CALIFORNIA INSTITUTE OF TECHNOLOGY
PHYSICS MATHEMATICS AND ASTRONOMY DIVISION
Sophomore Physics Laboratory
Physics 5 and 105 Course
Laboratory Notes
Academic Year 2012-2013
( Partial Revision December 2012)
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Copyright©Virgínio de Oliveira Sanníbale
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Caltech Physics Deparment, MS 103-33
1200 E. California Blvd.
91125 Pasadena,CA
http://www.ligo.caltech.edu/~vsanni/ph5/
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Contents
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Alternating Current Network Theory
1.1 Symbolic Representation of a Sinusoidal Signals, Phasors
1.1.1 Derivative of a Phasor . . . . . . . . . . . . . . . .
1.1.2 Integral of a Phasor . . . . . . . . . . . . . . . . . .
1.2 Network Definitions . . . . . . . . . . . . . . . . . . . . .
1.2.1 Series and Parallel . . . . . . . . . . . . . . . . . .
1.2.2 Active and Passive Components . . . . . . . . . .
1.3 Kirchhoff’s Laws . . . . . . . . . . . . . . . . . . . . . . .
1.4 Passive Ideal Components with Phasors . . . . . . . . . .
1.4.1 The Resistor . . . . . . . . . . . . . . . . . . . . . .
1.4.2 The Capacitor . . . . . . . . . . . . . . . . . . . . .
1.4.3 The Inductor . . . . . . . . . . . . . . . . . . . . .
1.5 The Impedance and Admittance Concept . . . . . . . . .
1.5.1 Impedance in Parallel and Series . . . . . . . . . .
1.5.2 Ohm’s Law for Sinusoidal Regime . . . . . . . . .
1.6 Two-Port Networks . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Bode Diagrams . . . . . . . . . . . . . . . . . . . .
1.6.2 The RC Low-Pass Filter . . . . . . . . . . . . . . .
1.6.3 The RC High-Pass Filter . . . . . . . . . . . . . . .
1.7 Ideal Sources . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Ideal Voltage Source . . . . . . . . . . . . . . . . .
1.7.2 Ideal Current Source . . . . . . . . . . . . . . . . .
1.8 Equivalent Networks . . . . . . . . . . . . . . . . . . . . .
1.8.1 Voltage Divider . . . . . . . . . . . . . . . . . . . .
1.8.2 Thévenin Theorem . . . . . . . . . . . . . . . . . .
1.8.3 Norton Theorem . . . . . . . . . . . . . . . . . . .
1.9 First Laboratory Week . . . . . . . . . . . . . . . . . . . .
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CONTENTS
4
Pre-laboratory Problems . . . . . . . . . . . . . . . . .
Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Resonant Circuits
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The LCR Series Resonant Circuit . . . . . . . . . . . . . . . .
2.2.1 Frequency Response with Capacitor Voltage Difference as Circuit Output . . . . . . . . . . . . . . . . . .
2.2.2 Frequency Response with Inductor Voltage Difference as Circuit Output . . . . . . . . . . . . . . . . . .
2.2.3 Frequency Response with the Resistor Voltage Difference as Circuit Output . . . . . . . . . . . . . . . .
2.2.4 Transient Response . . . . . . . . . . . . . . . . . . . .
2.3 The Tank Circuit or LCR Parallel Circuit. . . . . . . . . . . . .
2.3.1 LCR Circuit Frequency Response . . . . . . . . . . . .
2.3.2 Transfer Function . . . . . . . . . . . . . . . . . . . . .
2.3.3 Simplest Case . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 High Frequency Approximation . . . . . . . . . . . .
2.3.5 LCR Parallel Circuit Transient Response . . . . . . . .
2.4 Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Pre-laboratory Exercises . . . . . . . . . . . . . . . . .
2.4.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
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3 The Operational Amplifier
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Ideal Operational Amplifier . . . . . . . . . . . . . . . .
3.2.1 Ideal Op-Amp Fundamental Equation (Golden Rules)
3.2.2 Op-Amp Input Output “Logic” . . . . . . . . . . . . .
3.2.3 Op-Amp with a Feedback Network . . . . . . . . . .
3.2.4 The Virtual Ground . . . . . . . . . . . . . . . . . . . .
3.3 Commonly Used Op-Amp Circuits . . . . . . . . . . . . . . .
3.3.1 Non-Inverting Amplifier . . . . . . . . . . . . . . . . .
3.3.2 Inverting Amplifier . . . . . . . . . . . . . . . . . . . .
3.3.3 Differential Input Stage . . . . . . . . . . . . . . . . .
3.3.4 Voltage Follower (Unity Gain “Buffer”) . . . . . . . .
3.3.5 Integrator Amplifier . . . . . . . . . . . . . . . . . . .
3.3.6 Differentiator Amplifier . . . . . . . . . . . . . . . . .
3.4 Negative Feedback Amplifiers . . . . . . . . . . . . . . . . .
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1.9.1
1.9.2
CONTENTS
5
3.4.1
3.4.2
3.4.3
3.5
3.6
3.7
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Diodes and Transistors
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 The Semiconductor Junction (Diode) . . . . . . . . . . . . . . 85
4.2.1 Zener Diodes . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 Schottky Diodes . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Diode Dynamic Impedance . . . . . . . . . . . . . . . . . . . 88
4.4 Practical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.1 Rectifiers, AC to DC Conversion . . . . . . . . . . . . 89
4.4.1.1 Half-Wave Rectifier . . . . . . . . . . . . . . 89
4.4.1.2 Full-Wave Rectifier Bridge . . . . . . . . . . 91
4.4.2 Voltage Limiter (Diode Clamp) . . . . . . . . . . . . . 91
4.5 The Bipolar Junction Transistor (BJT) . . . . . . . . . . . . . . 92
4.5.1 The Collector Emitter Characteristic . . . . . . . . . . 94
4.5.2 The BJT as a Current-Controlled Current Source (CCCS)
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4.5.3 BJT Simplified DC Model . . . . . . . . . . . . . . . . 96
4.5.4 The BJT as an Amplifier . . . . . . . . . . . . . . . . . 97
4.5.4.1 BJT Amplifier Bias . . . . . . . . . . . . . . . 98
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Input Impedance Feedback Amplifier . . . . . . . .
Output Impedance Feedback Amplifier . . . . . . .
Simple Feedback Network . . . . . . . . . . . . . . .
3.4.3.1 Non-Inverting Configuration . . . . . . . .
3.4.3.2 Inverting Configuration . . . . . . . . . . .
The Real Op-Amp . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Bias Currents and Voltage and Current Offsets . . .
3.5.2 Compensated Op-Amp Transfer Function . . . . . .
3.5.2.1 Compensated Op-Amp with Constant Frequency Response Feedback . . . . . . . . .
3.5.2.2 Compensated Op-Amp in the Differentiator Configuration . . . . . . . . . . . . . .
3.5.3 The Common Mode Rejection Ratio (CMRR) . . . .
3.5.4 The Gain Bandwidth Product (GBWP) . . . . . . . .
3.5.5 The Slew Rate (SR) . . . . . . . . . . . . . . . . . . .
3.5.6 Ideal versus Real and Practical Considerations . . .
Problems Preparatory to the Laboratory . . . . . . . . . . .
Laboratory Procedure . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
6
4.5.4.2
4.5.5
4.5.6
4.5.7
BJT Amplifier Gain, Input and Output Impedance
(Low Frequency Model) . . . . . . . . . . . 99
BJT as Switch . . . . . . . . . . . . . . . . . . . . . . . 100
BJT as Diode . . . . . . . . . . . . . . . . . . . . . . . . 101
Current Mirror . . . . . . . . . . . . . . . . . . . . . . 101
5 Basic Op-Amp Applications
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Inverting Summing Stage . . . . . . . . . . . . . . . .
