltage controlled voltage source
(VCVS), the current controlled voltage source (CCVS), the voltage controlled current source (VCCS), and
36
v
v
i
i
i=0
v=0
nullator
i=?
v=?
norator
(a)
(b)
Figure 2.30: Characteristics of (a) nullator and (b) norator
nullor
+
-
Figure 2.31: The nullor: a combination of a nullator and a norator; in the nullor the polarity of the norator
and the current through it is chosen such that the norator provides energy for nonzero positive values of the
current.
the current controlled current source (CCCS).
Figure 2.32 shows the symbol and iv-characteristic for the dependent voltage source.
The terminal voltage of the dependent voltage source is exactly the same as that of an independent
voltage source, but its value is determined by another voltage or current in the circuit in which it is located.
If controlled by a voltage, vx the source is called a voltage controlled voltage source and a linear relationship
vc = μvx
(2.71)
specifies the value of the source. The constant μ is called the voltage gain.
When controlled by a current, ix , the source is called a current controlled voltage source and a linear
relationship
vc = rm ix
(2.72)
specifies the value of the source. The constant rm has the dimension of resistance and is called transresistance. The prefix “trans” is used because it “transfers” the effect of a current somewhere else in the circuit
to the dependent voltage source. (The subcript “m” denotes “mutual”, a synonym for “trans”.)
2.2.14
The dependent current source
Figure 2.33 shows the symbol and iv-characteristic for the dependent voltage source.
The terminal current of the dependent current source is exactly the same as that of an independent
current source, but its value is determined by another voltage or current in the circuit in which it is located.
If controlled by a voltage, vx the source is called a voltage controlled current source and a linear relationship
ic = g m v x
(2.73)
37
v
+
i
vc = μ vx
vc = rm ix
v
(VCVS)
(CCVS)
vc
i
vc
-
Figure 2.32: Symbol and characteristics of the dependent voltage source.
i
+
i
ic = gm vx
ic = β ix
v
(VCCS)
(CCCS)
ic
v
ic
-
Figure 2.33: Symbol and characteristics of the dependent current source.
specifies the value of the source. The constant gm has the dimension of conductance and is called transconductance.
When controlled by a current, ix , the source is called a current controlled current source and a linear
relationship
ic = βix
(2.74)
specifies the value of the source. The constant β is called the current gain.
2.3 Circuit analysis
For all the elements we considered so far, it holds that the voltage across any element and the current through
any element were always well-defined, circuits composed of only such elements are called lumped circuits.
In the theory describing lumped circuits, from now on called (electronic) circuit theory, the Kirchhoff laws
enter axiomatically. From a classical physics and electromagnetic theory point of view using Maxwell’s
equations, the lumped circuit is only an approximation. It assumes steady-state currents or a quasi-static
approach (only slowly changing electric and magnetic fields). Energy is conserved and if applied to circuits
that cannot radiate, Kirchhoff’s current and voltage laws can be obtained.
2.3.1
Kirchhoff’s laws
The topology of the a circuit specifies the location of nodes and elements or more precise: nodes and
branches; a branch may represent any two-terminal element, and the terminals are called nodes. In dis38
cussing circuits we describe the topology with nodes n , branch (element) currents in , and branch (element)
voltages vn . Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) are the two fundamental
postulates of circuit theory. They hold irrespective of the nature of the elements constituting the circuit.
Both reflect topological properties of the circuit.
Based upon the fundamental law of physics that electric charge is conserved, Kirchhoff’s current law
(KCL) for nodes states that, for all times t, the algebraic sum of the currents i1 , i2 , ..., in leaving any node
is equal to zero:
in (t) = 0.
(2.75)
n
Given any connected lumped circuit having n nodes, we may choose arbitrarily one of these nodes as
a datum node, i.e., as a reference for measuring (electric) potentials u1 , u2 , ..., un−1 . Let vi denote the
voltage (potential) difference across branch (element) i connected between node k and j. Kirchhoff’s
voltage law (KVL) states that, for all choices of datum node, for all times t, and for all pairs of nodes k and
j:
vi (t) = uk (t) − uj (t).
(2.76)
If we now consider a closed node sequence, thus starting and ending at the same node, the voltage difference
must be zero. Now, Kirchhoff’s voltage law (KVL) states that, for all closed node sequences (loops), for all
times t, the algebraic sum of all node-to-node voltages (voltage differences) around the chosen closed node
sequence is equal to zero:
vn (t) = 0.
(2.77)
n
2.3.2
Linear circuits and superposition
Any lumped circuit with the property that each of its elements is either a linear element or an independent
source is called a linear circuit. Important circuit theorems apply to linear networks. One of them is the
superposition theorem. The superposition theorem can be defined as follows: If the response of a system
to a linear combination of input signals always consists of the corresponding combination of the individual
output signals, then for this system the superposition principle applies. That is, if a system responds to
an input signal x1 with the output signal y1 and to the input signal x2 with the output signal y2 then the
superposition theorem holds if:
x(t) = Ax1 (t) + Bx2 (t) → y(t) = Ay1 (t) + By2 (t).
(2.78)
A and B can be any complex constants. For the analysis and design of (nanoelectronic) circuits this theorem
is of great importance. It allows us to decompose an original, more complicated, circuit analysis problem
into a number of simpler ones by considering partial responses.
As an example, consider Figure 2.34. The current i in the linear circuit can be found as the sum
of the partial responses i1 and i2 . In words, if i1 is the response with the voltage source deactivated
(reduced to zero, thus a short circuit) and i2 is the response with the current source deactivated (reduced
to zero, thus an open circuit) then i = i1 + i2 . We see that: i1 = −(2/3)is and i2 = (1/9)Ω−1 vs , so
i = (1/9)Ω−1 vs − (2/3)is .
Superposition cannot be used to calculate dissipated energy of individual elements (for example, you
can see this realizing that: ((1/9)Ω−1 vs − (2/3)is )2 = ((1/9)Ω−1 vs )2 + (−(2/3)is )2 ).
2.3.3
* Tellegen’s theorem
In all the circuits and networks that were considered upto here energy was conserved. This is no coincidence, it is a general property of lumped circuits and is based on a theorem called Tellegen’s theorem.
39
+
i
i1
3Ω
3Ω
vs
_
is =
6Ω
i2
3Ω
+
is + vs
_
6Ω
6Ω
Figure 2.34: Using the superposition principle it holds true that: i = i1 + i2
Tellegen’s theorem is general in that it depends solely upon Kirchhoff’s laws and the topology of the network. It applies to all (sub)circuits and elements, whether they be linear or nonlinear, time-invariant or
time variant, passive or active, hysteretic or nonhysteretic. The excitation might be arbitrary; the initial
conditions may also be arbitrary. It describes elements as ”black boxes” only assuming a circuit topology
and the validity of Kirchhoff’s laws.
-Tellengen’s theorem
Tellegen’s theorem (1952) can be formulated as follows. Suppose we have two circuits with the same
topologies, and all N elements are represented by a one-port as in Figure 1.1. For each element k in one
circuit (the unprimed circuit), there is a corresponding element k in the other circuit (the primed circuit).
If we now take the voltage across element k, vk , and multiply it with the current through the corresponding
element k , ik , and sum this for all N elements then:
N
vk ik = 0.
(2.79)
k=1
To fully understand this theorem consider the next example, see Figure 2.35.
We have a unprimed circuit and a primed circuit consisting of three elements a, b, and c. Using Kirchhoff’s voltage law the voltages v across the elements are expressed in potentials u with respect to the datum
node (ground). Now we can easily check that:
va ia + vb ib + vc ic = u1 ia + (u1 − u2 )ib + u2 ic =
u1 (ia + ib ) + u2 (−ib + ic ) = u1 0 + u2 0 = 0,
because according to Kirchhoff’s current law the currents at node 1 and 2 sum up to zero.
-Proof of Tellegen’s theorem
Consider a general element in a given circuit—the unprimed circuit—and a corresponding element in another circuit—the primed circuit—having exactly the same topology. Because the topologies are the same,
there is for any element between nodes α and β in the unprimed circuit a corresponding element between
1
ia
+
+
b
vb
_
2
+
i’b
u’1
u2
b’
1
i’a
+
ic
+
ib
u1
v’b
u’2
_
2
+
va a
vc c
v’a a’
v’c c’
_
_
_
_
Figure 2.35: Example of the Tellegen’s Theorem
40
i’c
the same nodes in the primed circuit. We assume that there are N elements and nt nodes and investigate the
sum
N
vk ik = 0.
k=1
Let us assume, for simplicity, that the circuit has no branches in parallel; i.e., there exists only one
branch between any two nodes. The proof can be easily extended to the general case. (If branches are in
parallel, replace them by a single branch whose current is the sum of the branch currents. If there are several
parts, the proof shows that Tellegen’s theorem holds for each of them. Hence it holds also when the sum
ranges over all branches of the graph.)
branch k
α
+
vk
_
iαβ
+
β
+
uα
uβ
_
_
1
Figure 2.36: An arbitrary branch, k, connects node α and node β
We first pick an arbitrary node as a reference node, and we label it node 1 . Thus, u1 = 0. Let uα
and uβ be potentials of the αth node and the βth node, respectively, with respect to the reference node. It
is important to note—in both the unprimed circuit and the primed circuit—that once the branch voltages
(v1 , v2 , ..., vN ) are chosen, then by KVL the node potentials u1 , u2 , ..., uα , ..., uβ , ... are uniquely specified.
Let us assume that branch k connects node α and node β as shown in Figure 2.36, and let us denote by
iαβ the current flowing in branch k from node α to node β in the primed circuit. The voltage vk is the
voltage across element k in the unprimed circuit, then
vk ik = (uα − uβ )iαβ
(2.80)
Obviously, vk ik can also be written in terms of iβα , the current from node β to node α in the primed
circuit, as
vk ik = (uβ − uα )iβα
Adding the above two equations, we obtain
vk ik =
1
(uα − uβ )iαβ + (uβ − uα )iβα
2
(2.81)
Now, if we sum the left-hand side of equation 2.81 for all the branches in the graph we obtain
N
vk ik
k=1
The corresponding sum of the right-hand side of equation 2.81 becomes, using iβα = −iαβ ,
41
t
t 1
(uα − uβ )iαβ
2 α=1
n
n
β=1
where the double summation has indices α and β carried over all the nodes in the graph. The sum leads to
the following equation:
N
k=1
t t
1
(uα − uβ )iαβ .
2 α=1
n
vk ik =
n
(2.82)
β=1
Note that if there is no branch joining node α to node β we set iαβ = iβα = 0. Now that equation 2.82
has been established, the right-hand side of equation 2.82 is split as follows:
N
k=1
⎛
⎞
n
nt
nt
nt
t
1
1
vk ik =
iαβ ⎠ −
iαβ .
uα ⎝
uβ
2 α=1
2
α=1
β=1
(2.83)
β=1
nt
iαβ is the sum of all branch currents entering node β . By KCL, each one of these
For each fixed α, β=1
sums is zero, hence,
N
vk ik = 0.
k=1
Thus we have shown that given any set of branch voltages subject to KVL only and any set of branch
currents subject to KCL only, the sum of the products vk ik is zero. This concludes the proof.
-Conservation of energy in lumped circuits
For a single arbitrary circuit we can take the same circuit for both the primed circuit and the unprimed
circuit. Tellegen’s theorem now states the conservation of energy in the circuit. We have, with the notations
of Tellegen’s theorem,
N
vk (t)ik (t) = 0
for all t.
k=1
Since vk (t)ik (t) is the power delivered at time t by the circuit to branch k, the theorem may be interpreted
as follows: at any time t the sum of the power delivered to each branch of the circuit is zero. Suppose the
circuit has several independent sources; separating in the sum the sources from the other branches, we
conclude that the sum of the power delivered by the independent sources to the circuit is equal to the sum
of the power absorbed by all the other elements of the circuit. This means that as far as lumped circuits are
concerned, the validity of KVL and KCL together imply conservation of energy.
2.3.4
Thévenin and Norton equivalents
The application of the superposition principle leads to the important linear Norton and Thévenin theorems
that are frequently used in linear circuit analysis to obtain Norton and Thévenin equivalent subcircuits. In
case of nonlinear circuits it is also possible to define Norton and Thévenin equivalent subcircuits but, in
general, the nonlinear resistances in the Norton and Thévenin equivalents will be different.
The linear theorems state that for most complex (passive) circuits there exists an equivalent circuit
consisting of just one resistance and one independent source. The Thévenin equivalent consists of a
resistance RTh in series with an independent voltage source voc (t). The voltage is called the open-circuit
voltage, and has the value of the voltage that appears across the two terminals when no other circuitry
42
Norton equivalent
NN
Thévenin equivalent
NTh
i
+
v
R Th
+
R Th
+
i sc
i
v
voc
-
-
-
(b)
(a)
i =0
+
v =0
i sc
R Th
R Th
+
i sc
voc
+
voc
-
-
-
(d)
(c)
i
i
NTh , N N
i sc
+
-
v
v oc
v
Slope =- 1/R Th
(e)
(f)
Figure 2.37: Definitions used in the linear passive Norton and Thévenin equivalents: (a) the Norton equivalent source model, (b) the Thévenin equivalent source model, (c) the short circuit current to obtain the
Norton equivalent, (d) the open circuit to obtain the Thévenin equivalent, (e) the equivalent circuits as an
element, and (f) the characteristic in the vi-plane for these models.
43
is attached. The resistance RTh is called the Thévenin equivalent resistance and has the value of the
equivalent resistance when all independent sources are deactivated.
The Norton equivalent consists of a resistance RTh in parallel with an independent current source isc (t),
see Figure 2.37. The current is called the short-circuit current, and has the value of the current that flows
through a short circuit of the two terminals. The resistance RTh is again called the Thévenin equivalent
resistance and has the value of the equivalent resistance when all independent sources are deactivated These
subcircuit concepts are well-known, for definitions see Figure 2.37.
Notice that it is not necessary to make assumptions concerning the load; it may be linear, nonlinear,
time-variant, or time-invariant. The network N is only required to be linear; it may include both dependent
and independent sources.
An important remark must be made. As the equivalent subcircuits are only equivalent to the original
subcircuit because they have the same characteristics in the iv-plane, they do not say much about absorbed
or generated power in the circuits. Generally, energy consumption is not the same in the subcircuit and its
Thévenin or Norton equivalent.
2.4 Real sources and equivalent circuits
Upto now we considered ideal components, that is, a resistor only has a resistance, a capacitor only has a
capacitance, an inductor only has a inductance, and energy sources were ideal. However, in reality a real
resistance will always have a capacitance to its environment, a capacitor will always have a resistance due
to isolation losses or an inductor will always have a resistance due the resistance of the wire itself and a
capacitance due to the winding of the coil. Additional elements can be used in addition to the ideal element
to describe a real component. These additional elements are often called parasitic elements.
V
I
+
Energy
source
Voc
Load
V
I
-
Isc
(b)
(a)
Ro
I
I
+
+
+
Voc
-
Go
Isc
V
V
-
-
(c)
(d)
Figure 2.38: Realistic source (a), its iv-characteristic (b), its voltage source equivalent (c), and its current
source equivalent
44
2.4.1
Real sources, sensors and actuators
Practical energy sources may approximate the ideal, but never reach it. For example, an ideal voltage
source must maintain its voltage even in the face of a zero-resistance load, which would require an infinite
current and an infinite amount of energy; this is obviously impossible. A more realistic source, with its
iv-characteristic, is shown in Figure 2.38 (a) and (b).
The characteristic shows an ideal source with voltage Voc on open circuit (I = 0) and a value of
terminal voltage V that decreases linearly as the current drawn from it increases, and sometimes used as a
characteristic for a real battery. This characteristic can be expressed mathematically by the equation:
V = Voc − Ro I,
(2.84)
where Ro is the ratio of the open-circuit voltage to the short-circuit current. An equivalent circuit for the
source is shown in Figure 2.38 (c). It is an equivalent circuit because the KVL equation for the circuit is
equation 2.84. Alternatively, we can write down an equation for the iv-characteristic in term of currents:
I = Isc − Go V.
(2.85)
The resulting equivalent circuit is shown in Figure 2.38 (d); its KCL is equation 2.85.
2.4.2
Solar cells
An important example of a nonideal current source is a solar cell. Basically, the solar cell is a diode
constructed such that photons (of light) can increase the reverse current Is ; that is, the current through the
diode when i and v are both negative. The solar cell operates in a so-called photovoltaic mode, that is, it is
unbiased and connected to a load impedance. The energy of the incoming light will cause a current through
the diode in additon to the current caused by the voltage across the diode. The solar cell is assumed to have
an ideal diode characteristic.
I
Dark
curve
0
diode curve
light intensity 1
Voc2
Isc1
Voc
Isc
V
1
2
light intensity 2
Figure 2.39: iv-characteristic of a solar cell showing short-circuit current and open-circuit voltage as the
light intensity increases. In the 4th quadrant the power is negative, meaning that we have the behavior of an
active nonlinear resistor that generates energy.
The total iv-characteristics of a solar cell under illumination is simply a summation of the dark current
Is (current in the dark) and the photocurrent given as:
qV
− 1 − ISC
(2.86)
I = Is exp
kT
45
The result is shown in Figure 2.39. As V = 0 on the vertical axis, the intersections with this axis are
called short-circuit currents and are indicated by ISC . The intersections with the horizontal axis (I = 0) are
the open-circuit voltages VOC representing the terminal voltage across the solar cell in absence of a load
(also called photovoltaic voltage). The shifted iv-characteristic shows that in the 4th quadrant the current
in the solar cell is negative (remember the passive sign convention) while the voltage across the solar cell
is positive; the solar cell is delivering energy to a load and can be used as an energy source for a subcircuit
(the solar cell behaves as an active nonlinear resistor in this quadrant).
In general, the short-circuit current is a linear function of the illumination, while the open-circuit current
is a logarithmic function of the illumination. The major increase in VOC occurs for lower-level increases in
illumination, thereby increasing the power capabilities. However, further increasing the voltage across the
solar cell decrease the power until the voltage reaches the open-circuit voltage where the solar cell cannot
provide any energy anymore. For a silicon single solar cell element the maximum voltage is typically 0.6
V. The series arrangement of solar cells permits a voltage beyond that of a single element. The typical
performance of such an arrangement is shown in Figure 2.40.
Output
current
increasing
intensity
optimum
power
Output voltage
Figure 2.40: Typical output characteristics of a multi-element solar cell.
We see that for small voltages the solar cell behaves as a current source. Figure 2.41 shows the equivalent circuit, where a constant current source, ISC is in parallel with the diode.
ISC
Figure 2.41: Simplified equivalent circuit of a solar cell under illumination.
Sensors, providing the inputs signals from the outside world, are often complicated devices including
electronic circuitry. In addition all sensors have an internal resistance, and will produce thermal noise, which
is added to their output signal; besides the sensors cannot provide an finite amount of energy. This represents a fundamental limit to the resolution of a measuring system. However, simple Norton and Thévenin
equivalents can in general give a satisfactory description. In this course we will represent the sensors by
their Norton and Thévenin equivalents, see Figure 2.42.
46
Rs
I
I
+
+
Vs
+
-
Rs
Is
V
-
V
-
(a)
(b)
Figure 2.42: Sensors will be represented by their Thévenin (a) or Norton equivalents (b).
Often the sensor resistance is unstable, temperature or frequency dependent, or nonlinear. The sensor
must then be modeled by a nonlinear Thévenin or Norton equivalent, as shown in Figure 2.43. This forces
us to minimize the influence of this resistance in electronic circuits if possible.
Rs
I
I
+
+
Vs
+
Is
V
-
Rs
-
V
-
(a)
(b)
Figure 2.43: More reaslistic sensors models.
47
2.5 Problems
1. Do a computer simulation of the sample-and-hold circuit of Figure 2.3 and remove the buffer at the
input. What do you notice if you look at the current through the ideal voltage source? Discuss the
result.
2. Consider a sinusoidal voltage source with an amplitude of 2 Volts. What is the average value of the
voltage source? And what is the effective value (or rms value) of the voltage source?.
3. Determine the average value and the rms value of a repetitive pulse voltage waveform width a duty
cycle of 50% symmetric around 0V, an amplitude of 4 Volts, and a period of 4 seconds? (Ans. 0V,
4V)
4. What is the differential resistance of a diode when the current through the diode is 2 mA? (Ans.
12.5Ω)
io
ii
+
μvi
Ri
vi
+
Ro
+
vo
−
−
−
(a)
io
ii
+
vs
Rs
+
vi
Ro
+
μvi
Ri
+
vo
Rl
−
−
−
−
(b)
Figure 2.44: (a) Circuit model for voltage amplifier. (b) Voltage amplifier with signal source and load.
5. Figure 2.44(a) shows a circuit model for a voltage amplifier, consisting of a voltage controlled voltage
source having a gain μ, an input resistance Ri that models the load the amplifier represents for the
input circuitry, and an output resistance R0 that indicates that the controlled voltage source is a
realistic source. Figure 2.44(b) shows the voltage amplifier in a simple circuit in which a sensor
voltage signal is fed into the amplifier, and the output is connected to a resistive load. Show that for
the overall voltage gain vo /vs holds:
Rl
vo
Ri
=μ
vs
Ri + Rs Rl + Ro
(2.87)
6. The voltage amplifier of figure 2.44(b) has an overall voltage gain of +20 dB, an input resistance of
100 kΩ, and an output resistance of 100 Ω. The signal voltage source has an internal resistance of
1kΩ and the amplifier drives a load of 1 kΩ. What is the value of μ? (Ans. 11.1).
48
B
C
+
gmvbe
rπ
vbe
ro
−
E
(a)
C
B
+
+
+
Rs
vs
gmvbe
rπ
vbe
ro
Rl
vo
−
−
−
E
(b)
B
ib
C
+
vbe
βib
rπ
ro
−
E
(c)
Figure 2.45: (a) Small-circuit model for a bipolar junction transistor (BJT). (b) The BJT used as a voltage
amplifier in a so-called common emitter amplifier with signal source and load. (c) An alternative smallsignal model for the BJT.
7. The small-signal model of the bipolar junction transistor (BJT) is shown in Figure 2.45. The three
terminals are the base (B), the emitter (E), and the collector (C). In Figure 2.45(a) in the model, we
see that the input resistance between the base and the emitter is modeled by a differential resistance
rπ , and that between collector and emitter a voltage controlled current source with transconductance
gm and a differential output resistance ro are modeled. Show that in the circuit of Figure 2.45(b):
vo
rπ
=−
gm (Rl ||ro ).
(2.88)
vs
rπ + Rs
Consider the equivalent small-model of the BJT in Figure 2.45(c), show that the relation
β = gm rπ
(2.89)
holds.
8. Figure 2.46 Shows a simplified small-signal model for an amplifier circuit with an emitter resistor
(will be treated later). Show that the input resistance Rin , defined as Rin = vx /ix , is
Rin = rπ + (β + 1)Re .
(2.90)
(Hint: use the KCL in the node next to E (the emitter) to express the current ix as a function of ib and
iE and use the constitutional relations of the resistances RE and rπ .)
9. Do a computer simulation of 2 circuits, one consisting of an ideal voltage source and two resistors in
series, and the other consisting of the ideal voltage source and a resistor in series with a diode. Check
Tellegen’s theorem.
10. If the power rating of a solar cell is roughly determined by the product VOC ISC , is the greatest rate
of increase obtained at lower or at higher levels of illumination? Explain.
49
ix
B
ib
+
C
βib
rπ
vx
Rl
Rin
E
Re
Figure 2.46: Simplified small-circuit model for a bipolar junction transistor (BJT) in a amplifier circuit with
an emitter resistor.
2.6 Simulation codes
Simulation code of the circuit in figure 2.3:
$ 1 5.0E-6 4.07 60 5.0 50
c 496 368 496 464 0 1.0E-8 -1.5
a 592 384 720 384 1 15.0 -15.0 1000000.0
O 720 384 768 384 0
w 720 384 720 432 0
w 592 432 592 400 0
g 496 464 496 496 0
w 592 432 720 432 0
w 496 368 576 368 0
w 576 368 592 368 0
f 448 304 448 368 0 1.5
w 400 368 432 368 0
w 464 368 496 368 0
R 448 304 448 240 0 5 600.0 20.0 -10.0 0.0 0.1
v 240 464 240 368 0 1 40.0 5.0 0.0 0.0 0.5
a 288 368 400 368 1 15.0 -15.0 1000000.0
w 400 368 400 432 0
w 400 432 288 432 0
w 288 432 288 384 0
w 288 352 240 352 0
w 240 352 240 368 0
w 496 464 240 464 0
o 13 64 0 35 9.3 9.1E-5 0 -1
o 12 64 0 35 18.7 9.7E-55 1 -1
o 2 64 0 34 9.3 9.7E-55 2 -1
50
Chapter
3
Processing of Voltage and Current Signals
Electronic circuits are ideally suited for processing signals. The information coded in voltage and current
signals can be processed as well. A number of signal processing circuits will be discussed. Completeness of
signal processing functions is not pursued; only those elementary signal functions important for this course
on amplifiers are treated.
What determines the choice of signal? Should I choose a current signal or a voltage signal? Or both?