5.1.2 Basic Instrumentation Amplifier . . . . . . . . . . . .
5.1.3 Voltage to Current Converter (Transconductance Amplifier) . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Current to Voltage Converter (Transresistance Amplifier) . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Logarithmic Circuits . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Logarithmic Amplifier . . . . . . . . . . . . . . . . . .
5.2.2 Anti-Logarithmic Amplifier . . . . . . . . . . . . . . .
5.2.3 Analog Multiplier . . . . . . . . . . . . . . . . . . . .
5.2.4 Analog Divider . . . . . . . . . . . . . . . . . . . . . .
5.3 Multiple-Feedback Band-Pass Filter . . . . . . . . . . . . . .
5.4 Peak and Peak-to-Peak Detectors . . . . . . . . . . . . . . . .
5.5 Zero Crossing Detector . . . . . . . . . . . . . . . . . . . . . .
5.6 Analog Comparator . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Regenerative Comparator (The Schmitt Trigger) . . . . . . .
5.8 Phase Shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Generalized Impedance Converter (GIC) . . . . . . . . . . .
5.9.1 Capacitor to Inductor Converter . . . . . . . . . . . .
5.10 Problems Preparatory to the Laboratory . . . . . . . . . . . .
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6 Active Filters
6.1 Classification of Ideal Filters . . . . . . . . . . . . . . . .
6.2 Filters as Rational Functions . . . . . . . . . . . . . . . .
6.2.1 Second Order Low-Pass Filter . . . . . . . . . .
6.2.2 Second Order High-Pass Filter . . . . . . . . . .
6.2.3 Band-Pass Filter . . . . . . . . . . . . . . . . . .
6.3 Common Circuit Filters Topologies . . . . . . . . . . . .
6.4 Infinite Gain Multiple Feedback Configuration (IGMF)
6.4.1 Low-pass Filter . . . . . . . . . . . . . . . . . . .
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CONTENTS
6.6
6.7
8
Basics on Oscillators
7.1 Introduction . . . . . . . . . . . . . . .
7.2 Barkhausen Criterion . . . . . . . . . .
7.2.1 Gain Stability . . . . . . . . . .
7.2.2 Automatic Gain Control (AGC)
7.2.3 Oscillation Kick-start . . . . . .
7.2.4 Frequency Stability . . . . . . .
7.3 Phase Shift Oscillator . . . . . . . . . .
7.4 The Wien-Bridge Oscillator . . . . . .
7.5 LC Oscillator . . . . . . . . . . . . . . .
7.6 Crystal Oscillator . . . . . . . . . . . .
7.7 Relaxation Oscillators . . . . . . . . . .
7.7.1 Square Wave Generator . . . .
7.7.2 Triangular Wave Generator . .
Phase Locked Loop (PLL)
8.1 Phase Locked Loop Basic Analysis .
8.2 Phase Detector . . . . . . . . . . . . .
8.2.1 Analog Mixer Phase Detector
8.2.2 Logic Gate Phase Detector . .
8.3 Voltage Controlled Oscillator (VCO)
8.4 Varactors or Varycap . . . . . . . . .
8.5 CMOS 4046 PLL Circuit . . . . . . .
8.6 Pre-lab Problems . . . . . . . . . . .
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6.4.2 High-pass Filter . . . . . . . . . . . . . .
6.4.3 Band-pass Filter . . . . . . . . . . . . . .
Generalized Sallen-Key Filter Topology (GSK)
6.5.1 GSK Second Order Low-pass Filter . . .
6.5.2 Simple Case . . . . . . . . . . . . . . . .
6.5.3 GSK Second Order High-pass Filter . .
6.5.4 Simple Case . . . . . . . . . . . . . . . .
6.5.5 GSK Band-pass Filter . . . . . . . . . . .
6.5.6 Simple Case . . . . . . . . . . . . . . . .
State Variable Filter Topology (SV) . . . . . . .
Practical Considerations . . . . . . . . . . . . .
6.7.1 Component Values . . . . . . . . . . . .
6.7.2 Components technology . . . . . . . . .
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6.5
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CONTENTS
8
9 Final Project, Some Useful Circuits
9.1 Solderless Breadboard . . . . . . . . .
9.2 Decoupling/Bypassing Capacitor . . .
9.3 Variable Resistors . . . . . . . . . . . .
9.4 Surface Mount Devices . . . . . . . . .
9.5 Switches . . . . . . . . . . . . . . . . .
9.6 Voice Coil Loud-speakers Driver . . .
9.6.1 Off the shelf Audio Amplifiers
9.7 Microphones . . . . . . . . . . . . . . .
9.8 Phototransistors . . . . . . . . . . . . .
9.9 Photodiodes . . . . . . . . . . . . . . .
9.9.1 Photovoltaic Mode . . . . . . .
9.9.2 Photocurrent Mode . . . . . .
9.10 Ultrasonic Transceivers . . . . . . . . .
9.11 Velocity Sensors . . . . . . . . . . . . .
9.12 Position Sensors . . . . . . . . . . . . .
A Decibels
A.1 Definition of Bel and Decibel . . . .
A.2 Generalization of the Use of Decibel
A.3 Useful Table and Properties . . . . .
A.4 Standard Power References . . . . .
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B Resistor Color Code
197
C The Cathode Ray Tube Oscilloscope
C.1 The Cathode Ray Tube Oscilloscope . . .
C.1.1 The Cathode Ray Tube . . . . . . .
C.1.2 The Horizontal and Vertical Inputs
C.1.3 The Time base Generator . . . . .
C.1.4 The Trigger . . . . . . . . . . . . .
C.2 Oscilloscope Input Impedance . . . . . . .
C.3 Oscilloscope Probe . . . . . . . . . . . . .
C.3.1 Probe Frequency Compensation .
C.4 Beam Trajectory . . . . . . . . . . . . . . .
C.4.1 CRT Frequency Limit . . . . . . . .
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CONTENTS
9
D Electromagnetic Field Noise
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.2 The Faraday Cage . . . . . . . . . . . . . . . . . . . . . . . . .
D.3 Practical Considerations . . . . . . . . . . . . . . . . . . . . .
211
211
211
212
E Common Emitter BJT Amplifier
E.1 Amplifier Design . . . . . . .
E.2 Resume . . . . . . . . . . . . .
E.3 Example . . . . . . . . . . . .
E.4 Amplifier Gain and Sign . . .
E.5 Input and Output Impedance
E.6 I/O Coupling Capacitors . . .
E.7 Emitter Bypass Capacitor . .
213
214
215
215
216
216
217
218
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DR
AF
T
F Push-Pull BJT Amplifier
219
F.1 Push-Pull BJT Configuration . . . . . . . . . . . . . . . . . . . 220
F.2 Dissipation and Efficiency . . . . . . . . . . . . . . . . . . . . 221
F.3 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
CONTENTS
DR
AF
T
10
Chapter 1
Alternating Current Network
Theory
11
DR
AF
T
In this chapter we will study the properties of electronic networks propagating sinusoidal voltages and currents (alternate current/AC regime).
In other words, voltage or current sources connected to the networks produce electromagnetic waves whose frequency can be ideally changed from
0 to ∞.
Considering an electromagnetic wave whose frequency f is 1MHz, (
typical maximum frequency f used in this course of analog electronics)
and the electromagnetic field propagates at a speed c = 3 · 108 m/s (1’/ns),
the wavelength λ = c/ f will be usually much greater than several hundred feet, hundreds to thousands of times greater than the physical sizes
of our electronic circuits. Consequently, each individual circuit component
will have at any instant, to a high degree of accuracy, zero net total current
flowing in or out of all its connections and components. Such components
are known as a lumped elements.
When fields wavelength becomes comparable to the size of the circuit
components, fields and currents can vary across the element, making it
a distributed element. Examples of distributed elements include antennas,
microwave waveguides, and the electrical power distribution grid.
For two-terminal lumped elements, we can conclude that at any instant, the current flowing out of one terminal must equal the current flowing into the other, so we can simply refer to the current flowing through the
element and the potential difference (voltage difference or voltage drop)
between the two terminals.
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
12
In general, if we have a sinusoidal signal (sinusoidal voltages, or currents) applied to a circuit having at least one input and one output, we will
expect a change in the amplitude and phase at the output. The determination of these quantities for quite simple circuits can be very complex. It
is indeed important to develop a convenient representation of sinusoidal
signals to simplify the analysis of circuits in this AC regime.
The AC analysis of such circuits is valid once the network is at the
steady state, i.e. when the transient behavior (such as those ones produced
by closing or opening switches) is extinguished.
1.1 Symbolic Representation of a Sinusoidal Signals, Phasors
A sinusoidal quantity (a sinusoidal current or voltage for example) ,
A(t) = A0 sin(ωt + ϕ),
or
~ = x + jy,
A
AF
T
is univocally characterized by the amplitude A0 , the angular frequency ω,
and the initial phase ϕ. The phase ϕ corresponds to a given time shift t∗ of
the sinusoid (ωt∗ = ϕ ⇒ t∗ = ϕ/ω ).
~ in the complex
We can indeed associate to A(t) an applied vector A
plane with modulus | A| = A0 ≥ 0, rotating counter-clockwise around the
origin O with angular frequency ω and initial angle ϕ (see figure 1.1). Such
vectors are called phasors.
The complex representation of the phasor is1
√
~ = A0 e j(ωt+ ϕ),
A
j = −1,
x = A0 cos(ωt + ϕ)
y = A0 sin(ωt + ϕ)
1 To
DR
Extracting the real and the imaginary part of the phasor, we can easily
compute its amplitude A0 and phase ϕ, i.e.
!
q
~]
ℑ[
A
~ ]2 + ℑ[ A
~ ]2
ϕ = arg [ A] = arctan
| A| = ℜ[ A
( t = 0 ),
~]
ℜ[ A
avoid confusion with the electric current symbol i, it is convenient to use the symbol j for the imaginary unit.
1.1. SYMBOLIC REPRESENTATION OF A SINUSOIDAL SIGNALS, PHASORS13
A
A0
A(t)
jy
ϕ+ω t
A(t)
0
t
A(0)
ϕ
t*=ϕ/ω
0
x
T= 2π/ω
~ at the
Figure 1.1: Sinusoidal quantityA(t) and its phasor representation A
initial time t = 0 and at time t.
and reconstruct the real sinusoidal quantity. It is worth noting that in general, amplitude A0 and phase ϕ are functions of the frequency.
The convenience of this representation will be evident, once we consider the operation of derivation and integration of a phasor.
1.1.1 Derivative of a Phasor
~ we get
Computing the derivative of a phasor A,
~
dA
~
= jωA0 e j(ωt+ ϕ) = jω A,
dt
1.1.2 Integral of a Phasor
~ is
The integral of a phasor A
Z t
t0
AF
T
i.e. the derivative of a phasor is equal to the phasor times jω.
i
h
~ ′ = 1 A0 e j(ωt+ ϕ) − A0 e j(ωt0+ ϕ) = 1 A
~ + const.,
Adt
jω
jω
DR
i.e. the integral of a phasor is equal to the phasor divided by jω plus a
constant. For the AC regime we can assume the constant to be equal to
zero without loss of generality.
14
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
Figure 1.2: Generic representation of a network
1.2 Network Definitions
DR
AF
T
To make easier the understanding of network basic theorems, some more
or less intuitive definitions must be stated.
An electronic network or circuit is a set of electronic components/devices
connected together to modify and transmit/transfer energy/information.
This information is generally called an electric signal or simply a signal.
To graphically represent a network, we use a set of coded symbols with
terminals for the network lumped elements and lines for connections. These
lines propagate the signal among the elements without changing it. The
elements change the propagation of the electric signals.
A network node is a point where more than two network lines connect.
A network loop, or mesh, is any closed network line. To determine a mesh
it is sufficient to start from any point of the circuit and come back running
trough the network to the same point without passing through the same
point.
Quantities defining signal propagation are voltages V across the elements and currents I flowing through them.
Solving an electronic network means determining the currents or the
voltages of each point of it.
Figure 1.2 shows a generic portion of a network with 4 visible nodes,
and 3 visible meshes. The empty boxes are the electronic elements of the
1.2. NETWORK DEFINITIONS
15
network and their size is just for convenience and don’t correspond to any
physical dimensions. These are the points in the network where voltages
and currents are modified.
1.2.1 Series and Parallel
Let’s consider the two different connection topologies shown in figure 1.3,
the parallel and the series connections.
A set of components is said to be in series if the current flowing through
them and anywhere in the circuit is the same .
A set of components is said to be in parallel if the voltage difference
between them is the same.
A
A
B
B
Figure 1.3: Considering the points A, and B, the components of the left circuit
1.2.2 Active and Passive Components
AF
T
are in parallel, and those in the right circuit are in series.
DR
Circuit components can be divided into two categories: active and passive
components. Active components are those devices that feed energy into
the network. Voltage and current sources are active components. Amplifiers are also active components.
Passive components are those components, that do not feed energy to
the network. Resistors, capacitors, inductors are typical passive components.
In general, both active and passive components dissipate energy.
16
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
1.3 Kirchhoff’s Laws
Kirchhoff’s laws, are fundamental for the solution of an electronic circuit.
They can be derived from Maxwell’s equations in the approximation of
slowly varying field.
Kirchhoff’s Voltage Law (KVL): The algebraic sum of the voltage difference vk at the time t around a loop must be equal to zero at all times, i.e.
∑ vk (t) = 0
k
Kirchhoff’s Current Law(KCL): The algebraic sum of the currents ik at the
time t entering and leaving a node must be equal to zero at all times, i.e.
∑ ik (t) = 0.
k
These laws hold for phasors as well.
1.4 Passive Ideal Components with Phasors
Let’s rewrite the I-V characteristic for the passive ideal components using
the phasor notation. For sake of simplicity, we remove the arrow above the
phasor symbol. To avoid ambiguity, we will use upper case letters to indicate phasors, and lower case letters to indicate a generic time dependent
signal.
1.4.1 The Resistor
v(t) = Ri (t).
jω
AF
T
For time dependent signals, Ohm’s law for a resistor with resistance R is
t=0
R
V
I
V
DR
I
1.4. PASSIVE IDEAL COMPONENTS WITH PHASORS
17
Introducing the phasor I = I0 e jωt (see figure above), we get
v(t) = RI0 e jωt ,
and in the phasor notation
V = R I.
The frequency and time dependence are implicitly contained in the
phasor current I.
1.4.2 The Capacitor
The variation of the voltage difference dv across a capacitor with capacitance C due to the amount of charge dQ, is
dv =
dQ
.