First, there are the electronic properties of the signal source. Can the signal source be described best by
an ideal or a nonideal source? Does a sensor provide a specified voltage that is completely independent of
the other circuit elements, or does the specified voltage drop when higher currents are required? Or is the
specified current a sensor produces independent of the load on the sensor?
Second, our signal processing (sub)circuit will be loaded by a subcircuit, for example an actuator, to
perform an action after the signals are processed. Can this load been represented by a linear resistor or a
nonlinear resistor? Does this load have requirements? Do minimum or maximum values exist? And how
does the load affect the signal processing circuitry? Third, in many sensors and actuators, or even in the
processing circuitry nonlinear elements are present. These nonlinearities cause distortion of the signals. In
such cases it may be useful to consider mainly either voltage signals or current signals in order to minimize
the effects the nonlinearities. Sometimes it is even useful to transform an accurate voltage signal to an
accurate current signal or vice versa.
Once we have chosen the signal, we can perform signal processing. Treated are: addition and subtraction of ideal current signal sources and ideal voltage signal sources, addition and subtraction of real current
signal sources and addition and subtraction of real voltage signal sources, division of voltage signals and
division of current signals, passive and active multiplication of current signals and of voltage signals, and,
finally, filtering of voltage signals and filtering of current signals.
3.1 Addition and subtraction of current and voltage signals
To find circuits to add or subtract current or voltage signals we use the Kirchhoff’s laws. To sum signals we
add instantaneously the various signal values at any point in time. Of importance are the directions of the
current signals and the polarities of the voltage signals.
When sinusoidal signals with the same frequency but different phase are added the resulting signal will
be sinusoidal with the same frequency. When sinusoidal signals with different frequencies (the signals may
or may not have different phases) are added the resulting signal will not be sinusoidal, however it will be
periodic. The repetition frequency will be equal to the greatest common divisor of the frequencies of both
to-be-added signals, an example is shown in Figure 3.1.
51
40 Hz
60 Hz
25 ms
16.7 ms
50 ms
Figure 3.1: Adding a 40 Hz signal to a 60 Hz signal.
3.1.1
Addition and subtraction of ideal current and voltage signal sources
More generally, based on the Kirchhoff current law we can sum currents in any node in the circuit and sum
voltages in any loop, see Figure 3.2. When the direction of the current to be added is of opposite direction,
this current will be subtracted. And when the polarity of voltage sources is not the same a subtraction will
take place.
i1+i2
i1
i1+i2
i2
i1
v2
v1
+
v2
_
v1+v2
Rl
i2
v1+
_
+
v1+v2
Rl
_
Figure 3.2: Adding currents in a node and adding voltages in a loop.
3.1.2
Addition and subtraction of real current signal sources
Currents can be summed in any node of the circuit due to the Kirchhoff current law. For the addition or
subtraction of nonideal current sources this result can be used to obtain an expression for the total current.
In general, if we consider the summation of n current signals from nonideal sources, represented by
their Norton equivalent circuit—an ideal current signal source in in parallel with a source resistor Rsn —as
in Figure 3.3; the current through the load resistor is proportional to the sum of the parallel ideal current
signals sources:
il =
Rs
(i1 + i2 + ... + in ),
Rs + Rl
(3.1)
where Rs is the equivalent resistance of all source resistors in parallel. Immediately, we see that if the load
52
il
Rs
Rs
1
Rs
2
il =
R‘s
Rl
n
in
i2
i1
il
Rs'
R' + R
s
Rl
i1 + i2 +...+ in
1
1
=
'
R
s
R
(i1 + i2 + ...+ in)
l
+
1
s
1
1
+... +
Rs2
Rs
n
Figure 3.3: Addition of currents signals from real (= nonideal) sources.
resistance is much smaller than Rs , the current through the load is the sum of the ideal current signal input
sources.
To gain some insight into the effects of the load resistors, we start examining the above result by studying
the circuit of Figure 3.4 to add two nonideal current signal sources with magnitude i1 and i2 . We are looking
A
il
+
R2
R1
i2
i1
vl
Rl
-
Figure 3.4: Addition of currents signals from two nonideal sources
for an equation for the current through the load Rl of the form:
il = α(i1 + i2 ).
(3.2)
The constant α is always smaller than 1, because part of the current will always flow through the source
resistors R1 and R2 . For obtaining α = 1, that is, to design a circuit in which il = i1 +i2 in case of nonideal
current signal sources, a current amplifier element has to be included in the circuit. Solving il in the circuit
of Figure 3.4:
1
1
+
(3.3)
il = i1 + i2 − v l
R1
R2
R1 + R2
(3.4)
Rl ,
= i1 + i 2 − i l
R1 R2
that is,
il = α(i1 + i2 )
with
α=
1+
R1 + R2
Rl
R1 R2
−1
.
We distinguish the following cases:
• R1 and R2 are both much larger than the load resistance Rl : in this case α ≈ 1.
53
(3.5)
We see that il ≈ i1 + i2 ; there is hardly any current flow through the source resistors R1 and R2 . For a
small load resistance and large source resistances, signal current sources can be summed in a single node.
We can consider two other cases:
• R1 = R 2 = Rl ,
in which case α = 1/3 and we note that
il =
1
(i1 + i2 ).
3
(3.6)
• Rl is much larger than both R1 and R2 .
In this case α becomes
α=
1
Rl
R1 +R2
1
Rl + R1 R2
≈ 0.
(3.7)
There is hardly any current through the load resistor Rl .
Can we only efficiently add currents if the load resistance Rl is small enough? What to do if Rl is not
small enough?
The problem can be identified by noting that, in case of a large load resistance, almost all the current is
flowing through the source resistors instead of flowing through the load resistor. To cope with this problem
we will force the current not to flow through the source resistors but only through the load resistorse, by
making the voltage across the source resistors zero. To do this we will need the ideal elements nullator and
norator; and we will see that circuits can be defined using the combination of both ideal elements in the
ideal two-port element nullor.
v=0
i=0
A
i1
-
il
+
vl
R2
R1
i = i1 + i2
v = Rl.i
i2
Rl
-
Figure 3.5: Efficient addition of current signals from real sources in case of large load resistance, based on
nullator and norator (also useful for any value of load resistance).
To evaluate this possibility, consider the circuit in Figure 3.5. By placing the nullator element (v =
0, i = 0) the voltage drop across the source resistors R1 and R2 will be zero. Consequently, no current will
be lost and the sum of the signal current sources i1 + i2 will flow towards the load resistor. This is only
possible if and only if the norator is included which produces the voltage necessary v = Rl (i1 + i2 ) for the
current (i1 +i2 ) to flow through the load. Effectively, this means that the influence of the source resistors on
the behavior of the circuit is neutralized independent of their values—they may even be nonlinear. Because
also the voltage across the current sources is zero, and thus these sources cannot deliver any power to the
load, the norator has to deliver this power; that is, the voltage v = Rl (i1 + i2 ) for the current (i1 + i2 ).
Point A, that is connected to earth by the nullator, is called virtual earth.
Now, we redraw the circuit using the nullor element, see Figure 3.6. The circuit with the nullor is
called a current follower. The circuit can excite any load with a current equal to (i1 + i2 ). The value
of this load may be large or small; however additional energy from a norator is necessary. Looking at the
current through and the voltage across the norator, we see that the norator delivers energy to the circuit.
This amount energy has to be supplied from the outside.
54
nullor
R2
R1
+
+
i2
i1
il
nullor
il = i1 + i2
vl = Rl.il
-
A
-
A
+
vl
Rl
+ il
vl
Rl
R2
R1
i2
i1
-
-
Figure 3.6: Addition of currents signals from real sources based on nullor. Both circuits are equivalent. The
circuit is called: current follower.
3.1.3
Addition and subtraction of real voltage sources
Addition and subtraction of voltage signals is more complicated. The Kirchhoff voltage law suggests to add
voltage signal sources according to Figure 3.7.
v2
vn
+
+
Rs2
Rs
il
n
R*s
Rs
il
+
1
Rl
Rl
+
v1 + v2 +...+ vn
v1
Rs* = Rs1 + Rs2 +... + Rsn
Figure 3.7: Addition of voltages signals from nonideal sources
However in most cases the signal sources are all defined with respect to a common reference (earth). To
add these sources we have to make all voltage sources floating by using a transformer with a turns ratio of 1,
or to convert the voltage signal sources to current signals, and add the current signals. This last conversion
can be done with a gyrator or a resistor.
Again, to understand the basics of adding real voltage (signal) sources defined with respect to a common
reference, we start examining the most simple circuit: two ideal voltage sources with their source resistors
Rs1 and Rs2 in parallel (that is, the voltage sources are defined with respect to earth) both exciting a resistive
load Rl , see Figure 3.8.
il =i1+i2
i1
i2
Rs1
Rs2
v1
-
vl
+
+
+
Rl
v2
-
-
Figure 3.8: Addition of voltage signals from two nonideal sources.
55
In case of the addition of voltage sources we are looking for an equation for the voltage across the load
Rl of the form:
vl = μ(v1 + v2 )
(3.8)
And again the constant μ is always smaller than 1, because there will always be a voltage drop across the
source resistors Rs1 and Rs2 . In order to obtain μ = 1, that is, to design a circuit in which vl = v1 + v2 in
case of nonideal voltage signal sources, an amplifier element has to be included in the circuit. Soving vl in
the circuit of Figure 3.8, we obtain using
i1 =
v1 − vl
Rs1
i2 =
and
v2 − vl
,
Rs2
(3.9)
for the current through the load we find:
v1
vl
v2
vl
−
+
−
il = i 1 + i 2 =
Rs1
Rs1
Rs2
Rs2
v2
Rs1 + Rs2
v1
+
−
vl
=
Rs1
Rs2
Rs1 Rs2
and thus for the voltage across the load vl :
Rl
Rs1 + Rs2
Rl
v1 +
v2 −
Rl v l
vl =
Rs1
Rs2
Rs1 Rs2
which can be rewritten as:
Rl
Rl
vl = β
v1 +
v2
Rs1
Rs2
(3.10)
(3.11)
(3.12)
with
β=
1+
Rs1 + Rs2
Rl
Rs1 Rs2
−1
.
(3.13)
The first important conclusion we have to draw is that for Rs1 = Rs2 there is no linear relation between
the sum of input voltages and the voltage across the load, that is,
vl = μ(v1 + v2 )
Rs1 = Rs2
if
(3.14)
To find a solution to this problem of the nonlinear relation, we place a large resistor Rc in series with
each nonideal voltage signal source, see Figure 3.9. The resistor Rc must be chosen such that its value is
large compared with all source resistances.
il
i1
i2
i3
in
Rc
Rc
Rc
Rc
+
+
+
v1
v2
+
Rl
vn
v3
Figure 3.9: Addition of voltage signals by linearisation with large equal resistances, compared to the source
resistances.
With Rs1 , Rs2 Rc (that is, Rc + Rs1 ≈ Rc ,and Rc + Rs2 ≈ Rc ) we obtain the following linear
relation:
−1
2Rl
Rl
and β = 1 +
.
(3.15)
vl = μ(v1 + v2 ) with μ = β
Rc
Rc
We distinguish three cases:
56
• Rc Rl : in this case β ≈ 1.
Equation 3.15 reduces to
vl ≈
Rl
(v1 + v2 ),
Rc
(3.16)
or more generally,
v Rl ≈
Rl
(v1 + v2 + ... + vn ).
Rc
(3.17)
With the restriction on Rc being large compared to the various Rs and Rl we could easily find the same result
using superposition, showing that we indeed converted the voltage signals into current signals. Obviously,
a disadvantage is that each voltage signal is reduced with a large fraction Rl /Rc .
• Rc = Rl : in this case μ = β ≈ 1/3, and the equation for the summation of two voltage signals
sources becomes:
1
(v1 + v2 ).
(3.18)
3
In order to obtain the sum of signal source voltages we have to multiply the result with a voltage amplifier
with a gain of 3.
vl ≈
• Rl Rc : in this case we obtain:
vl =
1
Rl
1
Rl
(v1 + v2 ) =
(v1 + v2 ) ≈ (v1 + v2 ).
2R
l
Rc 1 + R
Rc + 2Rl
2
(3.19)
c
And we notice that in order to obtain the sum of signal source voltages we have to multiply the result with
a voltage amplifier with a gain of 2.
From the above it is clear that the addition or subtraction of current and voltage signals has to be done
with care.
3.2 Division of voltage signals and division of current signals
Division of voltage signals to generate a voltage smaller than the “to-be-divided-voltage” is often used in a
variety of circuits. Also current dividers play an important role in circuit design. We first treat the simple
resistor voltage divider circuit that can be used to drive high ohmic loads. Second, we discuss the adaptation
necessary to drive small loads too. And third, the resistor current divider is dealt with.
3.2.1
Division of voltage signals
Division of voltage signals can be obtained by a voltage divider circuit. Due to the signals source resistance the addition of a load resistor already leads to a division of the voltage of the signal source, see
Figure 3.10(a).
Looking at Figure 3.10(a), applying KCL, KVL , and Ohm’s law leads to
i=
vs
Rs + Rl
and
vl =
Rl
vs .
Rs + Rl
(3.20)
Further division of voltage signals can be obtained by adding resistors in series as shown in Figure 3.10(b).
The circuit of Figure 3.10(b) can be represented by a single Thévenin equivalent that can help you to
determine the usefulness of the voltage divider when a load is attached to it.
The Thévenin equivalent of Figure 3.10(b) is voc = R2 vs /(Rs + R1 + R2 ) and RTh = R2 ||(R1 + Rs ),
and shows that the voltage across the load vl can only be voc if and only if the resistance of the load it is
exciting, is much larger than RTh ; that is, vl = voc iff Rl RTh .
57
Rs
i
Rs
i
+
R1
+
+
vl
vs
Rl
vs
+
-
vR
R2
(a)
2
(b)
Rl
vl =
Rs + Rl
vs
R2
vR2 =
R1 + R2 + Rs
vs
Figure 3.10: Division of voltage signals.
When the load resistance is comparable to or smaller than the Thévenin resistance RTh a substantial
drop in voltage across the load resistor will occur. This is often undesirable. What can we do to modify the
circuit in order to drive small loads with the voltage voc too?
The problem can be identified by noting that, in case of small load resistance, almost all the current at
the output is flowing through the load resistor instead of flowing through the parallel resistor, R2 , of the
voltage dividing subcircuit. To cope with this problem we will force the current not to flow through the load
resistor but only through the divider subcircuit by inserting a nullator between the divider subcircuit output
and the load resistor, see Figure 3.11.
R1*
vo
+
vs
i=0
v=0
+
+
v = vo
i = vo / Rl
+
R2
vo
-
Rl
-
Figure 3.11: Division of voltage signals using a nullator and norator. R1∗ includes the source resistance.
In addition to making the current zero from the divider subcircuit to the load, the nullator copies vo and
ensures the voltage across the load resistor to be vo , the desired value (the voltage drop across the nullator
is by definition zero). Of course, due to the fact that a current cannot be supplied by the divider subcircuit
(the current through the nullator is by definition zero too), a norator has to be attached to provide the current
necessary at the desirable voltage.
To find an implementation using a nullor, we redraw the circuit, see Figure 3.12. The subcircuit, see
Figure 3.13, with the nullor is known as voltage follower; the output follows the voltage across the resistor
parallel to the load. Because the nullor uses an (external) energy source, the norator, we call the circuit
including the nullor an active voltage divider circuit. The circuit is capable to deliver a constant voltage to
any load resistor.
58
R1*
nullor
-
vo
+
vs
+
+
v = vo
i = vo / Rl
+
R2
vo
Rl
-
-
Figure 3.12: Active division of voltage signals using a nullor.
nullor
-
Rs
+
vs
+
v = vo
i = vo / Rl
+
-
vo
Rl
-
Figure 3.13: The circuit called: voltage follower.
3.2.2
Division of current signals
Division of current signals can be obtained using a current divider circuit. Just as with the voltage signals,
the addition of a resistor, in this case in parallel, already leads to division of the current signal of the source,
see Figure 3.14(a).
Using
Rs iRs = Rl il
and
is = iRs + il
(3.21)
results in
Rs (is − il ) = Rl il
→
il =
Rs
is
Rs + Rl
(3.22)
Further division of current signals is obtained by adding more resistors in parallel, see Figure 3.14(b).
The circuits form a subset of the circuits discussed in the section on addition of current signals. The
analysis of how to use these circuits can be evaluated by the same methods as described in that section.
3.3 Multiplication of voltage and current signals
Both current and voltage signals can be multiplied by either passive or active multiplication. Multiplication
is important for compensating the loss of signal because of internal resistances, as was already discussed.
But it is also important for amplification of signals in general, due to reasons presented in the introductory
first chapter.
59
il
is
Rs
Rl
i2
is
Rs
(b)
(a)
il =
Rs
Rs + Rl
R2
R1
i2 =
is
Rl Rs
RlRs + R2Rs + R1R2
is
Figure 3.14: Division of current signals.
3.3.1
Passive multiplication of voltage and current signals
Passive multiplication of voltage signals or current signals can be done using a transformer as discussed
before. We obtained:
vo = nvi
and
ii = nio ,
(3.23)
n is the turns ratio, vi is the voltage across the primary coil, ii is the current through the primary coil, vo
is the voltage over the secondary coil, and io is the current through the secondary coil. If the voltage is
multiplied the current is divided by the same number, because energy is conserved in the ideal transformer.
In contrast to passive multiplication, active multiplication using an additional energy source may increase a
voltage or current without reducing the other. The next section will introduce this active multiplication.
3.3.2
Active multiplication of voltage signals
By definition, active signal multiplication uses an additional energy source; signal power must be amplified.
For the description of circuits that are able to perform active multiplication of voltage signals we start with
the voltage divider circuit and ask ourselves what we need to perform an inverse operation.
Figure 3.15(a) shows the voltage divider circuit. Clearly the voltage at the output vo is always smaller
than the voltage at the input vi . We now replace the input by the output, and the output by the input. To
obtain a physically possible circuit we add a norator in series with the resistors to provide the necessary
energy, see Figure 3.15(b). The norator has to provide a voltage vn = vo and a current in = vo /(R1 + R2 )
in order to obtain the output voltage vo and the voltage vi at the node between the resistors. Adding this
source will do the operation if we make sure that the input vi is coupled to the node without any voltage
drop and without providing any current. That is, we must add a nullator (i = 0 and v = 0) between the
input and the node. In fact, the combination of the input vi and the nullator condition determines exactly
how much energy must be delivered by the norator.
There is also another way of looking at the operation of this circuit. The nullator copies the input voltage
vi to the node in between the resistors. As a consequence a current of vi /R2 will flow through resistor R2 as
soon as an energy source, the norator, is present. Due to the fact that the nullator cannot pass any current, the
current through R1 must be the same as the one through R2 causing an increase in potential of (R1 /R2 )vi
2
vi . The nullator is used to feed forward the input voltage to the
leading to: vo = vi + (R1 /R2 )vi = R1R+R
2
node in between the resistors that causes a current through R2 and R1 resulting in a larger voltage at the
output; provided that an energy source (and thus a path for the current) is present.
The new circuit performs the inverse operation of voltage division, that is multiplication of the voltage
with a constant:
R1
R1 + R2
vi = (1 + R1 G2 ) vi
(3.24)
vi = 1 +
vo =
R2
R2
Redrawing Figure 3.15(b) gives Figure 3.16 and makes it possible to identify the nullor, consisting of
the nullator and the norator.
60
+
R1
vi
(a)
+
R2 vo
-
vo =
R2
R1 + R2
vi
-
+
norator to deliver energy
vn = vo
in = vo / (R1+R2)
+
vi
(b)
R1
nullator
vo
+
vi
i=0
v=0
vo =
R1 + R2
R2
vi
R2
-
-
Figure 3.15: (a) voltage divider, and (b) active voltage multiplication.
Now, we consider a special configuration. We replace the resistor R1 by a short circuit (R1 = 0) and
R2 by a open cicuit (G2 = 0). We immediately recognize the voltage follower as discussed in the section
on the division of voltage signals where the follower was used to “follow” the voltage from the divider
subcircuit. The voltage follower is thus a voltage multiplication circuit with a multiplication factor of
vo
= (1 + R1 G2 ) = (1 + 0 · 0) = 1,
vi
(3.25)
which, of course, is not a suprise.
3.3.3
Active multiplication of current signals
We can try to follow the same procedure to see whether we can perform current amplification. Therefore
we start with the normal current divider circuit, see Figure 3.17(a). Note that for the current divider it is
essential that both resistors are connected to the same nodes.
First, to make an active circuit doing the reverse we again replace the input by the output and the output
by the input and connect both resistors to an energy source: the norator, see Figure 3.17(b). Second, a
nullator is placed to force both resistors to share not only the central node between them (the node denoted
to have the voltage v) but also earth. This nullator also assures that no other current than ii (this is the
current through R2 ) is summed with the current through R1 at the central node. To obtain the required
output current, the sum of the current through R2 and through R1 , de norator produces the required voltage;
that is, vn = vR1 = vR2 . In this way we are sure that the voltage across both resistors is the same.
61
nullor
+
+
+
vi
R1
vi
vo
R1 + R2
vo =
R2
vi
R2
-
-
Figure 3.16: Redrawing the active voltage multiplication with the nullor. The circuit is called: voltage
amplifier.
(a)
R1
ii
io =
R2
R1
ii
R1 + R2
io
io
+
+
v
-
in = io
vn = vR1
io = ii + iR1
(b)
iiR2 = iR R1
R2
i=0
v=0
1
R1
io =
ii
R1 + R2
R1
}
=>
ii
-
Figure 3.17: (a) current divider, and (b) active current multiplication.
62
Looking carefully at the directions of the currents, we see that the potential at the central node, v, must
be negative (that is, below earth potential) for positive values of ii . The current through the norator is now
the sum of currents through both resistors and the voltage across the norator determines the voltage on the
central node. The new circuit performs the inverse operation of current division, that is multiplication of
the current with a constant:
io =
R1 + R2
ii = (1 + G1 R2 )
R1
(3.26)
io
+
nullor +
io =
R1 + R2
R1
ii
Rl
R2
R1
ii
-
Figure 3.18: redrawing the active current multiplication with the nullor. The ciruit is called: current amplifier.
A note on including a load is useful here. When the output is shorted, the norator produces exactly the
energy dissipated by both resistors (the current source does not provide any energy because vi = 0). To
produce this amount of energy vn must be vn = vR1 and the current in must be in = iR1 + iR2 . If a load
is included, the norator has to, and will!, produce more energy (remember Tellegen’s Theorem). That is,
the potential at the positive side of the load will always be larger than 0V (earth), for positive values of ii .
Consequently, the current through a load is always the sum of ii and iR1 .
Redrawing Figure 3.17(b) gives Figure 3.18 and makes it again possible to identify the nullor circuit. If
we replace R1 by an open circuit and R2 by a short circuit, we obtain the “current follower” of the section on
the addition of current signals; telling us that the current follower is an active current multiplication circuit
with amplification factor:
io
= (1 + G1 R2 ) = (1 + 0 · 0) = 1
ii
(3.27)
3.4 Conversion from current signals to voltage signals, and conversion from voltage signals to current signals
Conversion from currents to voltages or from voltages to currents can generally be done with a resistor.
However, we have to pay a price for the conversion, namely, that signal energy will be lost by dissipation
in the resistor. This can be very unwanted. The ideal solution for this loss is the proposed gyrator device.
With a gyrator the conversions can be done without energy loss. It is especially this last requirement what
makes it impossible to construct such a device; the nonenergetic single gyrator device does not exist.
In many cases it is useful to convert a current signal to a voltage signal or a voltage signal to a currrent
with an active circuit. The active circuit uses an external energy source that compensates for the energy loss
in the resistor. Such an active circuit is, as you probably already expect, based on the nullor too.
63
3.4.1
Conversion from current signal to voltage signal
+
To understand how a nullor based circuit can convert a current to a voltage, we again consider the “current
follower”. Realizing that the norator in this circuit besides passing an input current also determines the
voltage at the output (vl = Rl ·i), it is possible to modify the circuit such that not the load resistor determines
the norator voltage but a resistor between the nullator and the norator; see Figure 3.19(a).
To fully understand the circuit we note that node A is at earth potential (vA = 0V) and that, to conduct
the input current to the load, the potential of node B must be below earth potential, for positive values of ii ,
(vB < 0V) for any resistor value unequal to zero; that is, vB = −Ris . We see that the voltage across the
load, v, equals Ris ; so the current signal at the input is converted to a voltage signal. In Figure 3.19(b) the
circuit is redrawn to recognize the nullor; the circuit is called a transresistance amplifier.
v=0
i=0
i = is + il
v = R.is
-
is
R
B
A
vl
Rs
Rl
+
is
il
-
(a)
A
vl = R.is
-vl = -R.is
R
nullor
B
-
vl
+
is
Rl
-vl = vo
+
Rs
il
-
+
is
-
(b)
Figure 3.19: Active current to voltage converter circuit with the nullor. The circuit is called: transresistance
amplifier.
To avoid confusion due to the sign of the voltage across the load resistor, we will use the naming
convention that the output voltage (the voltage across the load) is the voltage vo as defined in a general
two-port description. In case of the current to voltage conversion:
vo = −Ris .