C
If the variation happens in a time dt and
i (t) =
dQ
,
dt
we will have
⇒
1
v(t) =
C
Z t
0
i (t′ )dt′ + v(0).
jω
t=0
I
C
V
I
DR
V
AF
T
dv(t)
1
= i ( t ),
dt
C
For sinusoidal time dependence , we introduce the phasor I = I0 e jωt
(see figure above), and get
18
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
1 t jωt′ ′
v(t) =
I0 e dt + v(0),
C 0
Using the phasor notation and supposing that for t = 0 the capacitor is
discharged, we finally get
Z
V=
1
I,
jωC
, v(0) = 0.
1.4.3 The Inductor
The induced voltage v(t) of an inductor with inductance L, is
v(t) = L
di (t)
.
dt
jω
V
I
L
t=0
I
V
Introducing the phasor I = I0 e jωt (see figure above), we get
v(t) = L
d
I0 e jωt ,
dt
V = jωL I.
AF
T
and in the phasor notation
1.5 The Impedance and Admittance Concept
Z (ω ) =
DR
Let’s consider a generic circuit with a port, whose voltage difference and
current are respectively the phasors V = V0 e j(ωt+ ϕ), and I = I0 e j(ωt+ψ).
The ratio Z between the voltage difference and the current
V
V
= 0 e j( ϕ−ψ) .
I
I0
1.5. THE IMPEDANCE AND ADMITTANCE CONCEPT
19
is said to be the impedance of the circuit.
The inverse
1
Y (ω ) =
Z (ω )
is called the admittance of the circuit.
For example, considering the results of the previous subsection, the
impedance of a resistor, a capacitor, and an inductor are respectively
ZR = R,
ZC (ω ) =
1
,
jωC
ZL (ω ) = jωL,
and the admittances are
YR =
1
,
R
YC (ω ) = jωC,
YL (ω ) =
1
.
jωL
In general, the impedance or admittance of a circuit port is a complex
function, which depends on the angular frequency ω. Quite often, they
are graphically represented by plotting the magnitude| Z(ω )| or |Y (ω )| in a
double logarithmic scale and the phase arg [ Z(ω )] or arg [Y (ω )] in a logarithmic scale.
For completeness, letŽs introduce some other definitions:
• The real part R of the impedance Z(ω ) is called resistance.
• The imaginary part X of the impedance Z(ω ) is called reactance. If
X > 0, then X is an inductive reactance (from the fact that the reactance
of an inductor is ωL ≥ 0). If X < 0 then X is a capacitive reactance (
the reactance of an capacitor is −1/ωC ≤ 0).
AF
T
• The real part G of the admittance Y (ω ) is called conductance.
• The imaginary B part of the admittance Y (ω ) is called susceptance. If
B < 0, then is B an inductive susceptance. If B > 0 then B is a capacitive
susceptance.
1.5.1 Impedance in Parallel and Series
Ztot = Z1 + Z2 + ... + ZN ,
DR
It can be easily demonstrated that the same laws for the total resistance of
a series or a parallel of resistors hold for the impedance
(impedances in series)
20
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
1
1
1
1
=
+
+ ... +
,
Ztot
Z1 Z1
ZN
(impedances in parallel)
It is left as exercise to derive the homologue laws for the admittance.
1.5.2 Ohm’s Law for Sinusoidal Regime
Thanks to the impedance concept, we can generalize Ohm’s law and write
the fundamental equation (Ohm’s law for sinusoidal regime)
V ( ω ) = Z ( ω ) I ( ω ).
1.6 Two-Port Networks
+
A
Io
Ii
H(ω)
Port 1 Vi
Ii
−
C
+
Vo Port 2
Io
B
D
−
Figure 1.4: Two-port network circuit representation. The voltage difference signs and current directions follow standard conventions.
DR
AF
T
A terminal port is a terminal pair of a linear circuit whose input current
equals the output current. A linear circuit with one pair of input terminals A, B and one pair output terminals C, D is called two-port network (see
figure 1.4). The electronic circuits we will consider here, are two-port network with terminals B and C connected together[2]. In this case, to characterize the behavior of a two-port network, we can study the response
of the output Vo as a function of the angular frequency ω of a sinusoidal
input Vi .
In general, we can write
Vo (ω ) = H (ω )Vi (ω ),
or
H (ω ) =
Vo (ω )
,
Vi (ω )
1.6. TWO-PORT NETWORKS
21
where the complex function H (ω ) is called the transfer function or frequency
response of the two-port network. The transfer function contains the information of how the amplitude and the phase of the input changes when it
reaches the output. Knowing the transfer function of this particular twoport network, we characterize the circuit2 . The definition of H (ω ) suggests
a way of measuring the transfer function. In fact, exciting the input with
a sinusoidal wave, we can measure at the output the amplitude and the
phase lead or lag respect to the input signal.
1.6.1 Bode Diagrams
To graphically represent H (ω ), it is common practice to plot the magnitude | H (ω )| (gain) in a double logarithmic scale, and the phase arg [ H (ω )]
using a logarithmic scale for the angular frequency. These plots are called
Bode diagrams. Units for ω are normally rad/s or Hz. The magnitude is
quite often expressed in decibels dB (see appendix A)
XdB = 20 log10 X
2A
AF
T
For example 20dB = 10, 40dB=100, etc... Practically, plotting a quantity
in dB (which is not a units symbol such as m,s,kg, Ω ) corresponds to plot
the quantity on a logarithmic scale.
The phase can be expressed in radians (rad) or in degrees (deg).
Logarithmic scales have the advantage of emphasizing asymptotic trends
and the disadvantage of flattening small variations. In other words, variations much smaller that the range of the plotted values become quite often
indistinguishable in a logarithmic scale. To find out such kind of behavior,
it is a good practice to look at magnitudes in both linear and logarithmic
plots.
The Asymptotic Bode diagram [1], a simplification of the frequency
response of a system is a convenient approximation of the characteristics
of H (ω ).
DR
thorough understanding of the transfer function of a circuit requires the concept
of the Fourier transform and the Laplace transform and the convolution theorem. See [2]
appendix C
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
22
I
R
Vin
I
Vout
C
Figure 1.5: RC low-pass filter circuit.
1.6.2 The RC Low-Pass Filter
Figure 1.5 shows the RC low-pass filter circuit. The input and output voltage
differences are respectively3
1
Vin = Zin I = R +
I,
jωC
1
I,
Vout = Zout I =
jωC
and the transfer function is indeed
H (ω ) =
Vout
1
,
=
Vin
1 + jτω
or
H (ω ) =
1
,
1 + jω/ω0
τ = RC.
ω0 =
1
.
RC
| H (ω )| = √
1
AF
T
Computing the magnitude and phase of H (ω ), we obtain
2
1 + τ2ω
arg( H (ω )) = − arctan
ω
ω0
3V
out
DR
Figure 1.6 shows the magnitude and phase of H (ω ). The parameter τ
and ω0 are called respectively the time constant and angular cut-off frequency
as function of Vin can be directly calculated using the voltage divider equation.
1.6. TWO-PORT NETWORKS
23
0
Magnitude (abs)
10
−1
10
Phase (deg)
0
−45
−90
8
10
9
10
Frequency (rad/sec)
10
10
Figure 1.6: RC low-pass filter circuit transfer function.
of the circuit. The cut-off frequency
is the frequency where the output Vout
√
is attenuated by a factor 1/ 2.
AF
T
It is worthwhile to analyze the qualitative behavior of the capacitor
voltage difference Vout at very low frequency and at very high frequency.
For very low frequency the capacitor is an open circuit and Vout is essentially equal to Vin . For high frequency the capacitor acts like a short
circuit and Vout goes to zero.
DR
The capacitor produces also a delay as shown in the phase plot. At
very low frequency the Vout follows Vin (they have the same phase). The
output Vout loses phase (ωt = ϕ ⇒ t = ϕ/ω) when the frequency increases. The output Vout starts lagging due to the negative phase ϕ, and
then approaches a maximum delay at a phase shift of −π/2.
24
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
I
C
Vin
I
R
Vout
Figure 1.7: RC high-pass filter circuit
1.6.3 The RC High-Pass Filter
Figure 1.7 shows the RC high-pass filter circuit. The input and the output
voltage differences are respectively
Vin
Vout
1
= Zin I =
R+
jωC
= Zout I = RI,
I,
and indeed the transfer function is
H (ω ) =
Vout
jωτ
=
,
Vin
1 + jτω
τ = RC.
or
jω/ω0
,
1 + jω/ω0
ω0 =
1
.
RC
AF
T
H (ω ) =
Computing the magnitude and phase of H (ω ), we obtain
τω
,
1 + τ 2 ω2
ω 0
arg( H (ω )) = arctan
ω
DR
| H (ω )| = √
Figure 1.8 shows the magnitude and phase of H (ω ). The definitions in
the previous subsection, for τ and ω0 , hold for the RC high-pass filter.
1.7. IDEAL SOURCES
25
0
Magnitude (abs)
10
Phase (deg)
90
45
0
0
10
1
10
Frequency (rad/sec)
2
10
Figure 1.8: RC high-pass filter circuit transfer function.
1.7 Ideal Sources
1.7.1 Ideal Voltage Source
DR
AF
T
An ideal voltage source is a source able to deliver a given voltage difference Vs between its leads independent of the load R attached to it (see
figure 1.9). It follows from Ohm’s law that a voltage source is able to produce the current I necessary to keep constant the voltage difference Vs
across the load R. The symbol for the ideal voltage source is shown in
figure 1.9.
Quite often, a real voltage source exhibits a linear dependence on the
resistive load R. It can be represented using an ideal voltage source Vs in
series with a resistor Rs called input resistance of the source. Applying
Ohm’s law, it can be easily shown that the voltage and current through
the load R are
R
Vs
V=
Vs ,
I=
.
R + Rs
R + Rs
26
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
If we assume
R ≫ Rs ,
V ≃ Vs ,
⇒
I≃
Vs
.
R
Under the previous condition, the real voltage source approximates the
ideal case.
V
+
I
Vs
R V
Vs
−
I
Figure 1.9: Ideal voltage source.
1.7.2 Ideal Current Source
I=
Rs
Is ,
R + Rs
V=
If we suppose
⇒
I ≃ Is ,
Rs R
Is .
Rs + R
V ≃ RIs ≫ 0
DR
Rs ≫ R,
AF
T
An ideal current source is a source able to deliver a given current Is that
does not depend on the load R attached to it (see figure 1.10). It follows
from Ohm’s law that an ideal current source is able to produce the voltage
difference V across the load R needed to keep Is constant. The symbol for
the ideal current source is shown in figure 1.10.
A real current source exhibits a dependence on the resistive load R,
which can be represented using an ideal current source Is in parallel with
a resistor Rs . Applying Ohm’s law and the KCL, it can be easily shown
that the voltage and current through the load R are
Under the previous condition, the real current source approximates the
ideal case.
1.8. EQUIVALENT NETWORKS
27
V
Is
I
R V
Vs
Is
I
Figure 1.10: Ideal current source.
1.8 Equivalent Networks
Quite often, the analysis of a network becomes easier by replacing part
of it with an equivalent and simpler network or dividing it into simpler
sub-networks.
For example, the voltage divider is an equation easy to remember that
allows to divide a complex circuit in two parts simplifying the search of
the solution.
Thévenin and Norton theorems give us two methods to calculate equivalent circuits which behave like the original circuit, as seen from two points
of it. The techniques briefly explained in this section, will be then extensively used in the next chapters .
1.8.1 Voltage Divider
VTot = (Z1 + Z2 ) I
V2 = Z2 I
V2 =
Z2
VTot ,
Z1 + Z2
which is the voltage divider equation.
DR
and indeed
AF
T
The voltage divider equation is applicable every time we have a circuit
which can be re-conducted to a series of two simple or complex components. Considering the circuit branch of figure 1.11 and applying Ohm’s
law, we have
28
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
VTot
Z1
I
V2
Z2
Figure 1.11: Voltage divider circuit.
1.8.2 Thévenin Theorem
Thévenin theorem allows us to find an equivalent circuit for a network
seen from two points A and B using an ideal voltage source VTh in series
with an impedance ZTh .
ZTh
+
Black Box
VTh
ZL
ZL
−
AF
T
Figure 1.12: Thévenin equivalent circuit illustration.
DR
The equivalence means that if we place a load ZL between A and B
in the original circuit (see figure 1.12) and measure the voltage VL and the
current IL across the load, we will obtain exactly the same VL and IL if ZL is
placed in the equivalent circuit. This must be true for any load we connect
between the points A and B .
The previous statement and the linearity of the circuit can be used to
find VTh and ZTh . In fact, if we consider ZL = ∞ (open circuit, OC), we
will have
VTh = VOC .
1.8. EQUIVALENT NETWORKS
29
The Voltage VTh is just the voltage difference between the two leads A
and B.
For ZL = 0 (short circuit, SC) we must have
ISC =
VTh
V
= OC .
ZTh
ZTh
and therefore
VOC
.
ISC
The last expression says that the Thévenin impedance is the impedance
seen from the points A and B of the original circuit.
If the circuit is known, the Thévenin parameter can be calculated in
the case for the terminals A and B open. In fact, VTh is just the voltage
across A and B of a known circuit. Replacing the ideal voltage sources
with short circuits (their resistance is zero) and ideal current sources with
open circuits ( their resistance is infinite), we can calculate the impedance
ZTh seen from terminals A and B.
Considering the previous results, we can finally state Thévenin theorem as follows:
Any circuit seen from two points can be replaced by an ideal voltage source of
voltage VTh in series with impedance ZTh . VTh is the voltage difference between
the two point of the original circuit. ZTh is the impedance seen from these two
points, short-circuiting all the ideal voltage generators and open-circuiting all the
ideal current generators.
ZTh =
Example:
VTh =
R1
Vs .
R1 + R2
AF
T
We want to find the Thévenin circuit of the network enclosed in the gray
rectangle of figure 1.13. To find RTh , and VTh we have to disconnect the
circuit in the points A and B. In this case, the voltage difference between
these two points, thanks to the voltage divider equation, is
RTh =
DR
Short circuiting Vs we will have R1 in parallel with R2 . The Thévenin
resistance RTh will be indeed
R1 R2
.
R1 + R2
30
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
R2
A
+
R Th
R0
Vs
−
+
R1
V0
A
R0
+
+
V0
VTh
−
−
−
B
B
Figure 1.13: Thévenin equivalent circuit example
Black Box
ZL
I No
ZNo
ZL
Figure 1.14: Norton equivalent circuit illustration.
1.8.3 Norton Theorem
DR
AF
T
Any kind of active network seen from two points A and B can by replaced by an
ideal current generator INo in parallel with a impedance ZNo . The current INo
corresponds to the short-circuit current of the two points A and B. The Resistance
ZNo is the Thévenin resistance ZNo = ZTh .
The proof of this theorem is left as exercise.
1.9. FIRST LABORATORY WEEK
31
1.9 First Laboratory Week
This first laboratory class is essentially intended to help the student to become acquainted with the laboratory instrumentation (function generator,
digital multimeter, circuit bread board, connectors), and with the use of
passive components and their mathematical descriptions.