3.4.2
(3.28)
Conversion from voltage signal to current signal
To understand how a nullor based circuit can convert a voltage to a current, we consider the “voltage
follower”. The nullator in this circuit passes the input voltage, but blocks any input current. Consider, now,
the circuit of Figure 3.20(a). The input voltage can be converted to a current by a resistor between the
central node and earth having a value R. This current, which also flows through the load and the norator is
iR = il = in = vs /R. The norator needs a voltage step vn of vn = vl + vR .
In Figure 3.20(b) the circuit is redrawn with the nullor; the circuit is called a transconductance amplifier.
64
-
+
v=0
i=0
-
in = iR = il
vn= vl + vR
il
il = vs/R
-il = -vs/R
+
vs
vl
R
Rl
+
vs
(a)
nullor
-
il
+
+
-
vs
Rl
iR
+
R
(b)
Figure 3.20: Active voltage to current converter circuit with the nullor. The circuit is called: transconductance amplifier.
65
3.5 Filtering signals
Filters are used to select ac signals with a specified frequency from a wide range of signals having many
different frequencies, or for selecting a certain frequency band and eliminating all others. The frequency
selectivity of a filter is based on the facts that the reactance of capacitors and inductors depend on frequency.
After studying the ac behavior of some basic components in Chapter 2, we are now in the position to
consider the frequency characteristics of simple circuits.
3.5.1
Impedances of series and parallel reactive circuits
Two impedances can be combined essentially in two ways: in series and in parallel. We will investigate the
characteristics of various combinations.
resistor and resistor
R2
R1
R s= R1 + R2
R2
R1
i
i1
i
v
v
(a)
i2
R ·R2
R p = R 1+ R
1
2
(b)
Figure 3.21: Series (a) and parallel (b) arrangement of resistors.
We first briefly review the combinations of two resistors. Figure 3.21(a) shows the series arrangement
of two resistors excited by an ideal voltage source. A current i flows through both resistors. Using the KVL
we obtain:
v = iR1 + iR2 = i(R1 + R2 ) = iRs .
(3.29)
The series combination can be replaced by an equivalent resistor Rs with value R1 + R2 . When R1 R2
the equivalent resistance Rs will be approximately equal to R1 . We note that in the series arrangement the
largest resistance dominates.
Figure 3.21(b) shows the arrangement of two resistors in parallel. When excited by an ideal voltage
source the voltage across the resistors R1 and R2 will be the same. According to the KCL we have:
i = i1 + i 2 =
v
v
R1 + R2
v
+
=
v=
.
R1
R2
R1 R2
Rp
(3.30)
The equivalent resistance in the parallel combination Rp is
Rp =
R1 R2
.
R1 + R2
(3.31)
When R1 R2 the equivalent resistance Rp will be approximately equal to R2 . Note that in the parallel
arrangement the smallest resistance dominates. Because the resistance of a resistor is independent of the
frequency the source the equivalent resistors Rs and Rp do also not change with frequency.
66
R
log |Z|
C
Δϕ
+90o
0o
log R
ωo
log ω
-90o
ωo
log ω
Figure 3.22: Impedance characteristics of a series RC arrangement.
resistor and capacitor
Figure 3.22 shows the series arrangement of a resistor and a capacitor and its impedance characteristic. For
the combination holds:
Z = ZR + ZC = R − j
1
ωC
(3.32)
and for the magnitude of the impedance:
1
R2 ω 2 C 2 + 1
2
|Z| = R + 2 2 =
.
ω C
ω2 C 2
(3.33)
For ω = ∞, ZC is zero and the resistor dominates. For ω21C 2 R2 ,the capacitor dominates. For
ω = 0 holds that YC is zero and leads to a serial combination of the resistor with an open circuit. When the
1
1
and we define ω0 = RC
.
impedance characteristics of R and C meet, holds: |ZR | = |ZC |, then R = ωC
√
For this value |Z| = 2R.
R
Δϕ
log | Z|
+90o
log R
C
0o
-90o
ωo
log ω
ωo
log ω
Figure 3.23: Impedance characteristics of a parallel RC arrangement.
Figure 3.23 shows the impedance characteristic of the parallel arrangement of a resistor and a capacitor.
For the combination holds:
1
1
=
+ jωC
Z
ZR
(3.34)
and for the magnitude of the impedance:
−1
1
2
2
+ω C
.
|Z| =
R2
(3.35)
67
For ω = 0 holds that YC is zero, the resistor dominates the parallel combination. For ω = ∞, ZC is zero,
the capacitor dominates. As the characteristics of the resistor and the capacitor meet, at frequency ω√
0 , the
1
the impedance of the parallel combination is |Z| = 21 2R.
capacitor starts dominating. For ω = ω0 = RC
resistor and inductor
R
L
log ⏐Z⏐
Δϕ
+90o
0o
log R
ωo
log ω
-90o
ωo
log ω
Figure 3.24: Impedance characteristics of a series RL arrangement.
Reasoning in the same way as in case of the combination of resistor and the capacitor we find for the
series arrangement of a resistor and an inductor, see Figure 3.24, that for ω = 0 holds that ZL is zero, the
resistor dominates the series combination. For ω 2 L2 R2 ,the the inductor dominates. Take over takes
place when |ZL | = |ZR |, that is as ω 2 L2 = R2 at frequency ω0 = R
L . For ω = ∞, YL becomes zero and
behaves like an open circuit.
R
log ⏐Z⏐
Δϕ
L
+90o
log R
0o
log ω
-90o
ωo
ωo
log ω
Figure 3.25: Impedance characteristics of a parallel RL arrangement.
Figure 3.25 shows the impedance characteristic of the parallel arrangement of a resistor and an inductor.
For ω = 0 holds that ZL is zero, the inductor dominates the parallel combination. For ω = ∞, YL is zero,
the resistor dominates. The characteristics of the resistor and the inductor meet at frequency ω0 = R
L .
capacitor and inductor
In combinations of a capacitor and an inductor something special occurs. Both at ω = 0 and at ω = ∞ one
of the components will have an zero admittance.
For a series arrangement, see Figure 3.26 the impedance of the combination will be infinite in these
cases.
Z = ZC + ZL = j(ωL −
1
)
ωC
(3.36)
1
We see that if ωL = ωC
the impedance of the series combination becomes zero. The frequency at which
68
ωo
log ⏐Z⏐
Δϕ
+90o
0o
ωo
log ω
-90o
log ω
L
C
Figure 3.26: Impedance characteristics of a series LC arrangement.
this happens is
1
ω0 = √
,
LC
(3.37)
that is as
f0 =
1
√
2π LC
(3.38)
and is referred to as resonant frequency.
L
log ⏐Z⏐
Δϕ
+90o
C
0o
log ω
-90o
ωo
ωo
log ω
Figure 3.27: Impedance characteristics of a parallel LC arrangement.
In a parallel arrangement, see Figure 3.27, the impedance in both ω = 0 and ω = ∞ will be zero,
because one of the components will have a zero impedance. Again we can define an angular resonance
1
, where the impedance of the parallel combination becomes infinite.
frequency ω0 = √LC
capacitor and inductor and resistor
In a series arrangement of a resistor, a capacitor, and an inductor the L − C combination will dominate the
impedance characteristic at both low and high frequencies. At the resonant frequency, where the impedance
of the L − C combination is zero the impedance of the total will be equal to R. The behavior is shown in
Figure 3.28.
For the parallel arrangement of a resistor, a capacitor, and an inductor we expect a zero impedance at
ω = 0 and ω = ∞, because one of the components has a zero impedance. At the resonant frequency, the
combination has an impedance value of R, see figure 3.29
69
R
L
C
log ⏐Z⏐
Δϕ
+90o
0o
log R
ωo
log ω
-90o
ωo
log ω
Figure 3.28: Impedance characteristics of a series RLC arrangement.
R
log ⏐Z⏐
Δϕ
C
+90o
log R
L
0o
log ω
-90o
ωo
ωo
log ω
Figure 3.29: Impedance characteristics of a parallel RLC arrangement.
3.5.2
Transfer functions and passive filters
The concepts voltage divider and current divider applied to the ac domain lead to the definition of transfer
functions. Figure 3.30 shows the frequency dependent voltage and current dividers.
Z1
Z2
Z2
vi
ii
vo
Z1
io
(b)
(a)
Figure 3.30: (a) Voltage divider and (b) current divider.
For the voltage divider holds:
vo = vi
Z2
.
Z1 + Z2
(3.39)
We can describe the behavior of this circuit by giving an expression for the output voltage divided by the
input voltage:
vo
Z2
=
.
vi
Z1 + Z2
(3.40)
This ration is the voltage gain of the circuit and is often referred to as transfer function.
70
A similar expression using an ac current divider leads to:
io
Z1
=
,
ii
Z1 + Z2
(3.41)
expressing the current gain of a circuit and is also called a transfer function. The transfer function tells us
how an input voltage or current is ‘transferred’ to the output of a circuit.
If we want to synthesize a circuit to select ac signals from a wide range of signals having many different
frequencies, we can use the frequency dependent voltage or current dividers. Based on the voltage divider
circuit, as an example, the synthesis goes as follows. When, at least, one of the impedances is frequency
dependent, the transfer function vo /vi will be frequency dependent too. Suppose we want to synthesize a
low-pass filter.
For a low-pass filter we require the transfer function, Eq. 3.40, to approach one for low frequencies, and
to approach zero for high frequencies. Now we can distinguish two topologies involving only one frequency
dependent element, shown in Fig. 3.31(a) and (c).
R
R
+
vi
Z
-
+
+
vo
vi
-
-
+
C
vo
-
(a)
(b)
Z
L
+
vi
R
-
+
+
vo
vi
-
-
(c)
+
R
vo
-
(d)
Figure 3.31: Synthesis of first order low-pass filters with passive elements based on the frequency dependent
voltage divider.
We consider the topology in Fig. 3.31(a). For low frequencies the transfer function should approach
one:
Z
→ 1,
Z+R
that is: |Z| R, for low frequencies. At the same time for high frequencies the transfer function should
approach zero:
Z
→ 0,
Z+R
that is: |Z| R for high frequencies. Looking to the impedance characteristics of RC arrangements,
Fig. 3.22 or 3.23, we come to the conclusion that the frequency dependent element must be a capacitor, see
Fig. 3.31(c).
Considering the topology in Fig. 3.31(b). For low frequencies the transfer function should approach
one:
R
→ 1,
Z+R
that is: |Z| R, for low frequencies. At the same time for high frequencies the transfer function should
approach zero:
R
→ 0,
Z+R
71
that is: |Z| R for high frequencies. Looking to the impedance characteristics of RL arrangements,
Fig. 3.24 or 3.25, we come to the conclusion that the frequency dependent element must be a inductor, see
Fig. 3.31(d).
R
R
+
vi
Z
-
+
+
vo
vi
-
-
+
vo
L
-
(a)
(b)
C
Z
+
vi
R
-
+
+
vo
vi
-
-
+
R
vo
-
(c)
(d)
Figure 3.32: Synthesis of first order high-pass filters with passive elements based on the frequency dependent voltage divider.
Similar reasoning will give us the basic topologies for high-pass filters Fig. 3.32(a) and (c). Now, we
will examine the transfer functions of various combinations of Z1 and Z2 in detail.
low-pass RC network
Suppose Z1 is a resistor and Z2 a capacitor in the voltage divider circuit, see Figure 3.33.
R
C
vi
v
log vo
i
vo
Δϕ
+90o
log 1
0o
-90o
ωc
log ω
ωc
log ω
Figure 3.33: Transfer characteristics of a RC low-pass filter.
The transfer function is:
1
−j ωC
ZC
1
vo
=
=
1 = 1 + jωRC .
vi
ZR + ZC
R − j ωC
(3.42)
We see that for ω = 0 the transfer function is 1 (the voltage amplification is 1). At high frequencies the
value of jωRC is large compared to 1 and the transfer function will be close to zero.
The magnitude of the voltage gain is given by:
vo 1
= .
(3.43)
vi 1 + (ωRC)2
72
√
1
When the value of ω is ω = RC
the voltage gain is 12 2 = 0.707 (on a log scale this is 0.15 below log 1).
This frequency is called the angular cut-off frequency, ωc :
1
,
RC
which equals
ωc =
(3.44)
1
(3.45)
2πRC
defining what is called the cut-off frequency fc .
1
we obtain a new expression
If we substitute in equation 3.42 the expressions ω = 2πf and RC = 2πf
c
for the gain in the circuit in terms of the signal frequency f and the cut-off frequency fc :
fc =
vo
1
1
=
=
.
vi
1 + jωRC
1 + j ff
(3.46)
c
We can investigate the behavior of this transfer function for different frequency ranges, see Figure 3.34.
Im vo
vi
Im vo
vi
vo
=1
vi
Im vo
vi
v
Re o
vi
(a) f << fc
v
Re o
vi
vo
= 0.7071
vi
(b) f = fc
vo fc
vi = f
v
Re o
vi
(c) f >> fc
Figure 3.34: Phasor diagrams of the gain of a low-pass filter at different frequencies.
• When f fc
When the signal frequency f is much lower than the cut-off frequency fc , the ratio ffc is much less than
unity and the voltage gain is approximately equal to 1. The imaginary part of the gain (transfer function) is
negligible and the gain is pure real, and the phase of the output voltage is the same as the input voltage, see
figure 3.34 (a).
• When f = fc
When the signal frequency equals the cut-off frequency the gain can be expressed by:
vo
1
1 1
=
= − j
vi
1+j
2 2
(3.47)
√
as illustrated in figure 3.34 (b). The magnitude is 12 2 the output voltage lags the input voltage by 45◦ .
The gain is therefore: 0.7071∠ − 45◦
• When f fc
In this case ffc is much greater than 1:
1
fc
vo
≈ f = −j .
vi
f
jf
(3.48)
c
The phasor is shown in Figure 3.34 (c). We note that the transfer function is purely imaginary. The magnitude of the gain is inversely proportional to the frequency. The output voltage lags the input voltage by
90◦ .
73
high-pass RC network
We can do the same analysis on the voltage divider if we reverse the positions of the resistor and the
capacitor, see Figure 3.35. We obtain the following results:
R
vi
vo
v
log vo
i
Δϕ
+90o
log 1
0o
-90o
ωc
log ω
ωc
log ω
Figure 3.35: Transfer characteristics of a RC high-pass filter.
vo
ZR
R
1
=
=
=
1
1 .
vi
ZC + ZR
R − j ωC
1 − j ωRC
√
A voltage gain of 21 2 at
fc =
(3.49)
1
.
2πRC
(3.50)
and rewriting equation 3.49 :
1
vo
=
.
vi
1 − j ffc
(3.51)
Analyzing this last expression shows, see Figure 3.36, that for:
Im vo
vi
Im vo
vi
vo
=1
vi
(a) f >> fc
v
Re o
vi
Im vo
vi
vo
= 0.7071
vi
v
Re o
vi
(b) f = fc
vo f
vi = fc
v
Re o
vi
(c) f << fc
Figure 3.36: Phasor diagrams of the gain of a high-pass filter at different frequencies.
• When f fc
vo
=1
vi
(3.52)
• When f = fc
74
vo
1 1
= + j
vi
2 2
(3.53)
• When f fc
vo
f
=j
vi
fc
(3.54)
and we obtain a filter that allows high frequencies to pass but filters out low-frequency signals.
high-pass RL network
Again similar analysis gives, see Figure 3.37:
R
L
vi
vo
v
log vo
i
Δϕ
+90o
log 1
0o
-90o
ωc
log ω
ωc
log ω
Figure 3.37: Transfer characteristics of a RL high-pass filter
vo
1
ZL
jωL
=
=
=
.
R
vi
ZR + ZL
R + jωL
1 − j ωL
(3.55)
The cut-off frequency is found to be
R
,
2πL
(3.56)
vo
1
=
,
vi
1 − j ffc
(3.57)
fc =
leading to
which is identical to the equation that described the high-pass RC filter. The frequency behavior of this
circuit is identical to the high-pass RC filter, differing only in the determination of the cut-off frequency.
comparison of RC and RL networks
Further investigation shows that the frequency characteristic of a RL low-pass filter is identical to that of a
1
RC low-pass filter. The RC circuits have a angular cut-off frequency of ωc = RC
; the RL circuit an angular
R
frequency of ωc = L .
In case of low-pass filters we will call the cut-off frequencies high-frequency cut-offs, because high
frequencies are attenuated; in case of high-pass filters we will call the cut-off frequencies low-frequency
cut-offs, because low frequencies are attenuated.
75
Transposing the components in a particular RL or RC filter will change it from a high-pass filter to a
low-pass filter, and vice versa. Replacing a capacitor by an inductor, or an inductor by a capacitor, will also
change the filter from a high-pass to a low-pass filter, and vise versa.
Doesn’t it matter whether we choose a RC or RL filter? No, it does. Other properties are different.
For example, the input impedance of a series arrangement of a resistor and a capacitor is different from the
input impedance of a inductor in series with a resistor. Also practical considerations may determine your
choice of filter circuit. In ICs, for example, inductors can be less easy realized.
filters in the current domain
The current divider circuit can be used to filter currents.
L
ii
C
ii
io
R
io
R
(a)
(b)
Figure 3.38: Current filtering: RL low-pass filter, (b) a RC high-pass filter.
The current transfer in Figure 3.38(a) has the characteristics of a low-pass filter. For frequencies below
the angular cut-off frequency ωc = R
L all current passes through the inductor L, while for frequencies
above the cut-off frequency the current passes increasingly through the resistor R. The cut-off frequency is
a high-frequency cut-off.
In contrast, the current transfer in Figure 3.38(b) has the characteristics of a high-pass filter. For frequen1
all current passes the capacitor, while for frequencies
cies above the angular cut-off frequency ωc = RC
below the cut-off frequency the current to a larger extend passes through the resistor. The cut-off frequency
is a low-frequency cut-off.
band-stop and band-pass filters
A series circuit of a resistor with a parallel LC network behaves as a band-pass filter, see Figure 3.39.
R
L
vi
C
vo
v
log vo
i
Δϕ
+90o
log 1
0o
-90o
ωo
log ω
ωo
log ω
Figure 3.39: Transfer characteristics of a band-pass filter.
1
The impedance of the parallel LC network increases to infinity at the resonant frequency ω0 = √LC
. In
this case it is clear that the transfer function is 1 (the voltage gain is 1). As soon as the frequency becomes
76
larger or smaller than the resonant frequency the impedance decreases and so does the voltage gain. The
current passes effectively a range of√frequencies. The frequency range, around the resonant frequency,
where the voltage gain is between 12 2 and 1 is called the bandwidth. The bandwidth B of the filter is
determined by the so-called quality factor Q:
B=
f0
.
Q
(3.58)
The quality factor Q quantifies the ‘narrowness’ of the bandwidth, and is measure of how much energy is
stored in the inductor or capacitor relative to how much is dissipated in the resistor during resonance. In
a series resonant circuit Q is increased by reducing the value of R, while in a parallel resonant circuit Q
is increased by increasing the value of R. In the above described filter, it can be found (easily checked by
simulation) that increasing the value of R increases the value of Q and decreases the bandwidth.
C
L
R
vi
vo
Figure 3.40: Another band-pass filter.
Another band-pass circuit is shown in Figure 3.40. In the voltage divider Z1 is chosen to be a series
combination of an inductor and a capacitor, and for Z2 a resistor is chosen. The series LC combination has
1
where the impedance is zero and thus the voltage gain 1. Above and
a resonant frequency at ω0 = √LC
below this frequency the amplification decreases. The quality factor can be increased by reducing the value
of R.
The transfer of a band-stop network consisting of a series arrangement of a capacitor and an inductor
is shown in Figure 3.41. The impedance of the LC combination becomes zero at the resonant frequency
1
, and thus the gain becomes zero. All frequencies outside the range near the resonant frequency
ω0 = √LC
are passed; it is a band-stop filter. Figure 3.42 shows another band-stop filter.
C
R
vi
L
vo
v
log vo
i
Δϕ
+90o
log 1
0o
-90o
ωo
log ω
ωo
log ω
Figure 3.41: Transfer characteristics of a band-stop filter.
Using the current divider circuit current band-pass and band-stop filters can be realized. Figures 3.43
and 3.44 show some examples.
77
C
L
vi
R
vo
Figure 3.42: Another band-stop filter.
C
L
R
ii
ii
io
R
C
io
L
Figure 3.43: Current filtering: band-pass filters.
3.6 Problems
1. What is the repetition frequency of the signal obtained by the addition of sinusoidal signals of frequencies 5Hz and 6Hz? And of sinusoidal signals of 60Hz and 40 Hz? (ans. 1Hz and 20Hz)
2. Consider the two-resistor voltage divider circuit of Figure 3.45(a). Figure 3.45(b) is the Thévenin
quivalent . Find expressions for Vo and RT h ; discuss the effect on the output voltage of a load Rl of
different values. Is the voltage divider a good battery? Explain.
3. Derive the two topologies for the high-pass filters based on a frequency dependent voltage divider
circuit with one frequency dependent element.
4. Derive the two topologies for the low-pass filters based on a frequency dependent current divider
circuit with one frequency dependent element.
5. Derive the two topologies for the high-pass filters based on a frequency dependent current divider
circuit with one frequency dependent element.
6. Derive equation 3.5.
7. Derive equation 3.13.
L
ii
ii
R
C
R
L
io
C
Figure 3.44: Current filtering: band-stop filters.
78
io
i
R1
RTh
+
+
+
R2
vs
=
vo
(a)
vo
(b)
Figure 3.45: Two-resistor voltage divider circuit.
3.7 Simulation codes
komt nog.
79
Intentionally left blank.
80
Chapter
4
Amplifier Models, Two-ports, and Nullor
As we discussed before electronic amplifiers amplify the signal power presented at their input. In order
to be able to provide power gain, electronic amplifiers are active. This means that they have (or need) an
external energy source that can be harnessed to produce an output having more power than the input, see
Figure 4.1.
Amplifier
energy source
+
ii
vi
pi=iivi
amplification
-
io
+
po=iovo
vo
-
Figure 4.1: Electronic amplifier: the signal power at the input is found amplified at the output.
Only amplifying an ac voltage signal or an ac current signal can be done with the aid of a transformer.
For example, an alternating voltage signal applied to the input of a step up transformer may generate a
larger voltage signal at the output. Although the voltage at the output is increased, the ability of the output
to provide current to an external load is reduced. The power supplied to the load is always less (or equal
in case of an ideal transformer) than the power absorbed at the input. Thus, a transformer may provide
voltage amplification but it cannot provide power amplification. The transformer is an example of a passive
amplifier. That is, the amplifier has no external energy source other than its signal inputs. Unless indicated
differently, for the remainder of this text, when we use the term amplifier we will be referring to an active
electronic amplifier.
Amplifiers are described with two-ports. We are only interested in the relation between the input current
and input voltage on the one side and the output current and the output voltage on the other side. After
introducing general amplifier two-ports, we will examine some properties of two-ports and will investigate
the two-port description of the two-port: nullor.
81
4.1 Modeling of amplifiers
Active electronic amplifiers may be of different types. The signal at the input may be represented by a
Thévenin equivalent circuit or by a Norton equivalent circuit, see Figure 4.2. The power gain in an amplifier
can be modeled by means of a controlled energy source—that is, by an ideal voltage source in series with
an internal resistor, or by an ideal current source in parallel with an internal resistor.
Rs
ii
+
vi
_
+
vs
il
io
+
+
Amplifier
Rl
vo
_
vl
_
(a)
is
ii
+
vi
Rs
io
il
+
+
Amplifier
Rl
vo
_
_
vl
_
(b)
Figure 4.2: Signal source as (a) Thévenin equivalent or (b) Norton equivalent of an amplifier system with a
load, in case of frequency independent amplification.
4.1.1
Sources and loads
Within electrical systems energy sources generate currents and voltages in the circuits. Figure 4.3(a) shows
a simple electronic system with a single source and a single load. In this figure, the source of energy is
some form of sensor and the load is some form of actuator. In other arrangement, like in Figure 4.3(b) the
source and load could be other electronic circuits. In both cases, the source is linked to the (sub)system by
input circuitry and the load is connected by an output circuit.
The division of a system in subsystems is called partitioning. It can be seen that the output of each
subsystem represents a source for the next, while the input represents a load for the previous.
4.1.2
Equivalent circuits for the amplifier
Real electronic amplifiers are complicated circuits, however we can model them with relatively simple
equivalent circuits for most purposes. To model the characteristics of the input of an amplifier we need to
describe the way it appears to other circuits that are connected to it, that is we have to model the amplifier as
a load of another circuit. Fortunately, the input circuitry in an amplifier can be modeled as a single resistor
in most cases.
Now, we can distinguish four different types of amplifier circuits. We start discussing the first two types:
the voltage amplifier and the transconductance amplifier, both shown in Figure 4.4.
In both amplifier circuits the sensor input is represented by its Thévenin equivalent, that is, the input is
a signal source formed by an ideal voltage source with value vs = vs (t) in series with a resistor Rs .