1.9.1 Pre-laboratory Problems
To complete the preparatory problems, it is recommended to read sections
1.1 to 1.6, and appendix C for the laboratory procedure.
CH1
CH2
AF
T
CH2
CH1
DR
Vertical Sensitivity [50 mV/div], 1div = distance between thick lines
1. Considering the figure below (a “snapshot” of an oscilloscope display), determine the peak to peak amplitude, the DC offset, the frequency of the two sinusoidal curves, and the phase shift between
the two curves (channels horizontal axis position is indicated by an
arrow and the channel name on the right of the figure).
Horizontal Sensitivity [1 ms/div], 1div= distance between thick lines
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
1. Sketch in a graph the magnitude and phase of the RC series circuit input impedance Zi . Determine the two asymptotic behavior
of | Zi | ,i.e. where the resistive and the capacitive behavior dominate.
2. Assuming that R2 ≫ R1 , repeat the previous problem for the following circuit
I
R1
Vi
C
R2
Vo
Hint: in this case there are three dominant behaviors, two resistive
and one capacitive, and two frequencies which separate the three
dominant behaviors.
3. The circuit below includes the impedance of the input channel of the
CRT oscilloscope, and Vs is indeed the real voltage measured by the
instrument.
I
R
Vin
C
VC
Cs
Rs
Vs
Oscilloscope Input
Stage
C ≫ Cs ,
AF
T
Find the voltage Vs (ω ) , and the angular cut-off frequency ω0 of the
transfer
function Vs /Vin ( i.e. the value of ω for which |Vs /Vin | =
√
1/ 2).
Show that for ω = 0 the Vs (ω ) formula simplifies and becomes the
resistive voltage divider equation.
Demonstrate that the conditions to neglect the input impedance of
the oscilloscope are the following :
R ≪ Rs
DR
32
(Hint: use the voltage divider equation to write VC .)
1.9. FIRST LABORATORY WEEK
33
4. Considering the previous circuit, calculate the value of R needed
to obtain Vin ≃ Vs with a fractional systematic error of 1%, if ω =
0rad/s and Rs = 1MΩ.
5. Determine the values of R, and C needed to get a RC series circuit
cut-off frequency of 20kHz. Choose values which make the oscilloscope impedance negligible (Rs = 1MΩ, Cs = 25pF).
1.9.2 Procedure
Whenever you work with electronic circuits as a beginner (all students are
considered beginners; it doesn’t matter which personal skills they already
have), some extra precautions must be taken to avoid injuries. These are
the main ones:
• N EVER
CONNECT INSTRUMENT PROBES OR LEADS TO AN OUTLET
OR MORE IN GENERAL TO THE POWER LINE .
• DO
NOT TRY TO FIX / IMPROVE AN INSTRUMENT BY YOURSELF.
• DO
NOT POWER AN INSTRUMENT WHICH IS NOT WORKING OR DIS -
ASSEMBLED .
• D O NOT
TOUCH A DISASSEMBLED OR PARTIALLY DISASSEMBLED IN -
STRUMENT, EVEN IF IT IS NOT POWERED .
• TO
PROTECTIVE GOGGLES EVERY TIME YOU USE A SOLDERING
AF
T
• W EAR
IRON .
AVOID EXPLOSIONS , NEVER USE A SOLDERING IRON ON A POW-
ERED CIRCUIT AND BATTERIES .
• P LACE
WORK .
A FAN TO DISPERSE SOLDER VAPORS DURING SOLDERING
DR
Read carefully the text before starting the laboratory measurements.
BNC cables and wires terminated with banana connectors are available
to connect the circuit under measurement to the instruments.
34
CHAPTER 1. ALTERNATING CURRENT NETWORK THEORY
BNC4 cables, a diffused type of radio frequency (RF) coaxial shielded
cable, have an intrinsic and quite well defined capacitance due to their
geometry as shown in the figure below
A
C
A
C
C0
B
D
B
D
They have typical linear density capacitance ∆C/∆l ≃= 98pF/m. Wires
terminated with banana connectors have smaller capacitance than BNC
cables but not as constant as BNC cables (Why?).
1. Build a RC low-pass or a RC high-pass filter with a cut-off frequency
between 10kHz to 100kHz. Choose components values which make
the perturbation of the input impedance of the oscilloscope negligible. Then, do the following
(a) Verify the circuit transfer function, and in particular for frequencies ν ≪ ν0 and ν ≫ ν0
(b) Find the cut-off frequency νo knowing the expected magnitude
and phase values, and compare with the theoretical values
(c) Change the capacitor value to be comparable to the oscilloscope
input capacitance or the BNC cable capacitance and experimentally find the new cut-off frequency ν1 . Compare with the theoretical value.
AF
T
(d) Drive the circuit input with a square wave and verify the transient response of the circuit.
2. Measure the output impedance of the function generator for a given
fixed sinusoidal frequency. Place at least three different loads to perform the measurement.
4 “BNC”
DR
seems to stand for Bayonet Neill Concelman (named after Amphenol engineer Carl Concelman). Other sources claims that the acronym means British Navy Connector. What is certain is that the BNC connector was developed in the late 1940’s as a
miniature version of the type C connector (what does the “C” stand for ?)
Bibliography
[1] Ref for Maxwell equations versus Kirchhoff’s laws.
35
DR
AF
T
[2] Microelectronics, Jacob Millman, and Arvin Grabel , Mac-Graw Hill
BIBLIOGRAPHY
DR
AF
T
36
Chapter 2
Resonant Circuits
2.1 Introduction
Resonators, one of the most useful and used device, are essentially physical systems that present a more or less pronounced peak in their transfer
function.
In general, their performance is measured by a dimensionless parameter named quality factor Q, which characterizes the sharpness of the resonant peak. The higher the quality factor the sharper is the peak and the
better is the resonator.
AF
T
Quite often, the major issues of building a resonator are to obtain very
high quality factors and good stability. For example, mechanical oscillators made of fused silica fibers under load, can achieve quality factors
above 108 in the acoustic band[?]. Very high quality factors in electronics
can be achieved using the mechanical resonances of piezoelectric materials
such as quartz. Lasers and resonant cavities made of mirrors can be used
to build resonators in the optical frequency range. The same principle can
be applied in the microwave range. Thermal stabilization is always a key
ingredient to obtain high stability.
37
DR
Resonators made with electronic passive components, reaching quality
factors values up to 10-100 or more, are quite easy to realize. In the next
sections we will study two typical resonant circuits, the LCR series and
LCR parallel circuits.
CHAPTER 2. RESONANT CIRCUITS
38
Vi
R
VR
L
VL
C
VC
Figure 2.1: LCR series circuit.
2.2 The LCR Series Resonant Circuit
Figure 2.1 shows the so called LCR series resonant circuit. Depending on
voltage difference, we are considering as the circuit output ( the capacitor,
the resistor, or the inductor), this circuit shows a different behavior. Let’s
study in the frequency and in the time domain the response of this passive
circuit for each one of the possible outputs.
2.2.1 Frequency Response with Capacitor Voltage Difference as Circuit Output
HC ( ω ) =
1
.
jωRC − ω2 LC + 1
DR
and the transfer function will be
AF
T
Considering the voltage difference VC across the capacitor to be the circuit
output, we will have
1
Vin =
R + jωL +
I,
jωC
1
I,
VC =
jωC
2.2. THE LCR SERIES RESONANT CIRCUIT
39
For sake of simplicity, it is convenient to define the two following quantities
r
1 L
L
1
2
,
Q=
= ω0
ω0 =
LC
R C
R
The parameter Q is the quality factor of the circuit, and the angular
frequency ω0 is the resonant frequency of the circuit if R = 0.
Considering the previous definitions, and after some algebra, HC (ω )
becomes
ω02
HC ( ω ) = 2
.
ω0 − ω2 + jω ωQ0
(2.1)
Computing the magnitude and phase of HC (ω ), we obtain
| HC (ω )| = r
ω02
ω02
2
− ω2
arg [ HC (ω )] = − arctan
+
ω ωQ0
1 ω0 ω
Q ω02 − ω2
2 ,
!
.
The magnitude has maximum for
1
2
2
ω C = ω0 1 −
,
2Q2
and the maximum is
| HC (ωC )| = q
Q
1−
1
4Q2
.
AF
T
If Q ≫ 1 then ωC ≃ ω0 , and | HC (ωC )| ≃ Q.
Far from resonance ωC , the approximate behavior of | HC (ω )| is
⇒
DR
| HC (ω )| ≃ 1 ,
ω2
ω ≫ ωC
⇒
| HC (ω )| ≃ 02 .
ω
Figure 2.2 shows the magnitude and phase of HC (ω ). In this case, the
circuit is a low pass filter of the second order because of the asymptotic
slope 1/ω2 .
ω ≪ ωC
CHAPTER 2. RESONANT CIRCUITS
40
Bode Diagram
20
Magnitude (dB)
10
0
−10
−20
−30
−40
0
Phase (deg)
−45
−90
−135
−180
4
10
5
6
10
Frequency (rad/sec)
10
Figure 2.2: Transfer function HC (ω ) of the LCR series resonant circuit with
a resonant angular frequency ωC ≃ 10.7krad/s.
2.2.2 Frequency Response with Inductor Voltage Difference
as Circuit Output
H L (ω ) = −
AF
T
Let’s considering now the voltage difference VL across the inductor as the
circuit output. In this case, we have
ω2 LC
.
jωRC − ω2 LC + 1
Using the definition of Q, and ω0 and after some algebra, HL (ω ) becomes
−ω2
ω02 − ω2 + jω ωQ0
DR
H L (ω ) =
(2.2)
2.2. THE LCR SERIES RESONANT CIRCUIT
41
Computing the magnitude and phase of HL (ω ), we obtain
| HL (ω )| = r
ω2
2
2
+ ω ωQ0
!
1 ωω0
Q ω02 − ω2
ω02 − ω2
arg [ HL (ω )] = arctan
The magnitude has a maximum for
ω2L = ω02
1
,
1
1 − 2Q
2
and the maximum is
| HL (ω L )| = q
Q
1−
1
4Q2
.
Again, if Q ≫ 1 then ω L ≃ ω0 , and | HL (ω L )| ≃ Q.
Far from resonance ω L , the approximate behavior of | HL (ω )| is
ω ≪ ωL
⇒
ω ≫ ωL
⇒
ω2
| HL (ω )| ≃ 2
ω0
| HL (ω )| ≃ 1
AF
T
Figure 2.3 shows the magnitude and phase of HL (ω ). In this case the
circuit is a second order high pass filter.
2.2.3 Frequency Response with the Resistor Voltage Difference as Circuit Output
DR
Considering the voltage difference across the resistor as the circuit output,
we will have
jωRC
HR (ω ) =
.
1 − ω2 LC + jωRC
Using the definition of Q and ω0 , and after some algebra, HR (ω ) becomes
CHAPTER 2. RESONANT CIRCUITS
42
Bode Diagram
40
Magnitude (dB)
20
0
−20
−40
−60
180
Phase (deg)
135
90
45
0
4
10
5
6
10
Frequency (rad/sec)
10
Figure 2.3: Transfer function HL (ω ) of the LCR series resonant circuit with
a resonant angular frequency ω L ≃ 10.7krad/s.
ω02 − ω2 + jω ωQ0
.
Computing the magnitude and phase of HR (ω ), we obtain
| HR (ω )| = r
ω0
Qω
2
+ ω ωQ0
!
ω02 − ω2
arg [ HR (ω )] = arctan Q
ωω0
2
DR
ω02 − ω2
(2.3)
AF
T
HR (ω ) =
jω ωQ0
2.2. THE LCR SERIES RESONANT CIRCUIT
43
The magnitude has maximum for
ω2R = ω02 ,
and the maximum is
| HR (ωR )| = 1 .
Far from the resonance ωR , the approximate behavior of | HR (ω )| is
ω ≪ ωR
⇒
ω ≫ ωR
⇒
1 ω
Q ω0
ω0
| HR (ω )| ≃
ω
| HR (ω )| ≃
Figure 2.4 shows the magnitude and phase of HR (ω ). In this case the
circuit is a first order band pass filter.
2.2.4 Transient Response
The equation that describes the LCR series circuit response in the time
domain is
Z
di
1 t ′ ′
vi = Ri + L +
i (t )dt ,
(2.4)
dt C 0
AF
T
where i (t) is the current flowing through the circuit and vi (t) is the input
voltage.
Supposing that
v0 , t > 0
,
vi ( t ) =
0, t ≤ 0
and differentiating both side of eq. 2.4, we obtain the linear differential
equation
t>0
DR
di
d2 i
1
+ L 2 + i = 0,
dt
C
dt
or, considering the definition of ω0 , and Q,
R
d2 i ω0 di
+ ω02 i = 0.
+
Q dt
dt2
CHAPTER 2. RESONANT CIRCUITS
44
Bode Diagram
0
Magnitude (dB)
−10
−20
−30
−40
−50
90
Phase (deg)
45
0
−45
−90
4
10
5
6
10
Frequency (rad/sec)
10
Figure 2.4: Transfer function HR (ω ) of the LCR series resonant circuit with
resonant angular frequency ωR ≃ 10.7krad/s.
The solutions of the characteristic polynomial equation associated with
the differential equation are
p
1 ω0 1 ± 1 − 4Q2 .
2 Q
AF
T
λ1,2 = −
As usual, we will have three different solutions depending on the discriminant value
∆ = 1 − 4Q2 .
DR
Under-damped Case: discriminant less than zero (Q > 1/2)
In this case we have two complex conjugate roots and the differential equation solution is the typical exponential ring down
2.3. THE TANK CIRCUIT OR LCR PARALLEL CIRCUIT.
i ( t ) = i0 e
ω
− 2Q0 t
ωC2
sin (ωC t + ϕ0 ) ,
=
ω02
1
1−
2Q2
45
.
Critically Damped Case: Discriminant equal to zero(Q = 1/2)
In this case we have a critically damped current and no oscillation
ω0
i (t) = i0 e− 2Q t
Over-damped Case: Discriminant greater than zero (Q < 1/2)
This is the case of two coincident solutions . We will have indeed, an
exponential decay (no oscillations)
i ( t ) = i0 e
ω
− 2Q0 t
Ae
−ωC t
+ Be
+ωC t
,
ωC2
=
ω02
1
1−
2Q2
,
Voltages across each single element can be easily computed considering the relation between v(t) and i (t).
Let’s just write the voltage across the capacitor for the under-damped
case. Considering that the integration operation in this case changes just
the phase and creates an offset, the voltage across the capacitor, neglecting
this offset, will be
AF
T
ω0
vC (t) = v0 e− 2Q t sin (ωC t + ψ) .
2.3 The Tank Circuit or LCR Parallel Circuit.
DR
Figure 2.5 shows the so called LCR parallel resonant circuit or tank circuit,
where the source depicted with an arrow inside a circle is an ideal current source. The resistor of resistance r accounts for inductor resistance.
Let’s study the frequency and the transient response using the Thévenin
representation shown in figure 2.6.