The amplifier input circuitry is represented by an input resistor Ri ; the amplifier output is an output
signal source represented by either a controlled voltage source or a controlled current source providing the
amplified signal energy to excite the load of the circuitry of the amplifier represented by Rl .
82
source
load
Electronic system
input
circuit
output
circuit
(a)
source
load
source
subsystem
load
source
subsystem
load
source
load
subsystem
(b)
Figure 4.3: (a) Source, load, input circuit, output circuit, and electronic system; (b) partitioning of the
system.
If the source is represented by a Thévenin equivalent and the amplifier output by a controlled voltage source the amplifier is called a voltage amplifier, see Figure 4.4(a). If the source is represented by
a Thévenin equivalent and the amplifier output by a controlled current source the amplifier is called a
transconductance amplifier, see Figure 4.4(b).
The voltage amplifier and transconductance amplifier are useful when the internal resistor of the signal
source Rs is nonlinear, temperature dependent or unstable in general, and its value is small compared to
the internal resistance of the amplifier Ri . Due to the potential divider formed by Rs and Ri , the resulting
voltage vi at the input of the amplifier is given by:
vi =
Ri
vs .
Rs + Ri
(4.1)
The effects of a varying value of Rs is minimized if Ri Rs , in which case vi reduces to
vi = vs ,
(4.2)
which is independent of the actual value of Rs .
In these cases the choice between a voltage or transconductance amplifier can be made by considering
the nature of the load.
If Rl is nonlinear, temperature dependent or unstable and Rl Ro , a voltage amplifier is preferable;
the voltage across the load resistor
vl =
Rl
vo
Ro + Rl
(4.3)
will in this case reduce to
vl = vo ,
(4.4)
which is independent of the actual value of Rl .
83
Rs
Ro
vi
vs
v o= μ v i
Ri
Rl
vl
(a)
il
Rs
vs
vi
Ri
io= gm·vi
Ro
Rl
(b)
Figure 4.4: (a) voltage amplifier; (b) transconductance amplifier
If Rl is nonlinear, temperature dependent or unstable and Rl Ro , a transconductance amplifier is
preferable: the current through the load resistor
il =
Ro
io
Rl + Ro
(4.5)
will in this case reduce to
i l = io ,
(4.6)
which is independent of the actual value of Rl .
Next, we discuss the other two types: the transresistance amplifier and the current amplifier, both shown
in Figure 4.5.
Now, in both amplifier circuits the sensor input is represented by its Norton equivalent, that is, the input
is a signal source formed by an ideal current source with value is = is (t) in parallel with the source resistor
Rs .
The amplifier input circuitry is represented by an input resistor Ri ; the amplifier output is an output
signal source represented by either a controlled voltage source or a controlled current source providing the
amplified signal energy to excite the load of the circuitry of the amplifier represented by Rl .
If the source is represented by a Norton equivalent and the amplifier output by a controlled voltage
source the amplifier is called a transresistance amplifier, see Figure 4.5(a). If the source is represented by
a Norton equivalent and the amplifier output by a controlled current source the amplifier is called a current
amplifier, see Figure 4.5(b).
The transresistance amplifier and current amplifier are useful when the internal resistor of the signal
source Rs is nonlinear, temperature dependent or unstable in general, and its value is large compared to the
84
ii
is
Rs
Ro
v o=rmii
Ri
vl
Rl
(a)
ii
is
Rs
il
Ri
io=β i i
Ro
Rl
(b)
Figure 4.5: (a) transresistance amplifier; (b) current amplifier
internal resistance of the amplifier Ri . Due to the current divider formed by Rs and Ri , the resulting current
ii at the input of the amplifier is given by:
ii =
Rs
is .
Ri + Rs
(4.7)
The effects of a varying value of Rs is minimized if Rs Ri , in which case ii reduces to
ii = i s ,
(4.8)
which is independent of the actual value of Rs .
In these cases the choice between a transresistance or current amplifier can be made by considering the
nature of the load.
If Rl is nonlinear, temperature dependent or unstable and Rl Ro , a transresistance amplifier is
preferable; the voltage across the load resistor
vl =
Rl
vo
Ro + Rl
(4.9)
will in this case reduce to
vl = vo ,
(4.10)
which is independent of the actual value of Rl .
If Rl is nonlinear, temperature dependent or unstable and Rl Ro , a current amplifier is preferable:
the current through the load resistor
il =
Ro
io
Rl + Ro
(4.11)
will in this case reduce to
il = io ,
(4.12)
85
which is independent of the actual value of Rl .
The choice of amplifier type has been based on the instabilities of Rs and Rl . If Rs and Rl are stable and
do not vary any type of amplifier can be used in principle. A choice, however, will then still be useful if
based on other criteria such as noise, bandwidth , or signal-to-noise ratio. Sometimes also the Ri and/or Ro
of the amplifier must have precise values, for example for transmission of high frequency signals through
cables.
4.1.3
Open-circuit voltage gain and loading
As we saw, all four types of amplifiers show two voltage or current divider subcircuits. These dividers give
rise to reduction of possible gain in these amplifiers, termed loading.
As an illustration we consider a voltage amplifier having a voltage gain μ of 10, an input resistor of
1 kΩ and an output resistor of 10 Ω. The amplifier is connected to a sensor represented by a voltage source
of 2 V and an internal resistor of 100 Ω, and a load resistor of 50 Ω, see Figure 4.6.
Source
Load
Amplifier
Rs = 100 Ω
Ro = 10 Ω
ii
+
Vs = 2 V
vi
io
+
Ri =
1 kΩ
μvi
vo
Rl =
50 Ω
_
_
Figure 4.6: loading in a voltage amplifier
We calculate the voltage gain vvoi of the circuit and the power gain ppoi of the circuit. The input voltage is
divided by Rs and Ri :
vi =
Ri
1kΩ
Vs =
2V = 1.82V.
Rs + Ri
100Ω + 1kΩ
(4.13)
This voltage is amplified by μ and divided by Ro and Rl . For vo at the output of the amplifier we obtain:
vo =
50Ω
Rl
10vi = 8.33vi = 15.2V,
μvi =
Ro + Rl
10Ω + 50Ω
(4.14)
and we find that the voltage gain of the amplifier circuit vvoi is reduced to
vo
15.2V
=
= 8.35
vi
1.82V
(4.15)
We see that when an amplifier is connected to a source and a load, the resulting output voltage may be
considerably less than one might have expected given the source voltage and voltage gain of the amplifier
in isolation. The input voltage is less due to the internal resistance of the source. Similarly the output
voltage is reduced due to the combination with the load resistor. This effect is known as loading, The lower
the resistance that is applied across the output of a circuit the more heavily it is loaded as more current is
drawn from it.
From the above example it is clear that the gain of the amplifier circuit is greatly affected by the circuits
that are connected to it. For this reason the voltage gain of an isolated amplifier μ is called open-circuit
voltage gain or unloaded voltage gain.
86
Now, we consider the power gain in this example. The power into the amplifier (input power) is
p i = ii v i =
vi2
1.82V2
= 3.3mW.
=
Ri
1kΩ
(4.16)
The power out of the amplifier (output power) is
po = i o v o =
vo2
15.2V2
= 4.62W.
=
Rl
50Ω
(4.17)
The power gain in this example is given by
power gain (Ap ) =
4.26W
po
=
= 1400.
pi
3.3mW
(4.18)
Of course the energy to obtain the power gain is supplied by the controlled voltage source.
Notice that even a circuit with a relative low voltage gain can have a relatively high power gain! The
power gain of modern electronic amplifiers may be very high— gains of 106 or 108 being common. With
these large numbers it is more useful to use logarithmic expressions. For this reason power gain is normally
expressed in decibel (dB) defined by:
power gain (dB) 10 log10
po
,
pi
(4.19)
where po is the output power and pi is the input power.
4.1.4
Ideal voltage amplifier
An ideal voltage amplifier is not affected by the circuits connected to it. It would draw no current from
the source and the amplification is independent of the load. This requires the input conductance and the
output resistance of the amplifier to be zero. Under these conditions the unwanted effects of the two voltage
dividers are removed and the voltage gain of the amplification circuit equals the unloaded gain. Figure 4.7
shows the previous circuit with an ideal voltage amplifier.
Source
Load
Amplifier
Rs = 100 Ω
Ro = 0
ii
+
vi
Vs = 2 V
io
+
Gi = 0
μvi
vo
Rl =
50 Ω
_
_
Figure 4.7: Amplifier circuit with an ideal voltage amplifier
It should be noted that although the voltage amplification of the circuit is equal to the unloaded voltage
gain, the amplifier is not able to amplify the input power because the input current is zero. The ideal voltage
amplifier alone is not capable of signal power amplification.
4.2 Two-ports
Figure 4.8 shows a general two-port subcircuit.
87
i2
i1
+
+
v1
v2
-
i1
i2
-
Figure 4.8: General two-port subcircuit.
The two terminals on the left are frequently called the input port, input terminals or simply input; the
terminals on the right the output port or just output. Often the two-port consists of only resistors in a
particular configuration. Then the two-port is called passive. In addition, often a signal source is connected
to the input, represented by its Norton or Thévenin equivalent circuit, while the output is driving a load
resistor.
Two-port theory is an extensive chapter in the network theory. We will only deal with some basics. Two
terminals of a subnetwork together form a port, if the current, entering at the network at one terminal has
the same value as the current, leaving the network at the other terminal. This is called the port constraint. A
two-port consists of two ports. If the two-port does not contain independent sources and all initial conditions
of the dynamic elements are zero (for example, all capacities are initially discharged), then the two-port can
be described by matrices that equate the port currents i1 and i2 and the port voltages v1 and v2 .
4.2.1
The two-port admittance matrix
We start considering the description with with the help of the admittance matrix, or Y-matrix with as entries
the admittance parameters, or y-parameters.
The two-port is described by:
i1 = y11 v1 + y12 v2
i2 = y21 v1 + y22 v2
(4.20)
in matrix notation:
i1
y y
v1
= 11 12
i2
y21 y22
v2
or I = YV with
i
I= 1 ,
i2
V=
v1
,
v2
(4.21)
Y=
y11 y12
y21 y22
(4.22)
Now there are two methods to determine the matrix entries.
1. find the equations for i1 en i2 directly from the subcircuit, or
2. find
i1 ,
y11 =
v1 v2 =0
i2 ,
y21 =
v1 v2 =0
i1 y12 =
v2 v1 =0
i2 y22 =
v2 v1 =0
(4.23)
88
A few remarks must be made. First of all, in case of a purely resistive two-port the admittance matrix
is called the conductance matrix G. Secondly, if we determine the y-parameters according to the second
method then the requirements v2 = 0, or v1 = 0 indicate, respectively, a short circuit at the output or just
a short circuit at the input of the two-port. For this reason the y-parameters are also called short-circuit
admittance parameters. Third, in all fractions the denominator may not be zero unless the numerator is zero
too; these cases have to be considered separately. Typically, problems do not occur in these cases.
An example
Determine the G-matrix of the two-port in Figure 4.9.
i2
i1
+
+
R
v2
v1
i1
-
i2
-
Figure 4.9: Two-port example to determine its G-matrix.
We find for i1 and i2
i1 = R1 (v1 − v2 ) = R1 v1 − R1 v2
i2 = −i1 = − R1 v1 + R1 v2
(4.24)
i1 1
⇒ y11 =
v1 v2 =0
R
(4.25)
or
y11 =
i1 1
⇒ y12 = −
v2 v1 =0
R
i2 1
y21 =
⇒ y21 = −
v1 v2 =0
R
i2 1
y22 =
⇒ y22 =
v2 v1 =0
R
y12 =
so
G=
1
R
− R1
− R1
1
R
=
1
R
1 −1
−1 1
(4.26)
(4.27)
(4.28)
(4.29)
Separate analysis of v1 = v2 = 0 provides directly that there will be no current through the resistor,
consequently i1 = −i2 = 0 which is in line with the description with the G-matrix.
4.2.2
The two-port impedance matrix
A similar description of a two-port with an impedance matrix or Z-matrix is possible. In this case we will
find as entries in the marix the so-called impedance parameters, or z-parameters. However in this case the
equations are given as:
89
v1 = z11 i1 + z12 i2
v2 = z21 i1 + z22 i2
(4.30)
in matrix notation:
v1
z z
i1
= 11 12
v2
z21 z22
i2
or V = ZI with
i
V= 1 ,
i2
I=
v1
,
v2
(4.31)
Z=
z11 z12
z21 z22
(4.32)
To determine the matrix entries we have two methods again.
1. find the equations for v1 en v2 directly from the subcircuit, or
2. find
v1 ,
z11 =
i1 i2 =0
v2 z21 =
,
i1 i2 =0
v1 z12 =
i2 i1 =0
v2 z22 =
i2 i1 =0
(4.33)
Again, some remarks. First of all, in case of only resistors in the two-port the impedance matrix is
called resistance matrix R. Secondly, if we determine the z-parameters according to the second method
then the requirements i2 = 0, of i1 = 0 indicate, respectively, an open circuit at the output or just an open
circuit at the input of the two-port. For this reason the z-parameters are also called open-circuit impedance
parameters.
An example
Determine the R-matrix of the two-port in Figure 4.10.
i2
i1
+
+
R
v1
-
i1
v2
i2
-
Figure 4.10: Two-port example to determine its R-matrix.
Looking at the two-port we find for v1 en v2
v1 = R(i1 + i2 ) = Ri1 + Ri2
v2 = v1 = Ri1 + Ri2
(4.34)
v1 ⇒ z11 = R
i1 i2 =0
(4.35)
or
z11 =
90
v1 ⇒ z12 = R
z12 =
i2 i1 =0
v2 z21 =
⇒ z21 = R
i1 i2 =0
v2 ⇒ z22 = R
z22 =
i2 i1 =0
such that
R=
R R
R R
11
=R
11
(4.36)
(4.37)
(4.38)
(4.39)
Analysis of the circuit if both i1 and i2 are zero tells us that in this case no voltage across the resistor
exists; a result which fits within the general solution.
Note that the determinant of R is zero; and an Y matrix of the two-port does not exist.
More matrices for describing two-ports exist. For example the hybrid H matrix and hybrid G matrix. For
this course, those matrices are not important; however for sake of completeness we give their definitions.
The hybrid H matrix and hybrid G matrix are defined as:
i
v1
=H 1
(4.40)
i2
v2
with
H=
and
i1
v2
with
4.2.3
(4.41)
v
=G 1
i2
G=
h11 h12
h21 h22
g11 g12
g21 g22
(4.42)
(4.43)
The two-port transmission or chain matrix
Of major importance, for the synthesis and analysis of the amplifier circuits, are the transmission or chain
matrices. First, we consider the definition of the transmission matrix as is done in most books on linear
circuit theory. The equations for the two-port are given as:
v1 = t11 v2 − t12 i2
i1 = t21 v2 − t22 i2
(4.44)
LOOK CAREFULLY AT THE MINUS-SIGN(S)!
in matrix notation:
v1
t11 t12
v2
=
i1
t21 t22
−i2
(4.45)
To determine the matrix entries we use the standard method:
v1 t11 =
,
v2 i2 =0
i1 t21 =
,
v2 i2 =0
v1 t12 =
−i2 v2 =0
i1 t22 =
−i2 v2 =0
(4.46)
As we notice the transmisssion matrix is defined with (-i2 ). Of course we could have taken (+i2 ) if we
had reversed the direction of the output current.
91
Chain matrix
To avoid confusions we define in electronics the so-called chain matrix, which has its own corresponding
two-port representation, see Figure 4.11. To distinguish between the two representations we shall change
the input and output parameters from numbers to in and out; that is instead of v1 , v2 , i1 en i2 we are going
to use vi , vo , ii en io .
io
ii
+
+
K=
vi
(
k11
k21
k12
k22
ii
-
)
vo
io
-
Figure 4.11: General two-port for use with the chain matrix description.
Note the direction of the currents! And the absence of a minus-sign in the output vector!
vi = k11 vo + k12 io
ii = k21 vo + k22 io
(4.47)
in matrix notation:
vi
k11 k12
vo
=
ii
k21 k22
io
(4.48)
The chain parameters are defined as follows:
vi k11 =
,
vo io =0
vi k12 = io vo =0
ii ,
k21 =
vo io =0
ii k22 = io vo =0
(4.49)
We see that, in the fractions to determine the chain parameters we require that either the current through
the output or the voltage across the output is equal to zero. We can often find the parameters or by considering an open circuit or just a short circuit.
Example 1
Determine the chain matrix of the two-port in figure 4.12.
We have
or
vi = vo + Rio
i i = io
(4.50)
vi k11 =
⇒ k11 = 1
vo io =0
(4.51)
vi ⇒ k12 = R
k12 = io vo =0
(4.52)
92
io
ii
+
+
R
vo
vi
-
ii
io
-
Figure 4.12: Two-port example to determine its chain matrix.
ii ⇒ k21 = 0
vo io =0
ii k22 = ⇒ k22 = 1
io vo =0
k21 =
(4.53)
(4.54)
such that
1R
K=
0 1
(4.55)
The analysis of the subcircuit when both io and vo are zero learns us that there is no voltage across the
resistor because of io = ii = 0, consequently: vi = v0 = 0. If at the output the current i is zero and the
voltage v is zero then there is no current or voltage on the input. These results fit within the description if
necessary.
Example 2
Determine the chain matrix of the two-port in Figure 4.13.
io
ii
+
+
R
vi
-
ii
vo
io
-
Figure 4.13: Two-port example to determine its chain matrix.
We find
vi = vo
ii − io = R−1 vo
or
(4.56)
vi ⇒ k11 = 1
k11 =
vo io =0
vi ⇒ k12 = 0
k12 = io vo =0
(4.57)
(4.58)
93
ii ⇒ k21 = R−1
k21 =
vo io =0
ii ⇒ k22 = 1
k22 = io vo =0
so
K=
4.2.4
1
0
R−1 1
(4.59)
(4.60)
(4.61)
Input and output resistances using the chain matrix
By considering the description of amplifier with the chain matrix we can derive expressions for the input
and output resistances of the circuit. Because the chain matrix relates output signals to input signals we
expect that the load resistor will appear in the expression of the input resistance, and that the source resistor
will appear in the expression for the output resistance. This can be seen by considering the realistic circuit
in Figure 4.20:
Rs
io
ii
+
+
vs
K=
A B
C D
( )
vi
_
il
+
vo
_
+
Rl
vl
_
Figure 4.14: Realistic amplifier circuit described by a chain matrix.
Starting from
vi = Avo + Bio
ii = Cvo + Dio
and using vo = Rl io we obtain:
vi = (ARl + B)io
ii = (CRl + D)io
from which follows by division that:
Ri =
vi
ARl + B
=
ii
CRl + D
and
Gi =
CRl + D
.
ARl + B
(4.62)
The calculation of Ro is a bit more involved. For the calculation of the output resistance we set the source
voltage equal to zero and use vi = −Rs ii . With the use of linear algebra, we write:
vo
io
=
A B
C D
−1 vi
,
ii
with
A B
C D
−1
=
1
AD − BC
D −B
−C A
94
Writing down the equations:
(AD − BC)vo = Dvi − Bii
(AD − BC)io = −Cvi + Aii
we obtain for the output resistance, using vi = −Rs ii :
Ro =
4.2.5
vo
DRs + B
=
−io
CRs + A
and
Go =
CRs + A
.
DRs + B
(4.63)
Combining two-ports
The importance of the description of a subcircuit with the aid of one of the two-port matrices comes from
how the different two-port matrices may be combined. For a parallel connection of admittance matrices
the new two-port admittance matrix is described by the sum of the individual admittance matrices; for a
series connection of impedance matrices, the new two-port impedance matrix is described by the sum of
the individual impedance matrices; and for a cascade of chain matrices the new two-port chain matrix is
described by the product of the individual chain matrices. We will investigate the above allegations.
YA
ZA
YB
ZB
NY
NK
NZ
(a)
KB
KA
(b)
(c)
Figure 4.15: Typical interconnections of two-ports, (a) parallel, (b) series, (c) cascade
Figure 4.15 shows three typical interconnection structures: the parallel, the series, and the cascade
interconnection.
Parallel interconnection of two-ports described by an admittance matrix
For the parallel interconnection of Figure 4.15(a), the network NY , the new admittance matrix is
Y = YA + YB
(4.64)
A derivation of this formula is straightforward.
i1
i2
i1A
i
+ 1B =
i2A
i2B
v
v
= YA 1A + YB 1B .
v2B
v2A
=
(4.65)
(4.66)
In the parallel combination the voltages at the input and the voltages at the output are the same, v1 =
v1A = v1B and v2 = v2A = v2B , from which we conclude:
v
i1
= (YA + YB ) 1 ,
(4.67)
i2
v2
and thus
Y = YA + YB .
(4.68)
95
Series interconnection of two-ports described by an impedance matrix
A similar reasoning leads to the expression for the sum of the impedance matrices in case of a serial
interconnection of two-ports:
v1
v2
v1A
v
+ 1B =
v2A
v2B
i
i
= ZA 1A + ZB 1B .
i2A
i2B
=
(4.69)
(4.70)
In the series combination the currents at the input and the currents at the output are the same, i1 =
i1A = i1B and i2 = i2A = i2B , from which we conclude:
v1
i
= (ZA + ZB ) 1 ,
(4.71)
v2
i2
and thus
Z = ZA + ZB .
(4.72)
Cascade connection of two-ports described by a chain matrix
Finally, from the definition of the chain parameters:
and
viA
iiA
viB
iiB
= KA
= KB
voA
ioA
voB
ioB
(4.73)
(4.74)
and knowing that viA = vi and iiA = ii ; voA = viB and ioA = iiB ; voB = vo and ioB = io we obtain:
vi
ii
= KA · KB
vo
io
(4.75)
and thus
K = KA · KB .
(4.76)
4.3 Transactors and nullor
An intimate relation exists between the nullor two-port element and ideal amplifiers with an infinite amplification factor. We will investigate this relation starting with a definition of the nullor using chain parameters.
4.3.1
Chain matrix of the nullor
If we carefully look at the definition of the chain parameters:
vi = k11 vo + k12 io
ii = k21 vo + k22 io ,
96
we see that if all the matrix entries are zero, that is, if
00
K=
00
then the output can take any desired current and voltage, like the norator part of the nullor two-port element,
while at the input both the input current and the input voltage will be zero (like the nullator part of the nullor
two-port element). We conclude that a zero K matrix describes the nullor two-port.
Now, to come to an interpretation of the nullor we are going to describe the K matrices of ideal amplifiers, the so-called transactors.
4.3.2
Transactors: ideal amplifiers
There are four transactors: the voltage-to-voltage transactor (vvt), the voltage-to-current transactor (vit), the
current-to-voltage transactor (ivt), and the current-to-current transactor (iit). All transactors are described
by two-ports consisting of a controlled energy source and its control unit. They describe ideal amplifiers or
ideal converters.
current-to-current transactor (iit)
Basically, this transactor is the two-port subcircuit representing an ideal current controlled current source
with its control current made explicitly, see Figure 4.16.
io
ii
+
+
β ii
vi
-
ii
vo
io
-
Figure 4.16: current-to-current transactor (iit)
At the input holds:
vi = 0
(4.77)
The input cannot simply be connected to a voltage source. This is a consequence of the properties of ideal
voltage sources. When a resistor is excited by an ideal voltage source, the current through the resistor is
determined by Ohm’s law: I = V /R. From a mathematical point of view, if we try to short an ideal voltage
source—the short can be modeled by a resistor with value 0Ω (see Section 2.2.1)—the current in the zero
valued resistor cannot be defined; R = 0 is excluded in the description of a resistor excited by an ideal
voltage source. The above argument leads to an exclusion rule in circuit theory: an ideal voltage source
cannot be shorted. The same reasoning with an ideal current source and the conductance of a resistor, leads
to another exclusion rule based on the excitation of a resistor with G = 0: an ideal current source may not
be connected to an open-circuit. According to this last exclusion rule the output of the current-to-current
transactor must be loaded.
We continue with the description of the current-to-current transactor: the output current is β times the
input current
io = βii ,
(4.78)
β is called the unloaded current gain of the transactor.
When ii en β are unequal to zero then io will be nonzero. As soon as a nonzero load, Rl = 0, is
connected, the voltage at the output vo will be nonzero, that is: vo = 0.
97
Under these conditions, the equations describing the transactor will be:
vi = 0vo + 0io
,
ii = 0vo + β −1 io
in matrix notation:
0 0
vo
vi
=
,
ii
0 β −1
io
(4.79)
And, we find the chain matrix of the current-to-current transactor to be:
0 0
.
Kiit =
0 β −1
(4.80)
The current ii and the voltage vo are determined by the circuit, which includes the transactor. We note
that the input resistance of the current-to-current transactor is zero (Rin = 0), and the output conductance
is zero (Gout = 0) too.
voltage-to-voltage transactor (vvt)
This transactor is the two-port subcircuit representing an ideal voltage controlled voltage source with its
control voltage made explicitly, see Figure 4.17.
io
ii
+
+
μvi
vi
-
+
vo
_
ii
io
-
Figure 4.17: voltage-to-voltage transactor (vvt)
At the input holds:
ii = 0
(4.81)
The input cannot simply be connected to an ideal current source.
For the voltage controlled voltage source holds:
vo = μvi ,
(4.82)
μ is called the unloaded voltage gain of the transactor.
Now, when vi en μ are unequal to zero then the output voltage will be nonzero. The controlled ideal
voltage source may not be shorted. As soon as a nonzero load, Rl = 0, is connected, the current at the
output io will be nonzero, io = 0.
Under these conditions, the equations describing the transactor will be:
vi = μ−1 vo + 0io
,
ii = 0vo + 0io
In matrix notation:
−1 vo
μ 0
vi
=
,
ii
io
0 0
(4.83)
98
And, we find the chain matrix of the voltage-to-voltage transactor to be:
−1 μ 0
.
Kvvt =
0 0
(4.84)
The voltage vi and the current io are determined by the circuit, which includes the transactor. We note
that the input conductance of the voltage-to-voltage transactor is zero (Gin = 0), and the output resistance
is zero (Rout = 0) too.
voltage-to-current transactor (vit)
The transactor is the two-port subcircuit representing an ideal voltage controlled current source with its
control voltage made explicitly, see Figure 4.18.
io
ii
+
+
gmvi
vi
-
ii
vo
io
-
Figure 4.18: voltage-to-current transactor (vit)
At the input holds:
ii = 0
(4.85)
For the voltage controlled current source holds:
io = g m v i ,
(4.86)
gm is the unloaded transconductance.
When vi en gm are unequal to zero the output current will be nonzero. The output may not be unloaded.
As soon as a nonzero load, Rl = 0, is connected, the voltage at the output vo will be nonzero, vo = 0.
Under these conditions, the equations describing the transactor will be:
−1
io
vi = 0vo + gm
,
ii = 0vo + 0io
In matrix notation:
−1
vi
0 gm
vo
=
,
ii
0 0
io
(4.87)
And, we find the chain matrix of the voltage-to-current transactor to be:
−1
0 gm
Kvit =
,
0 0
(4.88)
gm has a dimension of (resistance)−1 . The voltage vi and the voltage vo are determined by the circuit,
which includes the transactor. We note that the input conductance of the voltage-to-current transactor is
zero (Gin = 0), and the output conductance is zero (Gout = 0) too. And finally:
99
io
ii
+
+
rmii
vi
-
+
vo
_
ii
io
-
Figure 4.19: current-to-voltage transactor (ivt)
current-to-voltage transactor (ivt)
The transactor is the two-port subcircuit representing an ideal current controlled voltage source with its
control current made explicitly, see Figure 4.19.
At the input holds:
vi = 0
(4.89)
for the current controlled voltage source holds:
vo = rm ii ,
(4.90)
rm is called the unloaded transresistance of the transactor.
When ii en rm are unequal to zero, the output voltage is nonzero too. As soon as a nonzero load, Rl = 0,
is connected, then the current at the output io will be nonzero, io = 0. Again, the controlled ideal voltage
source may not be shorted.
Under these conditions, the equations describing the transactor will be:
vi = 0vo + 0io
,
−1
ii = rm
vo + 0io
In matrix notation
vo
0 0
vi
= −1
,
ii
io
rm 0
(4.91)
And, we find the chain matrix of the current-to-voltage transactor to be:
0 0
Kivt = −1
,
rm 0
(4.92)
rm has a dimension of resistance. The current ii and the current io are determined by the circuit, which
includes the transactor. We note that the input resistance of the current-to-voltage transactor is zero (Rin =
0), and the output resistance is zero (Rout = 0) too.
The transactors are active two-ports: the power at the input is always zero (the voltage or current is
zero), while loaded at the output, the power leaving the two-port is nonzero. A source of energy has to be
included in the implementation of the transactor.
Now, we can describe the nullor element in terms of transactors. In view of the discussion of the various
transactors a chain matrix with only zero entries can be considered to be (1) a current-to-current transactor
with an infinitely large unloaded current gain, (2) a voltage-to-voltage transactor with an infinitely large
unloaded voltage gain, (3) a voltage-to-current transactor with an infinite unloaded transconductance, or (4)
a current-to-voltage transactor with an infinitely large unloaded transresistance.
100
4.3.3
*Example: proof of vi = 0 and ii = 0 for the voltage-to-voltage transactor
with μ → ∞
A possible implementation of the nullor is the voltage-to-voltage transactor with μ = ∞. For this implementation we check the nullator condition.
The voltage-to-voltage transactor is defined by a two-port having:
• Gi = 0,
• Ro = 0,
• μ→∞
For the voltage-to-voltage transactor the chain matrix is defined as:
−1 μ 0
.
Kvvt =
0 0
(4.93)
With μ → ∞ the chain matrix becomes
00
K=
.
00
Zs
(4.94)
ii
+
+
vs
vi
_
io
K=
A B
C D
( )
il
+
vo
_
+
Zl
vl
_
Figure 4.20: Realistic voltage-to-voltage amplifier circuit described by a chain matrix.
For this implementation of the two-port element nullor the voltage transfer vl /vs , the voltage gain,
cannot be defined.
If we denote for the matrix entries: A, B, C, and D, then we have:
vi = Avo + Bio
(4.95)
ii = Cvo + Dio
(4.96)
At the output holds:
io =
vo
Zl
en
vo = vl
en
io = il
(4.97)
what, combined with Equation 4.95 results in:
vi = Avl +
B
io
Zl
(4.98)
At the input port holds (KVL):
v s = ii Z s + v i
(4.99)
with Equation 4.96 leads to:
ii Zs = CZs vl + D
Zs
vl = vs − vi
Zl
(4.100)
101
Equation 4.100 in Equation 4.98 gives:
vs − Avl −
B
Zs
vl = CZs vl + D vl ,
Zl
Zl
and leads to:
vs = (A +
B
Zs
+ CZs + D )vl
Zl
Zl
and finally to
vl
B
Zs
= (A +
+ CZs + D )−1 .
vs
Zl
Zl
(4.101)
Filling in A = B = C = D = 0 gives
vl
1
=
vs
0
and we have to conclude that vl /vs cannot be defined.
However, we do know that: (we rewrite Equation 4.95 and Equation 4.96)
vi = Avl + Bil
ii = Cvl + Dil
With A = B = C = D = 0, this immediately leads to the criterion that the nullator condition must hold:
vi = 0
and
ii = 0
(4.102)
as requirement for a voltage-to-voltage transactor with μ → ∞. Similarly, it can also be shown that the
nullator condition holds for the other transactors as a result of the infinity of the unloaded transconductance,
or the unloaded transresistance, or unloaded current gain.
4.4 Amplifier configurations with the nullor, revised
In Chapter 3 we derived different circuits that include a nullor. We have seen that the addition of current
signals can be done actively using a so-called current follower, voltage signals can be accurately reproduced
actively with a so-called voltage follower, voltage signals can be multiplied by a constant with the aid of
an active voltage amplifier, current signals can be multiplied by a constant with the aid of an active current
amplifier, current signals can be converted actively into voltage signals, and voltage signals can be converted
actively into and current signals; all these circuits included a nullor.
If we consider the current follower as a special case of the current amplifier and the voltage follower
as a special case of the voltage amplifier then we can view all of these circuits as circuits where the input
(a real voltage source or real current source) is offered on two parallel or two serial two-ports. For the
output something similar holds. We will now give a description with the aid of two two-ports. This is
more general than the description in Chapter 3, and we can win something with such a description. A
general description will enable us to associate the nullor circuits with the ideal amplifiers, the transactors.
Using such a description, we will able to make a judgment about the appropriateness of the many different
amplifier circuits. The description is therefore essential to come to a balanced judgment prior to the selection
of an amplifier circuit, and it is therefore important for the synthesis of circuits.
102
ii
_
+
io
+
+
Rs
+
vo
vi
vs
R2
_
R1
Rl
_
Figure 4.21: Active voltage multiplication, from Section 3.3.2
ii
+
in=0
vn=0
K=
0 0
0 0
( )
Rs
+
+
io
+
vo
vi
vs
_
_
Z2
Z1
Rl
_
Figure 4.22: Voltage-voltage converter, or voltage amplifier.
4.4.1
The voltage-voltage converter or voltage amplifier
In Section 3.3.2 we found the active voltage amplifier circuit including the nullor. Combined with an
realistic input and a resistive load and visualizing the two two-ports, the result is shown in Figure 4.21.
To come to a general description using the K matrix formalism, we consider the nullor as a black box
with the properties that it can provide any current and any voltage at its output and has the requirement
vn = 0 and in = 0 at its input. In addition we shall generalize the coupling resistances to impedances, the
transresistance (rm ) to transimpedance (Zm ), and transconductance (gm ) to transadmittance (Ym ).
Figure 4.22 shows the generalized circuit which converts a voltage to a voltage. We are going to proof
that this configuration behaves as a voltage-to-voltage transactor (vvt). We note that the two-port circuit is a
series interconnection of the two two-ports at the input, and a parallel interconnection of the two two-ports
at the output. The total two-port combination can be described by a new chain matrix:
vi = k11 vo + k12 io
ii = k21 vo + k22 io
For the nullor two-port we have vn = 0 and in = 0 (kij = 0 for all i and j). Using this, we see that we
can describe the total combination by the following equations:
ii = 0,
(4.103)
because ii = in = 0. And using KVL
vi =
Z2
vo ,
Z1 + Z2
(4.104)
103
because vn = 0.
We can compare these equations with the equations describing a voltage-to-voltage transactor (ii = 0
and vi = μ−1 vo ). When we assume that for the above two-port combination the same conditions hold—that
is, a load is always present and the input is always a Thévenin equivalent—then we come to the conclusion
that the voltage-voltage converter, or voltage amplifier, of Figure 4.22 is a voltage-to-voltage transactor with
a chain matrix of:
Z2
0
Z
+Z
1
2
K=
.
(4.105)
0
0
Z1
. And, correspondingly the
That is, the voltage gain, μ, of the two-port combination is μ = 1 + Z
2
voltage-to-voltage transactor, the two-port combination has a zero input admittance (Yin = 0), and the
output impedance is zero (Zout = 0).
4.4.2
The voltage-current converter or transadmittance amplifier
In Section 3.4.2 we found the nullor circuit for the voltage-current converter. Combined with realistic input
and a resistive load and drawn as two two-ports, the result is shown in Figure 4.23
ii
io
_
_
+
+
Rs
vo
vi
+
Rl
vs
iR
_
R
+
Figure 4.23: Active voltage-current converter, from Section 3.4.2
Figure 4.24 shows the generalized circuit which converts a voltage to a current. In the figure we have
generalized the coupling resistance to a impedance. We are going to prove that this two two-port configuration behaves as a voltage-to-current transactor (vit).
ii
+
in= 0
vn= 0
K=
0 0
0 0
( )
io
_
_
+
Rs
+
vo
vi
Rl
vs
_
iZ
Z
+
Figure 4.24: Voltage-current converter, or transadmittance amplifier.
104
We note that the two-port circuit is a series interconnection of the two two-ports at the input, and a series
interconnection of the two two-ports at the output. The total two-port combination is described by a new
chain matrix:
vi = k11 vo + k12 io
ii = k21 vo + k22 io
For the nullor two-port we have vn = 0 and in = 0. Using this, we see that we can describe the total
combination by the following equations:
ii = 0,
(4.106)
because ii = in = 0. And
vi = Zio ,
(4.107)
because vn = 0.
If we compare these equations with the equations describing a voltage-to-current transactor (ii = 0
−1
io ). Looking carefully at the polarity of the output voltage and the direction of the output
and vi = gm
current of the combination of the two-ports, we come to the conclusion that the voltage-current converter,
or transadmittance amplifier, of Figure 4.24 is a voltage-to-current transactor with a chain matrix of:
0 −Z
K=
.
(4.108)
0 0
That is, the transadmittance, Ym , of the two-port combination is Ym = − Z1 . And, correspondingly the
voltage-to-current transactor, the two-port combination has a zero input admittance (Yin = 0), and a zero
output admittance (Yout = 0).
4.4.3
The current-voltage converter or transimpedance amplifier
In Section 3.4.1 we found the nullor circuit for the current-voltage converter. Combined with realistic input
and a resistive load and drawn as two two-ports the result is shown in Figure 4.25
ii
io
_
_
+
+
+
is
Rs
_
vo
vi
Rl
R
_
+
Figure 4.25: Active current-voltage converter, from Section 3.4.1
Figures 4.26 (a) and (b) show the generalized circuit which converts a current to a voltage. In the figures
we have generalized the coupling resistance to a impedance. We are going to prove that this two two-port
configuration behaves as a current-to-voltage transactor (ivt).
We note that the two-port circuit is a parallel interconnection of the two two-ports at the input, and a
parallel interconnection of the two two-ports at the output. The total two-port combination is described by
a new chain matrix:
vi = k11 vo + k12 io
ii = k21 vo + k22 io
105
Z
ii
+
in= 0
vn= 0
K=
0 0
0 0
( )
ii
io
_
_
+
+
in= 0
vn= 0
K=
0 0
0 0
( )
io
_
_
+
Rs
is
Rs
vo
vi
Rl
is
vi
vo
_
+
Rl
Z1
_
+
(a)
(b)
Figure 4.26: (a): Current-voltage converter, or transimpedance amplifier, and (b) the current-voltage converter or transimpedance amplifier, redrawn.
For the nullor two-port we have vn = 0 and in = 0. Using this, we see that we can describe the total
combination by the following equations:
vi = 0,
(4.109)
because vi = vn = 0. And
ii = Z −1 vo ,
(4.110)
because in = 0.
If we compare these equations with the equations describing a current-to-voltage transactor (vi = 0
and vo = rm ii ). Looking carefully at the polarity of the output voltage and the direction of the output
current of the combination of the two-ports, we come to the conclusion that the current-voltage converter,
or transimpedance amplifier, of Figures 4.26 (a) and (b), is a current-to-voltage transactor with a chain
matrix of:
0 0
.
(4.111)
K=
−Z −1 0
That is, the transimpedance, Zm , of the two-port combination is Zm = −Z. And, correspondingly
the voltage-to-current transactor, the two-port combination has a zero input impedance (Zin = 0), and the
output impedance is zero (Zout = 0) too.
4.4.4
The current-current converter or current amplifier
Finally, we consider the current amplifier. In Section 3.3.3 we found the nullor circuit for the current-current
converter. Combined with realistic input and a resistive load and drawn as two two-ports the result is shown
in Figure 4.27
Figure 4.28 shows the generalized circuit which amplifies the current. In the figure we have generalized
the coupling resistances to impedances. We are going to prove that this two two-port configuration behaves
as a current-to-current transactor (iit).
We note that the two-port circuit is a parallel interconnection of the two two-ports at the input, and a
series interconnection of the two two-ports at the output. The total two-port combination is described by a
new chain matrix:
vi = k11 vo + k12 io
ii = k21 vo + k22 io
106
io
ii
+
+
is
_
+
_
+
Rs
vo
vi
R2
_
Rl
_
R1
Figure 4.27: Active current multiplication, from Section 3.3.3
ii
+
is
in= 0
vn= 0
K=
0 0
0 0
( )
io
+
_
+
Rs
vo
vi
_
Z2
Z1
Rl
_
Figure 4.28: Current-current converter, or current amplifier.
For the nullor two-port we have vn = 0 and in = 0. Using this, we see that we can describe the total
combination by the following equations:
vi = 0,
(4.112)
because vi = vn = 0. And
ii =
Z1
io ,
Z1 + Z2
(4.113)
because vi = 0 and both impedances share a common node (voltage).
We can compare these equations with the equations describing a current-to-current transactor (vi = 0
and ii = β −1 io ). When we assume that for the above two-port combination the same conditions hold—that
is, a load is always present and the input is always a Norton equivalent—then we come to the conclusion
that the current-current converter, or current amplifier, of Figure 4.28 is a current-to-current transactor with
a chain matrix of:
0
0
.
(4.114)
K=
1
0 Z1Z+Z
2
Z2
. And, correspondingly the
That is, the current gain, β, of the two-port combination is β = 1 + Z
1
current-to-current transactor, the two-port combination has a zero input impedance (Zin = 0), and a zero
output admittance (Yout = 0).
4.5 Problems
(Uit de inleiding van het boek Structured Electronic Design, wordt gebruikt bij 3de jaars vak in de elektronica)
107
Figure 4.29 shows a resistive two-port.
+
+
R
-
-
Figure 4.29: Resistive subcircuit
1. Calculate the chain matrix for the subcircuit of Fig. 4.29.
The chain matrix of the voltage amplifier of Fig. 4.30 can be calculated by matrix multiplication of the
chain matrices of the two smaller subcircuits that are indicated with dotted lines in the figure.
R2
R1
K=
0 0
0 0
+
( )
+
vsource
_
+
+
-
vout
-
Figure 4.30: A voltage amplifier
2. Calculate the chain matrix of the voltage amplifier.
3. Identify the entry in the chain matrix of the amplifier that introduces the amplification, and what is
amplification?
Figure 4.31 shows the same voltage amplifier, however the source impedance that will appear in any
real signal source is made explicitly.
4. Calculate the chain matrix of this realistic voltage amplifier.
When the source impedance Rs is inaccurate, the transfer of the amplifier becomes inaccurate.
5. Identify the entry in the chain matrix of the amplifier that introduces the influence of the source
impedance on the transfer, and what is the voltage amplification?
(Answers:
•
K=
1R
0 1
(4.115)
108
R2
Rs
R1
K=
0 0
0 0
+
( )
+
vsource
_
+
+
-
vout
-
Figure 4.31: Voltage amplifier with a realistic signal source.
•
1 R1
0 1
0 0
− R12 0
=
1
−R
R2 0
1
− R2 0
(4.116)
• K11 ;
vs = −
•
1 Rs
0 1
R1
R2
vo ⇔ vo = − vs
R2
R1
R1
0
−R
2
− R12 0
=
Rs
1
−R
R2 − R2 0
1
− R2
0
(4.117)
(4.118)
• K11 ;
vs = −
R1 + Rs
R2
vo ⇔ vo = −
vs
R2
R1 + Rs
109
(4.119)
Intentionally left blank.
110
Chapter
5
Amplifier configurations I: The op amp
nullor
In Chapter 3 we described properties of amplifier configurations based on the nullor. In Chapter 4 we
found that transactors with an infinite voltage gain, infinite current gain, infinite transconductance, or infinite transresistance have the same chain matrix as the nullor element. The first question is, of course,
whether we can find real implementations of the nullor element. This question is quite involved. Components or subcircuits that are able to amplify signal power with an infinite amplification factor can not exist,
because they would require an infinite amount of energy. Under certain conditions, however, components
and (sub)circuits exist that are able to amplify signal power enough to be a candidate for implementing
the nullor. In this chapter implementation by the operational amplifier is described. For applications, in
which the to-be-amplified-signal levels are such that the contribution of noise by the amplifying circuit is
not corrupting the signal-to-noise ratio, and in which the speed of information transport or processing is
not too high, standard amplifier building blocks called operational amplifiers (op amps) can be used successfully for implementing the nullor element. The ideal op amp is a voltage-to-voltage transactor with an
infinite voltage gain and one output grounded. As such, the ideal op amp can immediately be used as a
nullor implementation in those configurations in which one of the output leads is grounded. We will start to
synthesize circuits with the ideal op amp as a first step. Of course, this ideal op amp device does not exist.
And we will discuss the more realistic op amp, and some of its circuits further on in this chapter.
5.1 Circuits with the ideal operational amplifier
Figure 5.1 shows the ideal op amp: a voltage-to-voltage transactor with an infinite voltage gain and one
output grounded. The ideal op amp can immediately be used as a nullor implementation in those configurations in which one of the output leads is grounded. The two inputs are labeled + and − . The first
one is called the non-inverting input, the second is called the inverting input. Due to the fact that the op
amp must provide energy to the circuit it is included (this is done by the norator part), the op amp is usually
supplied with both a positive and a negative supply voltage. Because one of the output leads is grounded
the number of possible passive couple networks is limited to those that allow a parallel interconnection of
the op amp and the coupling two-port at the output. This leads to the following two ideal op amp amplifier
configurations: voltage amplifier based circuits and transresistance amplifier based circuits.
5.1.1
Voltage amplifier based circuits
Based on the amplifier configurations with the nullor in Section 4.4.1, we can immediately construct amplifier configurations with the ideal op amp. The first basic amplifier configuration is based on the voltageto-voltage converter.
111
μ(v+-v-)
μ→∞
+
v+
+
_
v-
∞
v+
+
v_o
_
_
+
+
_
vo
+
v_
_
(a)
(b)
Figure 5.1: The operational amplifier (op amp): (a) symbol, (b) ideal op amp
Non-inverting voltage amplifier
Based on the couple network of the nullor when used as a voltage-to-voltage converter, the passive couple
network of the voltage-to-voltage converter or voltage amplifier and the op amp is shown in Fig. 5.2. Based
on the nullor properties of the ideal op amp, the voltage across the non-inverting input and inverting input
is zero and the input current is zero too.
Therefore, because the voltage across the non-inverting input and inverting input is zero:
v R2 = v i .
(5.1)
We obtain the output voltage vo (KVL):
v o = v R 2 + iR 1 R 1 .
(5.2)
The current iR1 is provided by the op amp (at the norator side) and is also the current through R2 because
the current at the input of the ideal op amp is zero. That is: iR1 = iR2 = vR2 /R2 . Consequently, using
Equation 5.1:
R1
R1 + R2
vR
(5.3)
v R2 =
vi .
v o = v R2 + 2 R 1 = 1 +
R2
R2
R2
+
+
_
∞
R1
vi
+
vo R1 + R2
=
vi
R2
vo
_
R2
_
Figure 5.2: Non-inverting voltage amplifier; the arrows show the direction of the current for positive values
of the input voltage
If the voltage at the input is positive, the op amp delivers a positive current to excite the load (vo must
be positive too); if the input voltage is negative the op amp absorbs a positive current (delivers a negative
current). This explains why most op amps are connected to a positive and negative power supply. Also
note that a probable source resistance does not enter the transfer function, because the input resistance of
the ideal op amp circuit is infinite.
This circuit has a infinite input resistance and a zero output resistance, because of the fact that the circuit
is an ideal controlled voltage source with a gain of (R1 + R2 )/R2 ; the chain matrix of this ideal op amp
112
circuit is (see Section 4.3.2):
R2
0
K = R1 +R2
.
0
0
(5.4)
Voltage follower or buffer
In the special case of R1 equal to zero and R2 equal to infinity, the resulting circuit is shown in Figure 5.3.
We see that the infinity of R2 causes an open-circuit, that is, the current through it is zero and the voltage
can be any value and is, in this case, vi because the the voltage across the non-inverting input and inverting
input of the op amp is zero. The value zero of R1 results in a short-circuit, that is the voltage drop across the
resistor is zero allowing any current to flow through it and is, in this case, zero because the input current of
the op amp is zero. As overall result, the voltage at the output of the circuit equals the voltage at the input,
and the current to drive the load is provided by the energy sources attached to the op amp.
+
+
_
∞
vi
+
vo 1
=
vi
vo
_
Figure 5.3: Unit gain buffer amplifier or voltage follower
The result vo = vi is also obtained if we substitute the appropriate values for R1 and R2 in Equation 5.3:
R1
0
(5.5)
vi = 1 +
vo = 1 +
vi = vi .
R2
∞
The importance of this circuit is that it has a infinite input resistance and a zero output resistance,
because of the fact that the circuit is an ideal controlled voltage source with a gain of 1; the chain matrix of
this ideal op amp circuit is (see Section 4.4.1):
10
K=
.
(5.6)
00
As such, the circuit is very useful as a buffer.
Voltage divider
Once we understand the non-inverting voltage amplifier and the voltage follower, we can combine our
knowledge of them with the nullor based circuits for signal processing, as discussed in Chapter 3 on the
processing of voltage and current signals.
A first important circuit is the voltage divider circuit that is able to drive any load (that is, even one with a
very small resistance); as is shown in Figure 5.4.
113
+
vo
R2
=
vi R1 + R2
R1
+
_
vi
∞
R2
+
vo
_
_
Figure 5.4: Voltage divider
Addition of voltages with the non-inverting voltage amplifier
Also a couple of circuits for the addition of voltage signals sources can be derived from the circuits discussed
in Section 3.1.3 on the addition and subtraction of real voltage signals. Different amplification factors were
derived in Section 3.1.3 for adding voltage signals after a sufficiently large resistor Rc was placed in series
with each voltage signal source, resulting in three different circuit topologies—as shown in Figure 5.5 (a),
(b), and (c).
In Figure 5.5 (a) a non-inverting voltage amplifier is used to multiply the voltage across the load resistor Rl
with a factor
Rc
Rc
1+
≈
,
(5.7)
Rl
Rl
c
if we take care that R
Rl 1.
In Figure 5.5 (b) a non-inverting voltage amplifier is used to multiply the voltage across the load resistor
Rl = R with a factor of (1+n), where n is the number of input voltage signal sources. And in Figure 5.5 (c)
a non-inverting voltage amplifier is used to multiply the voltage across the load resistor Rl with a factor of
n, where n is the number of input voltage signal sources. Note that the condition Rc Rl is also fulfilled
if the load resistance Rl is left out, because of the infinite input resistance of the amplification circuit.
114
Rc
v1
+
+
_
Rc
v2
+
Rc
+
vn
Rc >> Rl
∞
vo
Rl
+
_ __
Rl
Rc
vo ≈ (v1+v2+...vn)
_
(a)
v1
Rc
+
v2
+
_
Rc
+
nR*
+
vn
Rl = Rc
∞
Rc
+
_ __
vo
vo = (v1+v2+...vn)
_
R* arbitrary
R*
Rc
(b)
v1
Rc
+
v2
+
_
Rc
+
(n-1)R*
+
vn
Rc << Rl
∞
Rl
+
_ __
vo
vo = (v1+v2+...vn)
_
R* arbitrary
R*
Rc
(c)
Figure 5.5: Addition of voltages with voltage amplifier: (a) Rc Rl ; (b) Rl = Rc ; (c) Rc Rl ; Rl is the
load of the circuit when the op amp subcircuit is removed, see also section 3.3.3
115
5.1.2
Transresistance amplifier based circuits
It is also possible to synthesize basic amplifier configurations based on the current-to-voltage converter.
Starting from the amplifier configurations with the nullor in Section 4.4.3, we can immediately construct
other amplifiers configurations with the ideal op amp.
Transresistance amplifier or current-to-voltage converter
The first circuit with the passive couple network of the current-to-voltage converter and the op amp is shown
in Figure 5.6.
R
_
ii
Rs
+
∞
+
vo
= -R
ii
vo
_
Figure 5.6: Transresistance amplifier or current-voltage converter; the arrows show the direction of the
current for positive values of the input current
Based on the nullor properties of the ideal op amp: the voltage across the non-inverting input and
inverting input is zero and the input current is zero, we derive that the input current ii flows through the
coupling resistor R completely. To achieve this the output voltage of the op amp must be negative with
respect to earth if the input current is positive. We obtain the folowing expression:
vo = −Rii .
(5.8)
Note that due to the virtual earth the source resistance does not enter the transfer, because there isn’t
any voltage difference across it. And because of the negative output voltage (in case of a positive input
current), current has to flow into the output port of the op amp. This is only possible if an energy source
with a negative voltage is attached to the op amp; the op amp must be connected to a positive and negative
power supply if both a positive input current and a negative input current must be processed.
The chain matrix of this current-voltage converter based on the ideal op amp circuit is (see Section
4.3.2):
0 0
.
(5.9)
K=
− R1 0
Addition of currents with the transresistance amplifier
Based on the current-to-voltage converter or transresistance amplifier various basic op amp circuits can be
synthesized. Almost trivial is the addition of currents in the virtual earth node, as is shown in Figure 5.7 in
case of two current signal sources.
The relation between the output voltage and the input current signal sources will be
vo = −R(i1 + i2 + ... + in ).
(5.10)
inverting voltage amplifier
If we replace the real current signal source in Figure 5.6 with a real voltage signal source, we obtain a circuit
know as inverting voltage amplifier, see Figure 5.8.
116
R
_
i1
i2
R1
R2
∞
+
+
vo= -R(i1+i2)
vo
_
Figure 5.7: Addition of currents with transresistance amplifier
R
Rs
+
+
vi
_
∞
+
vo
_
vo - R
vi = Rs
_
Figure 5.8: Inverting voltage amplifier
Due to the virtual earth, the current into the resistor R is vi /Rs . The transfer from input to output can
be described by:
vo = −
R
vi ,
Rs
(5.11)
and we see that the source resistance enters the transfer. Clearly, this can be a disadvantage.
Addition of voltages with the inverting voltage amplifier
Based on the inverting voltage amplifier a linear summation of voltage signals can also be obtained. Both
voltage signal sources are converted to currents and summed. The conversion is done by resistors that are
large compared to the source resistances of the signal sources. Figure 5.9 (a) and (b) show the circuits.
Due to the virtual earth we can write:
v1
(5.12)
i1 =
Rc
i2 =
v2
Rc
.
.
.
in =
vn
Rc
Using vo = −R(i1 + i2 + ... + in ) we obtain:
vo = −
R
(v1 + v2 + ... + vn ).
Rc
(5.13)
117
R
Rc
v1
+
_
Rc
v2
+
vn
∞
+
+
vo
+
_ __
Rc
vo = - (v1+v2+...vn) R
Rc
_
(a)
Rc
v1
Rc
+
_
Rc
v2
+
+
vn
∞
+
vo
+
_ __
Rc
vo = - (v1+v2+...vn)
_
(b)
Figure 5.9: Addition of voltages with a transresistance amplifier or inverting voltage amplifier: (a) linear
summation using an inverting voltage amplifier; (b) linear summation of voltages with an amplification
factor of 1.
Subtraction of voltages with the inverting voltage amplifier
By applying both the inverting and the non-inverting input of the ideal op amp we can obtain a circuit
capable of subtracting voltages. This can be expected if we realize that the circuit can be thought of as
a superposition of an inverting voltage amplifier, with a transfer proportional to −vi , and a non-inverting
voltage amplifier, with a transfer proportional to +vi . The circuit is shown in Figure 5.10 (a). Using
superposition we obtain from the inverting amplifier, Figure 5.10 (b):
vo = −
R
v1 ,
Rc
(5.14)
and from the non-inverting voltage amplifier, Figure 5.10 (c), (using voltage division at the non-inverting
input):
vo =
R + Rc
R
·
v2 ,
Rc
R + Rc
(5.15)
R
.
Rc
(5.16)
resulting in:
vo = (v2 − v1 )
5.1.3
Additional configurations
Other interesting circuits can be constructed using a transimpedance amplifier or inverting voltage amplifier
topology. In essence, these topologies convert a current into a voltage. Any current at the input can be
converted to a voltage.
118
R
v1
Rc
+
v2
_
Rc
∞
+
+
+
vo = (v2 - v1) R
Rc
vo
R
_ _
_
(a)
R
Rc
+
R
Rc
_
Rc
+
∞
_
Rc
+
+
+
v1
R
_
vo
+
R
v2
_
+
vo
_
_
(b)
∞
(c)
Figure 5.10: Subtraction of voltages with a transresistance amplifier or inverting voltage amplifier: (a) subtraction using an inverting voltage amplifier; (b) and (c):the transfer can be calculated using superposition
of (b) an inverting voltage amplifier and (c) a non-inverting voltage amplifier.
Integrator
An interesting circuit can be synthesized by replacing R in the inverting voltage amplifier circuit of Figure 5.8 with a capacitor. In this way we obtain a so-called integrator, see Figure 5.11 (a). The output of the
circuit is related to the incoming current by the constitutional relation of the capacitor
io = C
dvo
.
dt
(5.17)
The input current ii is determined by ii = vi /R. Using io = −ii the output voltage vo can be expressed by
1
vo = −
RC
t
−∞
vi (τ ) dτ
(5.18)
We see that the integrator circuit integrates an incoming voltage signal over time, and is as such often
applied in the design of electronic filters.
Differentiator
Exchanging the capacitor and the resistor in the integrator circuit gives rise to a so-called differentiator
i
circuit, see Figure 5.11 (b). The incoming current ii through the capacitor, ii = C dv
dt is directly multiplied
by (-R) to yield:
vo = −RC
dvi
.
dt
(5.19)
And we see that the output voltage is proportional to the derivative of the input voltage with respect to time.
119
C
R
_
∞
+
+
vi
_
t
+
vo
_
vo = - 1 ∫ vi (τ) dτ
RC -∞
+
vo
_
vo = - RC dvi
dt
+
vo
_
vo = - VT ln vi
RIs
(a)
R
C
_
∞
+
+
vi
_
(b)
D
R
_
+
+
vi
∞
_
(c)
Figure 5.11: (a): integrator; (b) differentiator; (c) logarithmic voltage amplifier
Logarithmic amplifier
It is also possible that the transfer is non-linear. For example consider the circuit of Figure 5.11 (c), the
coupling element is a diode. The current through the diode, in conduction, is approximately:
vd
id = Is exp
,
(5.20)
VT
Is being to saturation (reverse) current and VT being the thermal voltage. Taking the logarithm and realizing
that the current through the diode can be expressed as vi / R we obtain
vi
,
(5.21)
vo = −VT ln
RIs
when the diode is conducting.
120
5.2 Differential amplifier and operational amplifier with finite gain
A real implementation of an operational amplifier will have, amongst others, a high input resistance and
finite gain. Both input terminal voltages are then defined relative to a common node (earth). To say something useful about the difference of both voltages we encounter the problem of subtracting voltages, as we
have discussed in section 3.1.3. on addition and subtraction of voltage signals. We found that for summing
or subtracting real voltage signal sources with a common ground, the voltages have to be converted to well
defined currents first.
Special amplifiers have been developed that first convert voltage signals and then multiply the difference:
vo = A(v+ − v− ).
(5.22)
They are called differential amplifiers, see Fig. 5.12
v+
+
_ A
v_
+
vo
_
vo = A(v+ - v_)
Figure 5.12: Differential amplifier
Again, the two inputs are labeled + and − . The first one is called the non-inverting input, because a
positive voltage on this input with respect to the second will produce a positive output. The second is called
the inverting input. As the output of the differential amplifier is proportional to the difference of the two
input signals, we note that if the inputs are the same no output will be produced. Remember, for the real
thing the amplification factor is high but finite!
An important application of the differential amplifier is as both inputs have a small common noise signal
or common other unwanted signal component vn superposed on their input signal vs . In this case:
v+ = avs + vn
(5.23)
v− = bvs + vn
(5.24)
the differential amplifier gives the output:
v+ − v− = (a − b)vs
(5.25)
and the noisy parts cancel.
Voltages that are common to both inputs are called common-mode signals, while voltages that differ are
termed differential-mode signals. The ability to reject common-mode signals is expressed in the commonmode rejection ratio or CMRR, and is defined as the ratio of differential voltage gain AVD to commonmode voltage gain AVCM and expressed in decibels:
AVD
CMRR = 20 log10
dB.
(5.26)
AVCM
A common form of differential amplifier is the real op amp, which is non-ideal. Op amps are constructed
by fabricating all necessary components on one chip (IC). In general they have a very high input resistance,
a very small output resistance, and a very high gain. Often op amps tend to vary from one device to another,
and may vary with temperature and over time. We will now consider some circuits based on a non-ideal op
amp, and will discuss some of their properties.
121
5.3 Non-inverting voltage amplifier circuit
Based on a more realistic op amp, the first amplification circuit we are going to consider is that of the noninverting voltage amplifier, see Fig. 5.13. At the input of the op amp, the difference of the input voltage vi
and the voltage across R2 is amplified. If A is the amplification of the op amp, ri the input resistance, and
ro the output resistance then the amplifier circuit is described by:
vo∗ = A(v+ − v− ),
(5.27)
v+ = vi ,
(5.28)
v R2 = v −
(5.29)
and
vo*
ii
+
+
ro
ri
_
+
A
R1
vi
vo
R2
_
_
Figure 5.13: A more realistic non-inverting voltage amplifier
5.3.1
Open-loop gain and closed-loop gain
As a first step towards a more realistic op amp model and circuit description, we analyze the circuit using
a linear model of the op amp having an infinite input resistance, a zero output resistance, but a finite
amplification factor A. A is called the open-loop gain. The circuit is drawn in Fig. 5.14.
First, the to-be-amplified voltage vi is attached to the non-inverting input:
v+ = vi .
(5.30)
Next, we write the node equation (KCL) at the node between the two resistors, knowing that the model does
not allow a current from the inverting input:
v−
v− − vo
+
= 0,
R2
R1
(5.31)
or
vo =
R1 + R2
v− .
R2
(5.32)
The dependent-source relation yields:
vo = A(v+ − v− ).
(5.33)
Substituting and solving, we obtain the following expression for the circuit:
vo =
Avi
.
2
1 + A R1R+R
2
(5.34)
122
vi +
op amp
v+
A(v+-v-)
v-
+
+ vo
R1
R2
-
-
Figure 5.14: Model of a non-inverting voltage amplifier: finite amplification
Upon defining the fraction β, in this circuit, defined as:
β=
R2
R1 + R2
(5.35)
we obtain the expression:
vo =
A
vi .
1 + Aβ
(5.36)
The expression
G=
A
1 + Aβ
(5.37)
is called the closed-loop gain.
If we take the limit of A → ∞, we find that G is approaching β −1 :
A
lim
A→∞ 1 + Aβ
1
1
= .
A→∞ 1 + β
β
A
= lim
(5.38)
Also, for a very large open-loop gain A and β, for which holds that
Aβ 1
(5.39)
we readily find that
vo ≈
R1 + R2
vi .
R2
(5.40)
This is an important result. It tells us that the relation between vo and vi is almost independent of the gain
of the op amp under these conditions. The total gain G of the circuit is be determined by stable, reliable
resistor values. Of course the price we have to pay for this is that the overall closed-loop gain is much
smaller than the open-loop gain of the op amp.
We investigate the circuit with a numerical example. Suppose we have an op amp with open-loop gain
of 5x104 . We choose β to be 1/20. First, we check that Aβ = 2500 and fulfills the condition Aβ 1.
More precise we have:
G=
5x104
50000
A
=
= 19.99
4 =
1 + Aβ
2501
1 + 5x10
20
123
(5.41)
and
βvo = 0.9996vi .
(5.42)
The amplified input difference in the op amp is
A(v+ − v− ) = 5x104 (vi − 0.9996vi ) = 5x104 (0.0004vi ) ≈ 20vi .
(5.43)
Note that the actual voltage difference across the op-amp inputs is only 0.0004 V.
It is clear that the resistive couple network has profound effects on the overall gain of the system.
However, it also affects other system characteristics. For example, from Eqn. 5.37 taking the differential,
we find that for small deviations in open-loop gain A hold:
ΔG
ΔA
1
≈
.
.
G
A 1 + Aβ
(5.44)
We see that the relative error, in the amplification, is reduced by a factor (1 + Aβ). As a numerical example
consider β = 0.1 and A = 10, 000. The overall gain G = 10,000
1001 ≈ 10. For a variation of 10% in A we
find that the variation in G is only 0.01%. Note that, in the circuit with a given overall gain, increasing the
open-loop gain results in decreasing the actual input to the op amp. If the open-loop gain of the op amp
approaches infinity, the input voltage of the op amp decreases to zero (the nullator condition).
The importance of the couple network producing the ratio β cannot be underestimated. An accurate,
linear couple circuit produces a stable and precise overall gain G if Aβ 1. In electronics we will therefore always choose to implement the couple network with passive components of high quality.
For each of the circuits we studied, until now, there was a coupled network connected between the
output of the op amp and the inverting input terminal (the input terminal marked minus). This is no coincidence, and we will explore this in detail in the section on saturation, stability, and feedback, section
5.3.4. However, at this point in the description of the op amp circuits—where the open-loop gain of the op
amp is high enough to consider the op amp to be ideal: the input terminals were modeled with the nullator
that constraints the voltage across it and the currents through it to be zero—it is instructive to consider a
voltage amplifier circuit in which a couple network is connected between the output of the op amp and the
non-inverting input terminal. In this case the actual input is presented on the inverting input terminal. In
this case:
v− = v i .
(5.45)
v+
v+ − vo
+
= 0,
R2
R1
(5.46)
and
leading to
vo =
A
vi .
Aβ − 1
(5.47)
Taking the limit of A → ∞, we find that G is approaching β −1 in this case too:
lim
A
A→∞ Aβ − 1
=
1
.
β
(5.48)
This is, of course, not surprising because in case of an ideal op amp the voltage difference across the noninverting and inverting input terminal is zero. It is clear that the reason for choosing the couple network to
connect the op amp output with the inverting input terminal is not based on this ideal model of the op amp.
124
5.3.2
Output resistance and input resistance
Now, we consider the more realistic non-inverting voltage amplifier of Fig. 5.13, in which the more realistic
op amp is modeled using a (large) input resistance and a (small) output resistance. The existence of a
coupling network has a profound effect on the Thévenin-equivalent input and output resistance of the noninverting voltage amplifier circuit too. To illustrate this we first calculate the Thévenin output resistance. To
calculate this output resistance we apply a test current it , at the output terminals and calculate the resulting
voltage vt , setting all other independent sources, in this case vi , to zero, see Fig. 5.15.
vo*
ii
+
ro
ri
_
A
R1
+
vt
vi=0
it
R2
_
Figure 5.15: Calculation of the output resistance of the more realistic non-inverting voltage amplifier
We have the following equations:
v+ = 0
v− = βvt
vo = A(v+ − v− )
it =
vt
R1 + (R2
ri )
+
vt − vo
ro
(5.49)
To simplify the expression, we assume that ri R2 and find:
it
1
1
Aβ
= Ro−1 =
+
+
.
vt
R1 + R2
ro
ro
(5.50)
For large A and reasonable valued R1 and R2 the above expression leads to:
Ro ≈
1
ro
1 + Aβ
(5.51)
We see that the output resistance of the whole circuit is approximately the output resistance of the op amp
divided by (1 + Aβ), that is, divided by the same expression as the expression that reduced the open-loop
gain; or the expression that reduced the relative error.
Next we calculate the Thévenin input resistance of the circuit. The input resistance can be found by
applying a test voltage at the input and calculate the resulting current, see Fig. 5.16.
We obtain the following equations:
v+ − v− = ri it
vo = A(v+ − v− )
Defining the input resistance Ri :
Ri =
vt
vt
vt
=
ri = Ari
it
v+ − v−
vo
(5.52)
125
vo*
it
+
ro
ri
_
A
R1
vt
R2
Figure 5.16: Calculation of the input resistance of the more realistic non-inverting voltage amplifier
To find a simplified formula, we make the approximations that (1) ri R2 , and take ri = ∞ for the
calculation of the closed-loop amplification, (2) ro = 0, to make the approximation vo = vo . Using
Eq. 5.36
vt
1 + Aβ
vt
=
=
vo
vo
A
And we obtain for the input resistance of the non-inverting amplifier circuit:
Ri ≈ (1 + Aβ)ri .
(5.53)
A valid conclusion after studying the input and output resistances, is that, although the op amp is only a
rough approximation of the nullor, the transactor implementation, as a whole, is beyond expectation, due to
the fact that the application of the passive couple network has enormously improved the overall behavior.
At the cost of a lower (closed-loop) amplification factor.
5.3.3
Special case: the voltage follower
If in case of the non-inverting voltage amplifier, we choose R1 to be zero and R2 to be infinite (R1 =
0; R2 = ∞), we obtain the degenerate case drawn in Fig. 5.17.
ii
+
+
ro
ri
_
A
vi
+
vo
_
_
Figure 5.17: Voltage follower with more realistic op amp.
Because in this case β = 1, from Eq. 5.36 the input-output relation is
vo ≈ vi ,
(5.54)
that is, the output voltage follows the input voltage. At the same time we obtain from Eq. 5.53
Ri ≈ ri (1 + A) ≈ Ari .
(5.55)
126
From Eq. 5.51 we obtain that the output resistance of the circuit is
1
ro
Ro ≈ ro
≈ .
1+A
A
(5.56)
Now, the application of this circuit can become clear. We only need to look at the currents. The current
that must be supplied by the input source is extremely small. The maximum current that must be supplied
to some load is determined by the external energy source, and can easily be a few milli-amperes. Also, the
output resistance is extremely small, consequently loading effects will be minimal.
We notice that the output follows the input voltage, regardless of the external load. Consequently, the
circuit is called a voltage follower, a buffer, or an isolation buffer.
5.3.4
Dynamic model for the op amp: saturation, stability, and feedback
When discussing the effect of interchanging the inverting and non-inverting op amp inputs in case of ideal
op amps, Eq 5.48, we noticed that there was no difference in behavior when used in an ideal non-inverting
amplifier configuration. Practical op amps , however, do show a clear different behavior. Let us discuss this
different behavior. Any practical op amp will have maximum output values due to the voltage delivered to
the op amp from outside energy sources (generally, in case of the op amp: positive and negative voltages
from power supplies).
vo
+
+
_
∞
+
1
vi
1
vo
_
vi
_
(b)
(a)
vo
+
+
_
∞
+ Saturation
+
Linear region
vi
_
1
1
vo
-Esat
Esat
vi
_
- Saturation
(d)
(c)
Figure 5.18: Voltage follower (a,c); only having a linear regime (b), the complete transfer characteristic
including saturation (d).
When the output of the op amp cannot become higher, in case of a positive output voltage, or lower, in
case of a negative output voltage—because the external power supplies cannot deliver more voltage—the op
amp is said to be in +Saturation or -Saturation, respectively. A consequence of this is that in describing our
circuits we have to distinguish three modes of operation in three different regions of operation: the Linear
region where the output is just proportional to the input, the +Saturation region, and the -Saturation
region.
Using the voltage follower, as an example circuit, we start discussing the effects of the existence of the
saturation regions, simplifying the discussion by assuming that A = ∞, Gi = 0, and Ro = 0. In the absence
of saturation, the follower based on the ideal op amp is shown in Fig. 5.18(a,c). Its vi vo -characteristic is
127
shown in Fig. 5.18(b). The output will always be in the Linear region, because saturation will never occur,
and forces the nullator condition on the input vd , vd ≡ v+ − v− , of the op amp.
Introducing saturation will limit the possible output voltages. To ease the discussion we will assume a
symmetrical power supplies, VCC+ and VCC- : VCC+ = −VCC- . As a consequence of the maximum voltage
delivered by the power supplies, the output of the op amp is limited between Esat and -Esat . For positive
input values larger than the maximum positive output voltage, Esat , must hold that:
vd = vi − Esat > 0
(5.57)
vi > Esat .
(5.58)
that is
For negative input values larger than the maximum negative output voltage, -Esat must hold that
vd = vi − (−Esat ) < 0
(5.59)
vi < −Esat .
(5.60)
that is
We extending the voltage follower based on the ideal op amp model with the presence of saturation. The
complete transfer characteristic is shown in Fig. 5.18 (d). Starting from vi = 0 and increasing the input
voltage, the output follows the input (the linear region) until the op amp saturates. When vi exceeds the
saturation voltage Esat , the output goes into the +Saturation region. And, of course if the input voltage goes
from zero to negative voltages, the op amp will saturate in the -Saturation region. We see that a single
output voltage is found for every input voltage; the output is unambiguous.
What happens if we interchange the inverting and non-inverting terminals of the ideal op amp, as shown
in Fig. 5.19(a)? We, again, start from vi = 0 and increase the input voltage. As long as |vi | < |Esat | the
output is still in the linear region. Hence in the linear region, the transfer characteristic for this circuit, in
which the output is coupled to the non-inverting input terminal of the ideal op amp, is identical to that of
the circuit in which the output is coupled to the inverting input terminal.
We go further with analyzing the characteristic of the circuit, in which the output is coupled to the
non-inverting input terminal, by raising the input voltage above the linear region. Consider the input only
touching the value |Esat |. At this value the output of the ideal op amp will be determined by 0 × ∞, being
unknown, two trajectories are possible. The output may reduce as soon as the input decreases or the output
may stay in the +Saturation region as the input decreases. In this last case it will remain in saturation until
|Esat | is reached again. The characteristic results theoretically in the “Zorro” characteristic of Fig. 5.19(d).
Now, we consider the full characteristics allowing values of the input above the saturation condition.
Figure 5.19(f) shows what will happen if we start from vin = 0 and increase its value. After starting with
vin = 0 immediately after passing |Esat |, the output will jump to in the -Saturation region, because the input
of the op amp, vd = v+ − v− = vo − vi , will become negative (vi > vo ). By further increasing the input
voltage we stay in the -Saturation region (vd = −Esat − vi < 0) . However, if we, as soon as we touch the
-Saturation region, start decreasing vi we will stay in the -Saturation region until the value (−Esat ) is passed
(vd = −Esat − vi > 0) , and the circuit will jump to the +Saturation region. Extending this behavior by
starting from vi = 0 and decreasing the input voltage a similar reasoning starts jumping to the +Saturation
region, and after decreasing the negative input voltage jumping to the -Saturation region. Looking at the
characteristics we note that the output is not unambiguous; for vi = 0, the output can have three different
values for the same input voltage.
It is clear that the complete characteristics in Fig. 5.18(d) and Fig. 5.19(f) are quite different. And
in practice the circuits do not behave in the same way. One functions as a voltage follower the other in
general does not. To find the reason, we have to include the existence of the saturation regions and an even
more realistic model of the op amp than we have covered until now. We have to extend the model with a
(Thévenin) equivalent capacitor. This equivalent capacitance represents all capacitances that are present in
a real op amp. The inclusion of capacitances will make the op amp model a dynamical model, in which
time plays a role.
128
vo
_
+
∞
+
-Esat<vin<Esat
vi
+
Linear region
-Esat
(b)
(a)
vo
_
+
∞
+
vi
+ Saturation
+
vo
-Esat
_
_
vi
(d)
vo
_
+
Esat
_
- Saturation
(c)
allowing vin to have
values beyond Esat
vi
Esat
_
_
allowing vin to have
values ≤ Esat
1
1
vo
∞
+
vi
+ Saturation
+
vo
_
_
-Esat
Esat
vi
_
- Saturation
(e)
(f)
vo
_
+
including
dynamics
∞
+
vi
+ Saturation
+
vo
_
_
vi
- Saturation
(h)
(g)
Figure 5.19: Op amp circuit with the non-inverting input terminal coupled to the output (a,c,e,g); without
saturation, only having a linear regime (b), allowing vin to have values ≤ Esat , including saturation (c), the
complete transfer characteristics including saturation, but without dynamics (f), transfer characteristics including dynamics in the op amp model (h). The arrows indicate the trajectories followed when an sinusoidal
input is presented.
129
Looking at Fig. 5.19(d and f) we notice that even if the input of the op amp is operating between its
saturation values there are three distinct output voltages possible for each value of vi , one on the linear
vo = vi line, one on the +saturation line, and one on the -saturation line. The theory of Dynamic Systems,
not treated in this text, tells us in this case that not all output values are stable: all operating points in the
linear region line are unstable. If the values of vi causes the output to be in the Linear region the output will
quickly move to either the +Saturation region, or to the -Saturation region, see Fig 5.19(h).
Intuitively, we can understand the presence of stable and unstable output values. The presence of capacitances in the op amp causes a small delay in the change of the output as soon as the input changes. As
a consequence a small change in the voltage difference,vd , between the input terminals of the op amp will
appear: vd = v+ − v− = vi (t2 ) − vo (t1 ) with t2 > t1 .
First, we consider the voltage follower in which the output is coupled to the inverting input terminal.
Suppose we have a positive vd whereupon a positive vi is increasing slightly; from vi (t1 ) → vi (t2 ). Since
vd = vi (t2 )−vo (t1 ), vd will decrease a short time later, because vo (t2 ) is larger than vo (t1 ) (an increasing vi
will cause an increasing vo ). The operating point moves into the direction of its original position (remember
vo (t2 ) always larger than vo (t1 )). The circuit is said to be stable. In the theory of Dynamic Systems this
kind of coupling is called negative feedback.
On the other hand if vo is negative for a positive vi , the new vd would become larger and the operating
point moves away from its original position towards one of the saturation regions. The coupling is called
positive feedback and the circuit could be unstable. If the increase in |vd | leads to a further increase in |vd |
the op amp will go into saturation very fast.
Second, we consider the circuit in which the output is coupled to the non-inverting input terminal. In
this case vd = vo − vi , and starting from a positive and increasing vi , the new vd will be negative a short time
later, and will become more negative later on, driving the circuit into -Saturation. The linear region can not
be maintained in any case. Figure 5.19(h) shows the complete characteristic in this case. The characteristic
is said to exhibit hysteresis (switching from -Saturation to +Saturation at a different value of the input than
from +Saturation to -Saturation). Because the coupling of the output to the non-inverting input terminal
leads to saturation of the op amp circuit too, circuits using this coupling are often called “positive feedback
circuits” too.
5.3.5
Gain-Bandwith product
As a consequence of the dynamic behavior, the open-loop gain of an op amp is dependent on the frequency.
As the frequency increases, the internal capacitances (often from transistors) become more significant. To
maintain a well-defined open-loop gain most op amps are internally compensated to obtain a frequency
response that is essentially equivalent to a simple low-pass filter, by adding capacitors in the op amp design.
If A is the dc open-loop gain of the op amp, then the frequency dependent open-loop gain, A for an
internally compensated op amp is defined as:
A=
A
,
1 + j ffh
(5.61)
in which fh is the upper cut-off frequency of the op amp.
We, now, consider the non-inverting voltage amplifier circuit. The overall gain (the closed-loop gain),
G, will be a function of frequency too, given by:
A
1+j ff
A
h
=
G=
1 + Aβ
1 + Aβf
1+j f
(5.62)
h
Simplifying Eq. 5.62, we obtain:
G=
A
A
=
f
1 + Aβ + j fh
(1 + Aβ) 1 + j
f
(1+Aβ)fh
130
G=
1
A
f
1 + Aβ 1 + j
=
(1+Aβ)fh
G
,
1 + j ffhG
(5.63)
in which G = A/(1 + Aβ) is the dc gain of the non-inverting voltage amplifier, and fhG = (1 + Aβ)fh is
the upper cut-off frequency for the non-inverting voltage amplifier. We note that for the the non-inverting
voltage amplifier the dc gain is the dc open-loop gain of the op amp divided by the factor 1 + Aβ, and that
the upper cut-off frequency (and, hence, the bandwidth) is multiplied by the factor 1 + Aβ. The use of the
coupled network can significantly increase the bandwidth of the non-inverting voltage amplifier.
It is now most useful to define the gain-bandwidth product fT of an op amp as being the product of
its dc gain A and its upper cut-off frequency (bandwidth) fh :
fT = Afh .
(5.64)
The importance of this gain-bandwidth product becomes clear when we consider the gain-bandwidth
product of the non-inverting voltage amplifier:
GfhG =
A
(1 + Aβ)fh = Afh = fT .
1 + Aβ
(5.65)
In order words, the gain-bandwidth product of the non-inverting voltage amplifier is constant and is equal
to the gain-bandwidth product of the op amp. Related to the gain-bandwidth product is the unity-gain
bandwidth if the amplifier circuit, and is defined as the bandwidth of the amplifier when the gain drops to
unity. For an op amp and the non-inverting voltage amplifier circuit the two measures are the same. Figure
5.20 shows the voltage gain against the frequency for non-inverting voltage amplifier for different values of
the closed-loop gain.
voltage gain
105
Open-loop gain
104
103
G=1000
102
10 G=10
1
1
10 100
1k 10k 100k 1M
frequency (Hz)
Figure 5.20: Gain against bandwidth characteristic for the non-inverting voltage amplifier.
5.4 Inverting voltage amplifier circuit
A second amplifier circuit we are going to consider, in more detail, is the inverting voltage amplifier circuit.
Before discussing the amplifier circuit based on a more realistic op amp model, we discuss the input resistance of the ideal inverting voltage amplifier circuit. We still assume the amplification of the op amp to be
infinite. To find the input resistance of the inverting voltage amplifier circuit, we describe the circuit with
chain matrices, see Fig. 5.21.
First, we split the circuit in two parts and consider the circuit as the cascade of the two parts. Both parts
will be described by their chain matrix. By cascading the two parts the chain matrices will be multiplied.
The first part consists of the single resistor R2 , and is described by
v i − v o = R 2 ii
131
R1
R2
vo = - R1 vi
R2
_
+
+
+
vi
vo
_
_
(a)
R2
ii
+
+
+
vi
- R1 ii
_
_
_
+
+
vo
vi
_
_
R2
+
- R1 vi
R2 _
+
vo
_
(c)
(b)
Figure 5.21: Input resistance of an inverting voltage amplifier based on an ideal op amp
and
i i = io ,
leading to the following chain matrix KR2 :
1 R2
.
KR2 =
0 1
(5.66)
The second part is the transresistance amplifier subcircuit described by the chain matrix KT A :
0 0
KT A =
.
− R11 0
(5.67)
Cascading both parts we obtain the complete inverting voltage amplifier circuit described by the following
chain matrix KIV A :
R2 − R1 0
0 0
1 R2
.
(5.68)
=
KIV A = KR2 · KT A =
− R11 0
0 1
− R11 0
Writing down the matrix equations, we obtain
R2
vo
R1
(5.69)
1
vo .
R1
(5.70)
vi = −
and
ii = −
132
In the first equation, 5.69, we recognize the amplification factor of the inverting voltage amplifier circuit
−R1 /R2 . Combining both equations 5.69 and 5.70, we can obtain an expression for the input resistance
of the inverting voltage amplifier circuit:
Ri =
vi
−R2 −R1
=
vo
= R2
ii
R1
vo
(5.71)
and the circuit can be represented by Fig. 5.21(c).
5.4.1
Open-loop gain and closed-loop gain
Now, we consider more realistic op amp models, starting assuming the finite open-loop gain A of the op
amp. The circuit is shown is Fig: 5.22.
R1
vo
+
R2
vi
v-
A(v+-v-)
+
v+
-
-
Figure 5.22: Model of an inverting voltage amplifier: finite amplification
The following equations hold:
v+ = 0
and (KCL)
vi − v−
vo − v−
+
= 0,
R2
R1
leading to
v− =
R2
R1
vi +
vo .
R1 + R2
R1 + R2
(5.72)
Introducing a finite open-loop gain:
vo = A(v+ − v− )
we find
vo =
1
−A R1R+R
2
2
1 + A R1R+R
2
(5.73)
vi .
The expression
G=
1
−A R1R+R
2
(5.74)
2
1 + A R1R+R
2
133
is the closed-loop gain for the inverting voltage amplifier circuit. As soon as
A
R2
1
R1 + R2
(5.75)
Equation 5.73 reduces to
vo ≈ −
R1
vi ,
R2
(5.76)
so, we can use criterion Eq. 5.75 to evaluate the validity of the closed-loop gain −R1 /R2 .
5.4.2
Input resistance and output resistance
The input resistance of the inverting voltage amplifier circuit can be calculated in much the same way as for
the non-inverting voltage amplifier circuit. We apply a test source at the input of the circuit, see Fig. 5.23.
R1
R2
it
_
ro
ri
vt
A
+
Figure 5.23: Calculation of the input resistance for the inverting voltage amplifier
The input resistance Ri consists of the resistor R2 in series with a complicated circuit that can be reduced
to the Thévenin equivalent input resistance of the transresistance amplifier circuit with the more realistic op
amp. We can calculate the Thévenin equivalent. To do this, we consider the circuit in Fig. 5.23 setting R2
equal to zero. We obtain the following equations:
v+ = 0,
v− = vt
Applying KCL we obtain:
it =
vt − A(v+ − v− )
vt
+
.
ri
R1 + ro
(5.77)
We rewrite this last equation:
1
it
1
1
A
=
= +
+
,
RTh
vt
ri
R1 + ro
R1 + ro
or
RTh = ri
(R1 + ro )
R1 + ro
A
(5.78)
(5.79)
.
For a large open-loop gain A, RTh is dominated by the last term:
RTh ≈
R1 + ro
A
(5.80)
and we expect the Thévenin input resistance of the transresistance amplifier to be small.
134
The input resistance of the inverting voltage amplifier will, thus, be
Ri = R2 + RTh ≈ R2 .
(5.81)
for realistic values of R1 and R2 .
The calculation of the output resistance is straightforward and can be found easily by noting that when
we apply a test current at the input, we obtain a similar circuit as the circuit we used to analyse the output
resistance of the non-inverting voltage amplifier circuit. By comparison we immediately find:
Ro ≈
1
ro .
2
1 + A R1R+R
2
(5.82)
Concerning the dynamic properties of the inverting voltage amplifier, due to the presence of capacitances in the op amp design, the general behavior discussed in the sections on the non-inverting voltage
amplifier circuit applies to the inverting voltage amplifier circuit too: the behavior of the circuit is frequency
dependent and we should always couple the coupling resistance R1 to the inverting input terminal of the op
amp for amplifier applications. Because of the coupling of the couple circuit to the inverting input terminal,
for stability reasons, these circuits are usually called negative feedback amplifier circuits.
5.4.3
Gain-bandwidth product
Finally, we consider the inverting voltage amplifier with a frequency dependent op amp. Again, we assume
if A is the dc open-loop gain of the op amp, then the frequency dependent open-loop gain, A for an internally
compensated op amp is defined as:
A=
A
,
1 + j ffh
(5.83)
in which fh is the upper cut-off frequency of the op amp. The expression for the closed-loop gain, for the
inverting voltage amplifier circuit, becomes:
G=
1
−A R1R+R
2
2
1 + A R1R+R
2
=
1
1
−A R1R+R
f
2 1+
fh
1+
(5.84)
R
2
A R +R
1
1+ ff
2
h
1
−A R1R+R
2
G=
2
1 + A R1R+R
+
j ffh
2
G=
1
−A R1R+R
2
2
)
(1 + A R1R+R
2
=
1
−A R1R+R
2
2
(1 + A R1R+R
) 1+j
2
1
1+j
1
2
f
R
2
(1+A R +R
)fh
2
with
R
2
)fh
(1+A R +R
1
And we can write:
G
,
G=
1 + j ffhG
f
fhG =
1+A
R2
R1 + R2
fh
(5.85)
It becomes clear immediately that the gain-bandwidth product of the inverting voltage amplifier circuit is
unequal to the gain-bandwidth product of the single op amp (fT ):
1
−A R1R+R
R1
R1
R2
2
fh
=
fT
(5.86)
|GfhG | = | 1 + A
| = Afh
R
2
R1 + R2
R1 + R2
R1 + R2
(1 + A R +R )
1
2
We notice that the gain-bandwidth product is not conserved in case of the inverting-voltage amplifier, and
we do not have a simple graphical way to determine gain versus bandwidth, as in case of the non-inverting
voltage amplifier circuit. Besides this, we note that the gain-bandwidth product of the inverting voltage
amplifier is always lower than that of the non-inverting voltage amplifier. In case of a gain of one (R1 = R2 )
the gain-bandwidth product of the inverting amplifier is only half of that of the non-inverting amplifier.
135
Intentionally left blank.
136
Chapter
6
Amplifier configurations II: The single
bipolar junction transistor nullor
In Chapter 1 we briefly examined an amplifier circuit based on a single transistor: the bipolar junction transistor (BJT) in a so-called common-emitter (CE) configuration. We are going to consider this amplifier in
more detail. Again, we start by examining the transistor characteristics; they can be found by measurement
or by simulation.
6.1 npn transistor characteristics in CE configuration
For a typical bipolar transistor we examine the characteristics. The characteristics are obtained in the
common-emitter configuration, see Fig. 6.1.
IC
c
IB b
+ VCE
+
VBE
IE
e
Figure 6.1: npn transistor in common-emitter (CE) configuration for measuring or simulating the characteristics.
The first characteristics we look at are the IC (IE ) characteristic in Fig. 6.2(a) and the IC (IB ) characteristics in Fig. 6.2(b). From the IC (IE ) characteristic we learn that the collector current is almost equal to
the emitter current:
IC = αIE ;
α ≈ 1.
(6.1)
Looking at the IC (IB ) characteristic we note that collector current is proportional to the base current:
I C = hF E I B ,
(6.2)
in which hF E is called: the current gain of the transistor. The use of hF E in electronic design is limited
due to the fact that hF E is a not well-defined quantity in the manufacturing process of bipolar transistors,
and can typically vary between 35 and 300.
137
IC (IB)
IC (IE)
VCE = 10 V
IC 14
(mA)
12
15
IC
(mA)
VCE = 5 V
10
10
8
6
5
4
2
0
0
0
5
10
15
0
20
40
IE (mA)
(a)
60
80
100
IB (μA)
(b)
Figure 6.2: Characteristics showing (a) IC (IE ) and (b) IC (IB ).
We also note that the collector current IC is larger for larger values of the collector-emitter voltage VCE .
This is because the collector current increases for increasing voltage across the collector-base voltage VCB ;
the voltage difference across the collector and emitter,VCB , in the CE-configuration of Fig. 6.1 is determined
by VCB = VCE − VBE . This effect, which is also visible in the IC (VCE ) characteristic (yet to come) is
called “Early effect”, mentioned after James Early who first studied this effect. The “Early effect” will lead
to the definition of the “Early voltage”.
IC (VBE) for VCE = 10V
20
IC
(mA) 18
16
14
12
10
8
6
4
2
0
0.5
0.55
0.6.
0.65
0.7
0.75
VBE (V)
Figure 6.3: Characteristic showing IC (VBE )
We follow the theoretical prediction for the emitter current, see also Chap. 1—Eq. 1.14:
VBE
IE = Is exp
,
VT
(6.3)
where VT is the thermal voltage and has a value VT ≈ 25 mV at room temperature. Using the definition of
α we obtain the following equations for the collector current:
IC = αIs exp
VBE
VT
(6.4)
.
For VCE constant and such that VCB > 0, the IC (VBE ) characteristic is shown in Fig. 6.3 and follows the
theoretical prediction.
Knowing the relation between the input voltage VBE and the output current IC , Eq. 6.4, we can consider
the relationship between a small change in VBE , Δvbe , and the corresponding change in IC , Δic . This is
138
gm (VBE)
gm 700
(mS)
600
500
400
300
200
100
0
0.52
0.56
0.6
0.64
0.68
0.72
VBE (V)
Figure 6.4: The transconductance gm as a function of the base-emitter voltage VBE
important when we consider the effect of adding a small-signal source to the constant input source, VBE , as
we did in Chapter 1. Looking at Fig.6.3 we note that the curve clearly is nonlinear. The slope of the curve,
at any point, represents the transconductance of the device gm :
gm =
dIC
dVBE
(6.5)
Figure 6.4 shows the value of gm as a function of the base-emitter voltage.
Using the expression for the collector current IC , Eq. 6.4, we derive for the transconductance:
VBE
VBE
dIC
d
IC
1
1
gm =
=
αIs exp
IC =
αIs exp
=
=
dVBE
dVBE
VT
VT
VT
VT
VT
(6.6)
This equation shows explicitly the dependence of gm on the collector current in the circuit. Because
VT = kT
q , we can write:
gm =
q
IC ,
kT
(6.7)
and see that gm is determined solely by physical constants, the temperature, and the collector current. For
given T and IC , gm is the same for all bipolar transistors!
In the common-emitter configuration the external voltage source VCE provides the energy for power amplification, as we saw in Chapter 1. Therefore important characteristics in the common-emitter configuration
are the IC (VCE ) characteristics for various values of the base-emitter voltage VBE , see Fig. 6.5. These
characteristics are often referred to as the common-emitter output characteristics.
For a given VBE , the collector current initially rises rapidly with VCE as this increases from zero. However, it soon reaches a steady value. The section of the characteristic over which the collector current is
approximately linearly related to VCE is called the saturation region. The section over which the collector
current is almost constant is called the active region. It shows that in this region the output current is almost
independent of the collector-emitter voltage, and only depends on the base-emitter voltage as describe by
Eq. 6.4. Most amplifier circuits operate in this region. Looking at the IC (VCE ) characteristic in the active
region we note that the behavior can be approximate described by a current source; leading to the simplified
dc transistor model of Fig. 6.6. Note, however, that this current source absorbs energy—as lying in the first
quadrant the current goes from the higher voltage at the collector to the lower voltage at the emitter—the
behavior is that of a nonlinear resistor.
139
IC(VCE)
IC
(mA)
14
VBE = 0.7V
12
active region
10
saturation
8
VBE = 0.68V
6
4
VBE = 0.66V
2
VBE = 0.64V
VBE = 0.62V
0
0
0.5
1
1.5
2
2.5
3
VCE (V)
Figure 6.5: Characteristics showing IC (VCE )
IB
B
IC = gm(VBE) VBE
+
C
+
IC
VBE
E
ro
R
_
_
E
Figure 6.6: Simplified large-signal dc model of the bipolar transistor in CE configuration in the active
region.
6.2 Small-signal equivalent circuits in CE configuration
In the active region, small changes in VBE lead to small changes in IC . Based on the voltage controlled
current source we can create a two-port description of the bipolar transistor for small voltage and current
signals in the common-emitter configuration. This description must model at the input port the base-emitter
voltage and the base current, at the output port the collector current and collector-emitter voltage, see
Fig 6.7.
B
E
ic
ib
+
vbe
+
vce
_
_
C
E
Figure 6.7: Small-signal two port description of the bipolar transistor in CE configuration
To model the output port we note that looking carefully at the IC (VCE ) characteristics, Fig 6.5, we see
that the lines are not horizontal but have a slight gradient. The slope of these lines indicate that the output
current is increasing with increasing output voltage and can, therefore, be modeled by an output resistance,
140
ro . For the small-signal description, ro can be defined as:
ro =
dVCE
dIC
(6.8)
Figure 6.8 shows ro as a function of VCE . First, the figure shows that the magnitude of the output resistance
is large, that is, typically between 10kΩ and 1MΩ. Second, it shows that the output resistance increases
as VBE decreases and, thus, IC decreases. Generally, it can be found that the product ro times IC is
approximately constant:
ro IC = VA ,
(6.9)
for ro we obtain:
ro =
VA
IC
(6.10)
where the constant VA is called the Early voltage. VA typically has values in the order of 50-100 V.
ro (VCE) in active region
450
ro
(kΩ) 400
VBE = 0.58 V
350
300
250
VBE = 0.6 V
200
150
VBE = 0.62 V
100
VBE = 0.64 V
0.66 V
50
0
0
0.5
1
1.5
2
2.5
3
VCE (V)
Figure 6.8: The output resistance ro as a function of VCE
To model the input port, we have to take the base current into account, that is, we have to model an input
resistance. For the small-signal equivalent circuit we define the small-signal input resistance:
rπ =
dVBE
.
dIB
(6.11)
Figure 6.9 shows rπ as a function of VBE .
From the definition of rπ and Eq. 6.6 we can derive:
−1
−1
dIB
hF E
hF E V T
dIC
= hF E
=
=
.
rπ =
dVBE
dVBE
gm
IC
(6.12)
We note that rπ is inverse proportional to IC , and may vary for different transistors because hF E varies.
The considerations above lead to a small-signal equivalent circuit for the bipolar transistor in a commonemitter configuration as shown in Fig. 6.10. The equivalent circuit is known as the small-signal lowfrequency hybrid equivalent circuit.
For the small-signal collector current we find:
vce
(6.13)
ic = gm vbe +
ro
Looking at Fig. 6.10 it is clear that the hybrid equivalent circuit only provides a good description of
the transistor’s operation at low frequencies, but does not take into account the effects of the device capacitances. These capacitances greatly alter the behavior at high frequencies. A better model that takes these
141
r (VBE)
2200
r (Ω)
2000
1800
1600
1400
1200
1000
800
600
400
200
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.71 0.72
VBE (V)
Figure 6.9: The input resistance rπ as a function of VBE
ib
B
ic
gm vbe
r
vbe
C
ro
E
E
Figure 6.10: The small-signal hybrid equivalent circuit.
capacitance into account in the so-called hybrid-π equivalent circuit, is shown in Fig 6.11. Looking at
this figure, it is clear that the capacitance Cπ will produce a fall in collector current at high frequencies and
leads to the definition of an fT , in case of a transistor termed transition frequency. It is the frequency
where the current gain is 1.
B
ib
vbe
rb
rμ
r
ic
gm vbe
C
C
ro
E
E
Figure 6.11: The small-signal hybrid-π equivalent circuit; rb is the resistance of the base, Cπ the capacitance
between base and emitter, and rμ the differential resistance between base and collector (in general rμ is very
high).
6.2.1
Approaching the nullor with BJT in CE configuration
We can consider the small-signal equivalent circuit as a amplifier two port. For this purpose we write down
the chain matrix of the small-signal hybrid model, as shown in Fig. 6.10, with vi = vbe , vo = vce , ii = ib ,
and io = −ic .
We have:
vi = k11 vo + k12 io
ii = k21 vo + k22 io
(6.14)
142
with
vi ⇒ k11 = (−gm ro )−1
vo io =0
vi ⇒ k12 = (−gm )−1
k12 = io vo =0
ii k21 =
⇒ k21 = (−gm ro rπ )−1
vo io =0
ii ⇒ k22 = (−gm rπ )−1 = (−hF E )−1
k22 = io vo =0
k11 =
(6.15)
(6.16)
(6.17)
(6.18)
We know that r0 is typically very large (10 kΩ—1MΩ). So if we have a transistor with high current gain
hF E , the chain matrix approaches:
KBJT−CE ≈
0 −1/gm
0
0
(6.19)
and represents a voltage-to-current transactor (vit). If we also increase gm (by increasing IC ), the chain
matrix approaches the nullor:
00
KBJT−CE ≈
(6.20)
00
Of course, the approximation is only roughly. The value of both hF E and gm will never go to infinity.
However, we can use the approximation as a starting point for synthesis of circuits. Then we have to
analyze these circuits after synthesis to see how their behavior deviates from the ideal case (the nullor
description with ib = 0, hF E = ∞, and gm = ∞) .
6.3 Simple voltage amplifier
We can make a simple voltage-to-current transactor by assuming ro = ∞, rπ = ∞, and 1/hF E ≈ 0,
described by the K matrix of Eq. 6.19. To make the circuit we offer a small voltage signal to the commonemitter configuration and introduce a load resistor. The small-signal equivalent circuit will look as in
Fig. 6.12.
Rs
B
C
ic
transistor
+
+
vs
ib
il
+
vbe
gm vbe
vl
_
_
E
E
Rl
Figure 6.12: Equivalent circuit for small signals in a bipolar transistor based voltage-to-current transactor.
From Fig. 6.12 we derive:
vl = −gm vbe Rl ;
with
vbe = vs .
(6.21)
And obtain the voltage gain Av :
Av =
vl
= −gm Rl
vs
(6.22)
143
We see that Av is proportional to gm , and thus to the collector current IC . Besides this it depends on the
temperature T .
To synthesize the simple voltage amplifier based on the voltage-to-current transactor, we have to design
circuitry that provides the current IC and the collector-emitter voltage VCE that sets (biases) the transistor
in the active region. Biasing can be done by introducing voltage or current sources that specify IB or VBE ,
to determine IC , and VCE . As being local sources only, they are separated from the signals by capacitors.
Figure 6.13 (a) and (b) show two possible schemes: in (a) the base-emitter voltage and the collectoremitter voltage are specified (VCE being VCC − IC RC ), in (b) the base current (thus the collector current)
and the collector-emitter voltage are specified. Both topologies reduce to the small-signal signal model of
the transactor, Fig. 6.12; what can easily be seen by making the bias sources zero.
+
c
~
IB
VBE
+b
Rl
VCE+vce
+
vbe
-
VCE+vce
+
e
VCC
-
VCC
-
-
(b)
RB
RC
RC
+
+
Rl
~
+
vbe
-
-
-
(a)
R1
Rl
b
v~be
e
+
c
vl
~
+
VCC
-
VCC
-
vl
vbe
+
R2
Rl
+
-
-
(c)
-
(d)
Figure 6.13: Biasing the simple voltage amplifier based on a voltage-to-current transactor: (a) biasing VBE
and VCE ; (b) biasing IB and VCE ; (c) implementation of the bias of (a); (d) implementation of the bias of
(b). Note that by both implementations the small-signal models changes, and the small-signal circuit does
not represent a transactor anymore.
For actually doing the implementation, we have to place the components necessary in such a way that
the bias sources are effectuated. By going back to small-signal descriptions, we can better include the
transistor impedances too; by this diverge from the ideal transactor structure. We obtain a simple voltage
amplifier.
The circuits in Fig. 6.13 (c) and (d) can do the job. In Fig. 6.13(c) the base-emitter voltage is determined
by voltage division using R1 and R2 , and the collector-emitter voltage by voltage division using RC and
the transistor (acting as a nonlinear resistor); in (d) a large resistor RB in series with the voltage source
VCC act as a current source. However, by doing the implementation the small-signal models are changed.
144
They are changed into those of Fig. 6.14 and Fig. 6.15.
Rs
B
ib
transistor
ic
C
gmvbe
+
vs
R2 vbe
R1
r
RC
ro
vo
Rl
_
E
E
Figure 6.14: Small-signal equivalent circuit of a simple transistor amplifier-(c)
For the circuit of Fig. 6.13(c) we obtain the small-signal equivalent circuit of Fig. 6.14; we derive:
vo = −gm vs
(R1
R1 R2 rπ
(RC
R2 rπ ) + Rs
ro ).
(6.23)
And obtain the voltage gain Av :
Av =
vo
R1 R2 rπ
=−
gm (RC
vs
(R1 R2 rπ ) + Rs
ro ),
(6.24)
it depends on the source resistance Rs . It is clear that to minimize loading the resistors R1 and R2 should
be higher than rπ . We can also assume RC to be much lower than ro , ro being very large. This leads to a
voltage amplification of
Av ≈ −gm RC
rπ
rπ + Rs
(6.25)
The small-signal input resistance of the amplifier is, assuming (R1
R2 ) rπ :
ri ≈ rπ
(6.26)
The input resistance rπ varies for various transistors because hF E varies for various transistors; making rπ
not well defined in the circuit. The small-signal output resistance of the amplifier is, assuming ro RC
which will be normally the case,
ro ≈ RC
(6.27)
Rs
B
ib
transistor
+
vbe
RB
C
+
gmvbe
+
vs
ic
r
RC
ro
vo
Rl
_
_
E
E
Figure 6.15: Small-signal equivalent circuit of a simple transistor amplifier-(d)
For the circuit of Fig. 6.13(d) we obtain the small-signal equivalent circuit of Fig. 6.15; we derive:
vo = −gm vs
RB rπ
(RC
(RB rπ ) + Rs
ro ).
(6.28)
145
And obtain the voltage gain Av :
Av =
vo
RB rπ
=−
gm (RC
vs
(RB rπ ) + Rs
ro )
(6.29)
As in the circuit the serial combination of VCC and RB represents a current source, it is clear that RB will
be much higher than rπ . Again, we can also assume RC to be much lower than ro , leading to a voltage
amplification of
Av ≈ −gm RC
rπ
rπ + Rs
(6.30)
The small-signal input resistance of the amplifier is, assuming RB rπ
ri ≈ rπ
(6.31)
And the small-signal output resistance of the amplifier is:
ro ≈ RC
(6.32)
As we expect the basic properties of both implementations are the same, because we only changed the
biasing of the transistor circuit.
To find values for the various components of the implementations we can use the output characteristics
of the npn transistor and construct a load line, just as we did with the diode in Chapter 2, to find the
operating point for small signals. Figure 6.16 shows a load line for the simple amplifier. The load line
passes through VCC and has a slope equal to the reciprocal of the collector resistance. The load line
represents possible currents through RC as the voltage VCC changes. The intersection of the load line and
one of the characteristics represents a current IC through both the collector resistor RC and the transistor
for a given base-emitter voltage.
possible
IC
active region
load line
operating point
actual
IC
VBE
0
0
actual
VCE
ICRC
VCC
possible VCE
Figure 6.16: Common-emitter output characteristics with load line.
Now, we consider the accuracy of the simple amplifier. Looking at Eq. 6.30 we see that at the input
voltage division takes place by rπ /(Rs + rπ ). If the source resistance Rs is not known accurately and of
the same order of magnitude as rπ , then the voltage amplification is not accurate too. In order to improve
this, we might increase the value of rπ , by decreasing IC , but thereby decreasing gm . Another problem is
146
the nonlinear distortion of small-signals due to the fact that gm is nonlinear. A remedy is to increase the
collector current such that the transistor in CE-configuration approaches the nullor. In this case we can use
coupling circuits to obtain a gm independent amplification and improve on the input and output resistances
of the transistor.
6.4 Amplifiers based on a bipolar transistor nullor implementation
To improve on the nonlinear amplification by the transistor and the relatively low input resistance of the
described simple voltage amplifier, we will employ the nullor based topologies as described in Chapter 4.
To do this we have to synthesize the circuits such that the collector current is high enough. By doing so
the chain matrix of small-signal description of the common-emitter configuration approximates the chain
matrix of the nullor, Eq. 6.20.
6.4.1
Transconductance based voltage amplifier
The first nullor based design is the voltage-current converter or transconductance (or transadmittance) amplifier treated in Section 4.4.2. Figure 6.17(a) shows the small-signal equivalent circuit of the transconductance amplifier; Fig. 6.17(b) shows a possible implementation.
B
Rs
vbe
il
C
r
ro
gm vbe
E
Rl
+
vs
RE
(a)
R1
RC
+
Rl
+
vs
-
vl
R2
+
RE
VCC
-
-
(b)
Figure 6.17: Voltage amplification circuit based on the transconductance amplifier, using the nullor description of the BJT common-emitter configuration; (a) small-signal equivalent circuit; (b) circuit implementation.
The ideal transfer we expect from the transconductance amplifier is
1
io
=− .
vi
RE
(6.33)
147
In the absence of a load and neglecting the base current, the collector current is equal to the emitter current.
Therefore, we expect the small-signal voltage amplification to be
RC
vo
≈−
.
vi
RE
(6.34)
The small-input can indeed be increased. Assuming ro RE the input resistance of the unbiased follower
is:
ri (unbiased) ≈ rπ + hF E RE ,
(6.35)
see Chapter 2, problem 8. For the actual implementation of Fig. 6.17, we find for the small-signal input
resistance assuming that rπ + hF E RE (R1 R2 ):
ri ≈ R1
(6.36)
R2 ,
and can be much higher than in the amplifier not based on the nullor. In general, for the implementation the
high output resistance will be the same, and thus:
ro ≈ RC ,
(6.37)
because the emitter resistance will appear in series with the already high, and even further increased, ro .
Use of a decoupling capacitor
As we compare the results of the simple amplifier and the nullor based amplifier, we see that we improved
on stability of the amplification and on the input resistance of the circuit. However, we have achieved this
at the expense of a considerable reduction in voltage gain. In some applications this reduction in gain is
not acceptable. In these cases we may consider to use a decoupling capacitor to increase the ac signal
amplification, while maintaining the stability of the bias conditions of the circuit. Figure 6.18 shows the
circuit. The actual amplification of the signals is increased to −gm RC for sufficiently high frequencies.
R1
RC
+
vl
+
vs
-
Rl
CE
R2
+
VCC
RE
-
-
Figure 6.18: Voltage amplification circuit based on the transconductance amplifier, using a decoupling
capacitor.
6.4.2
Voltage follower or emitter follower
The next nullor based design is the voltage follower, the voltage-voltage converter or voltage amplifier with
an open to earth at the nullor output, and a short from the output of the nullator to the output the nullor—
treated in Section 4.4.1. Figure 6.19(a) shows the small-signal equivalent circuit of the voltage follower;
Fig. 6.19(b) shows a possible implementation.
148
B
r
vbe
Rs
C
ro
gm vbe
il
E
+
Rl
vs
(a)
R1
+
vs
+
R2
-
Rl vl
VCC
-
(b)
Figure 6.19: Voltage follower circuit or emitter follower; (a) small-signal equivalent circuit; (b) circuit
implementation.
The ideal transfer we expect from the voltage follower is
vo
= 1.
vi
(6.38)
To describe the small-signal behavior of the circuit, we consider the chain matrix of the follower. In
Fig. 6.20 the circuit is redrawn. By inspection we can find the entries.
il
B
E
r
gm vbe
ro
Rl
C
Figure 6.20: emitter follower; small-signal equivalent circuit, redrawn.
⎞−1
⎛ 1
r
+
g
m
o
rπ
vi ⎠ ≈ 1;
k11 =
⇒ k11 = ⎝
vo io =0
r
1+ 1 +g
rπ
m
(gm ro 1)
(6.39)
o
This expression can easily be derived by using the KCL on the node together with vbe = vi − vo and
vo = ro ir0 The other entries are
−1
1
vi ⇒ k12 = gm +
k12 = ≈ (gm )−1
(6.40)
io vo =0
rπ
149
ii −1
⇒ k21 = (gm rπ + 1)ro ) ≈ (hF E ro )−1
k21 =
vo io =0
ii −1
⇒ k22 = ((−gm rπ + 1) ≈ (hF E )−1
k22 = io vo =0
(6.41)
(6.42)
For a good voltage follower the amplifier should have a high small-signal input resistance and a low
small-signal output resistance. For the implementation we need to know the input and output resistances
of the unbiased small-signal circuit. The input resistance is the same as in case of the transconductance
amplifier:
ri ≈ R1
(6.43)
R2
The calculation of the output resistance is more involved. It requires the use of Eq. 4.63 of Chapter 4,
section 4.3.4., which gives a general expression of the output resistance for a circuit described by a chain
matrix. Doing this calculation gives:
ro ≈
1
Rs
+
,
gm
hF E
(6.44)
we see that ro depends on the source resistance, however if Rs is not too large, the output resistance can
be very small. Looking at Fig. 6.20 we note that the small-signal equivalent circuit is a common-collector
topology.
6.4.3
Common-base amplifier
Next, we look at the current-to-current configuration using the nullor. What we can synthesize from the
common-emitter small-signal circuit is a current follower, see Fig. 6.21.
il
C
B
gm vbe
r
ro
Rl
E
is
+
Rs
(a)
+
RE
is
RC
Rs
Rl
+
-
VCC
VEE
-
+
-
(b)
Figure 6.21: Common-base amplifier; (a) small-signal equivalent circuit; (b) circuit implementation. Note
that vBE is negative.
150
The ideal transfer we expect from the current follower is
io
= 1.
ii
(6.45)
To describe the small-signal behavior of the circuit, we consider the chain matrix of the current follower. In
Fig. 6.22 the small-signal equivalent circuit is redrawn. By inspection we can find the entries.
gm vbe
E
is
C
Rs
r
il
ro
+
Rl
-
B
Figure 6.22: Common-base; (a) small-signal equivalent circuit, redrawn.
vi ⇒ k11 = (1 + gm ro )−1 = (gm ro )−1
k11 =
vo io =0
−1
vi 1
k12 = ≈ (gm )−1
⇒ k12 = gm +
io vo =0
ro
ii −1
⇒ k21 = (rπ + gm ro rπ ) ≈ (gm ro rπ )−1
k21 =
vo io =0
⎞−1
⎛ 1
ro + gm rπ
ii hF E
⎠ ≈
k22 = ≈1
⇒ k22 = ⎝
io vo =0
hF E + 1
r
1+ 1 +g
ro
m
(6.46)
(6.47)
(6.48)
(6.49)
π
For a good current follower the amplifier should have a small-signal input resistance and a high smallsignal output resistance. We can find the input and output resistance by using the general expressions of
Chapter 4, Eq. 4.62 and Eq. 4.63. Performing the calculations we find:
ri ≈
Rl
1
gm ro + gm
Rl
h F E ro + 1
ro ≈
1
gm + R s
Rs
1
g m ro + hF E ro
≈
1
gm
(6.50)
≈ hF E ro ,
(6.51)
and see that the input resistance is quite low, while the output resistance is quite high.
6.4.4
Transresistance based amplifier
Finally, we consider the current-voltage converter or transresistance (or transimpedance) amplifier treated
in Section 4.5.3. Figure 6.23(a) shows the small-signal equivalent circuit of the transresistance amplifier;
Fig. 6.23(b) shows a possible implementation.
The ideal transfer we expect from the transresistance amplifier is
vo
= −R3 .
ii
(6.52)
151
R3
B
C
r
vbe
ro
gm vbe
E
is
Rl
Rs
(a)
R3
RC
+
Rl
vl
is
+
VCC
-
-
(b)
Figure 6.23: Transresistance voltage amplifier; (a) small-signal equivalent circuit; (b) circuit implementation.
To find expressions for the chain matrix entries is a bit more involved and will not be done here. Its
application is often found in voltage amplifiers made by cascading either a transconductance amplifier
and a transresistance amplifier or current amplifiers made by cascading a transresistance amplifier and
a transconductance amplifier. In such a configuration we either first convert a voltage to a current and
thereafter a current to a voltage, or first convert a current to a voltage and thereafter a voltage to a current.
6.4.5
Basic building blocks
Now we have examined the small-signal circuits of the common-emitter configuration, the common-collector
configuration (the emitter follower), and the common-base configuration, we note that they can be seen as
basic building blocks, having many chain matrix entries in common. The common-emitter small-signal
circuit is the only circuit that can approach the nullor, while the common-collector and common-base
circuits—obtained by suitable coupling in the common-emitter circuit, results in a good voltage follower
circuit and a good current following circuit, respectively. As such, the three configurations can be used as
building block in synthesis of more complex or cascaded designs. Figure 6.24 summarizes these results.
152
C
E
B
E
C
B
E
C
CE
B
CC
CB
gm vbe
B
C
E
B
C
E
r
r
gm vbe
gm vbe
ro
r
ro
ro
B
C
E
CE
CC
CB
-1
gmro
-1
gm
1
1
gm
1
gmro
1
gm
-1
gmror
-1
hFE
1
gmror
1
hFE
1
gmror
1
CE
CC
CB
Figure 6.24: Small-signal diagrams of basic amplifying stages composed of a single bipolar (npn) transistor
and their approximated chain matrix.
6.4.6
Differential amplifier
Another often used configuration is that of a differential pair. The differential pair produces an output
proportional to the difference between two input signals. A common form of the differential amplifier is the
long-tailed pair, which is commonly used in the input stage of operational amplifiers (op amps). To make
it possible to have two input signals the differential pair consists of two common-emitter configurations in
anti-series as shown in Fig. 6.25.
To obtain an expression for the chain matrix of the new two-port we have to convert first the chain
matrices into two-port impedance matrices; sum both impedance matrices; and then reconvert the resulting
impedance matrix into a chain matrix.
We start expressing the entities of the chain matrix in entities for a impedance matrix. Suppose the chain
matrix is expressed as:
vi = Avo + Bio
ii = Cvo + Dio .
From the second equation we obtain
1
(ii − Dio ).
C
Substituted into the first:
BC − AD
A
v i = ii +
io
C
C
or
A
Δ
v i = i i − io ,
in which Δ is the determinant.
C
C
vo =
153
B
A B
C D
+
+
-
E
C
E
C
A B
C D
+
+
B
(a)
(b)
Figure 6.25: (a) Two chain matrices anti-series connected; and (b) two CE-configurations anti-series connected.
Together with
vo =
D
1
ii − io
C
C
We can identify the corresponding impedance matrix:
⎡ A −Δ ⎤
Z=⎣
C
C
1 −D
C C
⎦.
Taking twice this matrix (we sum two identical matrices) we obtain:
⎡ 2A −2Δ ⎤
Z+Z=⎣
C
C
2
C
−2D
C
⎦.
Now, we have to convert this matrix back into a chain matrix. The conversion from the impedance
matrix to the chain matrix can be found in a similar way as we found the conversion from the chain matrix
to the impedance matrix:
⎡ E −Δ∗ ⎤
G
G
E F
⎦.
Z=
⇒K=⎣
G H
1
−H
G
G
Calculating the new entries and working out the new determinant:
∗
Δ =
2
C
2
(AD − (AD − CB)) =
2
C
2
CB =
we find the chain matrix for the differential pair:
2A
k11 =
C
2
C
−1
=A
−1
4B 2
k12 =
= 2B
C C
−1
2
C
=
k21 =
C
2
154
4B
C
k22 =
2D
C
2
C
−1
that is
=D
Kanti−series =
A 2B
.
C
D
2
(6.53)
The resulting chain matrix of the differential pair is:
⎤
⎡
− gm1ro − g2m
⎦
Kanti−series−CE ≈ ⎣
− 2gm1ro rπ − hF1E
(6.54)
and approaches a nullor in almost the same way as the single CE-stage did. As such the differential pair
can be used as an amplifier with the property that, in the unbiased configuration, the output voltage approximates:
vo ≈ −gm ro (v1 − v2 ),
(6.55)
or when biased with two the same collector resistors, see Fig. 6.26:
v3 − v4 ≈ −gm RC (v1 − v2 ).
(6.56)
+VCC
RC
RC
v3
v1
v4
v2
I
-VEE
Figure 6.26: A bipolar long-tailed pair amplifier.
6.5 pnp transistor characteristics in CE configuration
Bipolar transistors are made of three layers of doped semiconductor material. In addition to the npn transistor, the one we discussed until now, a second type exists, named pnp transistor. Both types are widely in
use and are often combined in circuits. The operation of the pnp transistor is similar, differing mainly in the
polarities of the voltages and currents involved. The small-signal equivalent circuits are similar and also the
circuits are similar, but the polarities of the various voltages and currents are reversed.
We will not treat them separately, however, Fig. 6.27 shows an example of a simple pnp voltage amplifier
for illustration.
155
R2
~
+
vbe
-
+
R1
+
VCC
-
RC
-
Figure 6.27: Simple voltage amplifier circuit based on the pnp transistor
156
Chapter
7
*Amplifier configurations III: The
field-effect transistor nullor
In Chapter 6 we examined amplifier circuits based on the bipolar junction transistor (BJT) in a so-called
common-emitter (CE) configuration. Similar configurations and circuits can be realized with field-effect
transistors (JFET, NMOS, PMOS, or CMOS transistors). Looking at the output characteristics of the fieldeffect transistors (FET) we can see strong relationship between the bipolar transistor and the FET’s. In case
of field-effect transistors the base is called gate, the emitter is called source, and the collector is called
drain. Typical expressions for the behavior of small-signal equivalent circuits, amplification, and other
transfer functions, can be obtained by the following substitutions:
vbe → vgs ,
rπ → ∞,
ro → rd ,
hF E → ∞.
(7.1)
Again, we start by the transistor characteristics; they can be found by measurement or by simulation.
7.1 FET transistor characteristics in CE configuration
7.1.1
Junction field-effect transistor (JFET)
The characteristics of a n-channel Field-Effect Transistor are shown in Fig. 7.1. The drain current is described by the following equation:
2
VGS
,
ID = IDSS 1 −
VP
(7.2)
in which IDSS is the drain-to-saturation-current, that is, the drain current when the gate-source voltage is
zero; and VP is the pinch-off voltage, the gate-source voltage at which the channel stops conducting.
7.1.2
Metal oxide field-effect transistor (MOSFET)
The characteristics of a n-channel depletion MOSFET are shown in Fig. 7.2. The drain current is described
by the following equation:
2
ID = K (VGS − VT ) ,
(7.3)
in which K is a constant that depends on the physical parameters and geometry of the device; and VT is the
threshold voltage, the gate-source voltage at which the channel stops conducting.
157
ID
(mA) 16
ID
(mA) 16
D
G
14
14
IDSS
12
VGS = 0V
S
12
10
10
8
8
6
6
VGS = -1V
4
4
VGS = -1.5V
2
2
VGS = -0.5V
VGS = VP
0
0
-1.5
-2
-0.5
-1
0
0.5
0
10
5
VGS (V)
15
VDS (V)
Figure 7.1: Characteristics of a n-channel junction FET
ID 0.8
(A)
0.7
0.6
D
D
ID 0.8
(A)
S
0.6
0.7
G
G
S
0.5
0.5
0.4
IDSS
VGS = 0V
0.4
0.3
0.3
0.2
0.
0.2
0.1
0.1
0
0
-1
VGS = 0.5V
-0.5
0
1
0.5
VGS = -0.5V
VGS = -1V
0
5
10
15
20
25
VGS=VT
30
VDS (V)
VGS (V)
Figure 7.2: Characteristics of a n-channel depletion MOSFET
The characteristics of a n-channel enhancement MOSFET are shown in Fig. 7.3. The drain current is,
just like the depletion MOSFET, described by the following equation:
2
ID = K (VGS − VT ) ,
(7.4)
in which K is a constant that depends on the physical parameters and geometry of the device; and VT is the
threshold voltage, the gate-source voltage at which the channel stops conducting.
7.2 Basic building blocks
As the treatment of the small-signal behavior of the field-effect transistor fits within the description of the
small-signal behavior of the bipolar junction transistor, the basic building blocks will behave in a similar
way. Figure 7.4 shows the building blocks and corresponding small-signal diagrams. Due to this correspondence, we expect to synthesize circuits in the same way, having more or less the same topologies (the
components to bias the transistors may differ), as the amplifier circuits using a bipolar junction transistor.
Meer later!
158
ID 200
(mA)
180
D
ID 200
(mA) 180
D
G
G
160
160
S
S
140
140
120
120
100
100
80
80
60
60
40
40
20
VGS = 2.5V
VGS = 2V
VGS = 1.5V
20
VT
0
0
0
1
0.5
1.5
2
3
2.5
10
5
0
VGS (V)
15
VDS (V)
Figure 7.3: Characteristics of a n-channel enhancement MOSFET
D
S
G
D
S
G
S
D
CS
G
CD
CG
gm v
G
D
D
S
S
G
v
v
gm v
gm v
rd
rd
v
rd
G
D
S
CS
CD
CG
-1
gmrd
-1
gm
1
1
gm
1
gmrd
1
gm
0
0
0
0
0
1
CS
CD
CG
Figure 7.4: Small-signal diagrams of basic amplifying stages composed of a single field-effect transistor
and their approximated chain matrix.
159
BJT circuit met feedback - 1
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
1
2.7 V
2.0 V
BJT circuit met feedback - 2
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
Vo ( quiescent ) | VC
2 mA
VE
RE
5.6 V
IC | I E
2.0 V
1 k:
2
2.2 k:
1.0 k:
R
v e C
RE
2.2
R
v i C
RE
BJT circuit met feedback - 3
R
C
RE
v
e RC
RE
Voor de kleinsignaal
spanningen geldt:
i c RC | i eRC
ve | vb | vi
vo
voltage gain
vo
vi
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
3
BJT circuit met feedback - 4
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
4
BJT circuit met feedback - 5
Kleinsignaal equivalent circuit wanneer een ontkoppel
capaciteit gebruikt wordt
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
5
FETs – Circuit voorbeelden 1
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
6
FETs – Circuit voorbeelden 2
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
7
FETs – Circuit voorbeelden 3
Klein-signaal gedrag van het FET circuit
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
8
voltage gain
ro | RD
ri | RG
v
o
| g m RD
vi
FETs – Circuit voorbeelden 4
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
9
FETs – Circuit voorbeelden 5
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
Circuit heeft
“automatic
bias”
10
FETs – Circuit voorbeelden 6
Tegengekoppelde n-channel enhancement MOSFET
versterker circuit.
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
11
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
12
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
13
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
14
15
Kan ook met BJT
Andere transistor circuits (optioneel)
• Bufferschakelingen (A=1)
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
v1 v 2
CMRR | g mRS
16
vo v3 v 4
vo
voltage gain
vi
v3 v4
|
g
m RD
v1 v2
vi
Andere transistor circuits (optioneel)
v3 v 4
• Differential amplifier
vo
vo
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
17
Andere transistor circuits (optioneel)
• Stroombronnen en stroomspiegel
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
•Stroomspiegel - circuit 1g.txt
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
18
19
Andere transistor circuits (optioneel)
• schakelaar
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen
versterking
Active load
20
Andere transistor circuits (optioneel)
• simplyfied CMOS opamp
Active loads
Aftrekken van
v+ - vm.b.v stroom
uit
stroomspiegel
Cursusjaar 2010/2011
EE1310 Halfgeleiders en versterkerschakelingen