CHAPTER 2. RESONANT CIRCUITS
46
L
Is
R
C
Vo
r
Figure 2.5: The tank circuit.
2.3.1 LCR Circuit Frequency Response
Using Thévenin theorem for the current source and R, the LCR parallel
circuit considering the equivalent circuit as shown in figure 2.6 where the
current source and the resistor R have been replaced with the Thévenin
circuit.
Considering that the current I of the current source can be written as
Vi
,
R
I = Y Vo =
we have
Vi
=
R
1
1
+
+ jωC Vo ,
R r + jωL
1
1
+
+ jωC Vo
R r + jωL
Defining the following complex quantity as
1
r ∗ (ω )
+
and
1
1
=
,
∗
jωL (ω )
r + jωL
R∗ = R || r ∗ ,
Vi
=
R
1
1
+
+ jωC Vo
R∗
jωL∗
DR
eq. 2.5 becomes
(2.5)
AF
T
I =
(2.6)
2.3. THE TANK CIRCUIT OR LCR PARALLEL CIRCUIT.
47
R
+
L
Vo
C
Vs
−
r
Figure 2.6: The tank circuit with the current source and the resistance R
replaced with the Thévenin equivalent circuit.
After some algebra, we will have
jωL∗
Vo
R∗
.
= ∗
Vi
R − ω2 CL∗ R∗ + jωL∗ R
(2.7)
Generalizing the definition of ω0 , and Q
ω0∗ = p
1
L∗ (ω )C
Q ∗ = R∗ (ω )
,
s
C
L∗ (ω )
,
and substituting in eq. 2.7 we finally obtain
jωω0∗ /Q∗
R∗
H (ω ) =
(ω0∗ )2 − ω2 + jωω0∗ /Q∗ R
∗
"
r (ω ) = r 1 +
ωL
r
2 #
,
r 2
ωL
i.
r 2 L (ω ) = L 1 +
ωL
DR
and finally
1
1
1
i+
h
= h
2
r + jωL
r 1 + ωL
jωL 1 +
r
AF
T
Let’s find the implicitly defined functions r ∗ , L∗ . Using the term containing the inductance L in eq. 2.6, we obtain
∗
CHAPTER 2. RESONANT CIRCUITS
48
Bode Diagram
0
Magnitude (dB)
−20
−40
−60
−80
−100
90
Phase (deg)
45
0
−45
−90
1
10
2
10
3
10
4
10
Frequency (rad/sec)
5
6
10
10
7
10
Figure 2.7: Typical bode plot of a LCR parallel circuit with resonant angular frequency near the acoustic band. As expected , if r is not zero, then
the magnitude doesn’t go to zero for ω = 0.
2.3.2 Transfer Function
| H (ω )| = rh
ωω0∗
Q∗
2
ω0∗
i2
+
ω2 − ω2
Q∗ 0
ωω0
| R∗ |
∗ 2 R
ωω
0
Q∗
)
,
DR
arg( H (ω )) = arctan
(
− ω2
AF
T
From the solution of the LCR parallel circuit we have
whose bode plots are shown in figure 2.7.
2.3. THE TANK CIRCUIT OR LCR PARALLEL CIRCUIT.
49
2.3.3 Simplest Case
If r = 0, then we will have much simpler expressions for the thank circuit
formulas, i.e.
r
1
C
ω0 = √ ,
Q=R
,
L
LC
and
H (ω ) =
ω02
jωω0 /Q
− ω2 + jωω0 /Q
.
The magnitude and the phase will be
ωω0
Q
| H (ω )| = r
2
2
,
ω02 − ω2 +
)
(
ω02 − ω2
.
arg( H (ω )) = arctan Q
ωω0
ωω0
Q
2.3.4 High Frequency Approximation
For high frequency ω ≫ r/L, we have
and ω0 becomes
2
,
⇒
L∗ ≃ L
1
ω0 ≃ √ .
LC
AF
T
L
r (ω ) ≃ r ω
r
∗
Evaluating the several defined quantities at ω0 , we will have
DR
L
,
rC
LR
R ∗ ( ω0 ) ≃
RCr + L
r
C
LR
Q ∗ ( ω0 ) ≃
RCr + L L
r ∗ ( ω0 ) ≃
CHAPTER 2. RESONANT CIRCUITS
50
Step Response
0.04
0.03
Amplitude
0.02
0.01
0
−0.01
−0.02
−0.03
0
0.2
0.4
0.6
Time (sec)
0.8
1
1.2
−3
x 10
Figure 2.8: Typical step response of a LCR parallel circuit near the acoustic
band.
2.3.5 LCR Parallel Circuit Transient Response
γ=
1
,
2Q
AF
T
Let’s briefly analyze the response to a step of the LCR parallel circuit for
the under-damped case.
If we define the following quantity
DR
called damping coefficient, and if 0 < γ < 1, then we will have at the
circuit output
2γ −ω0 γt
v ( t ) = v0
e
cos
γ−1
q
1 − γ2 ω0 t + ϕ 0
+ v1 .
2.3. THE TANK CIRCUIT OR LCR PARALLEL CIRCUIT.
51
DR
AF
T
p The voltage output v(t) is a damped sinusoid with angular frequency
1 − γ2 ω0 and time constant τ = 1/ω0 γ. The DC offset v1 depends on
the inductor resistance r and the initial step.
Figure 2.8, a typical step response of the LCR circuit shows a ring-down
with a DC offset.
52
CHAPTER 2. RESONANT CIRCUITS
2.4 Laboratory Experiment
Real inductors have not negligible resistance. To build a LCR series circuit
with a highest quality factor it is indeed necessary to minimize the resistance of the circuit by mounting in series the inductor and the capacitor
only. Typical effective resistance of the inductors used in the laboratory is
about 10Ω to 80Ω at resonance .
Because of the internal resistance of the function generator (the best
scenario gives ∼ 50Ω) is then comparable at some frequencies to LCR
load, we will expect that the approximation of ideal generator will be no
longer valid.
Moreover, harmonic distortion of the function generator will be quite
evident in the LCR series circuit because of the dependence of the load on
the frequency.
An estimation of a ring-down time constant τ can be obtained as follows. From the ring-down equation we have that after a time t = τ the
amplitude is reduced by a factor 1/3 (e ≃ 1/2.718). This means that we
can easily estimate τ by just measuring the time needed to reduce the amplitude down to about 1/3 of its initial value. A similar but quite coarse
way is to count how many periods n∗ the amplitude takes to decrease to
1/3 of its initial value. Then the estimation will be
n∗
∗
τ ≃ Tn =
,
νres
where T, and νres are respectively the period and the frequency of the oscillation. Considering that Q = πνres τ then
Q ≃ πn∗ .
DR
2.4.1 Pre-laboratory Exercises
AF
T
The quality factor can also be estimated from the frequency response
considering that
νres
Q=
,
∆ν
where ∆ν is the Full Width at Half Maximum (FWHM) of the peak resonance.
It is suggested to read the appendix about the electromagentic noise to
complete the pre-lab problems and the laboratory procedure.
2.4. LABORATORY EXPERIMENT
53
1. Determine the capacitance C of a LCR series circuit necessary to have
a resonant frequency νC = 20kHz if L = 10mH, and R = 10Ω. Then,
calculate Q, τ, ν0 , (ω = 2πν) of the circuit.
2. Find the LCR series input impedance Zi and plot its magnitude in a
logarithmic scale. Determine at what frequency is the minimum of
| Zi | .
3. Supposing that the internal resistance of the function generator is
Rs = 50Ω, and using the previous values for L, C, and R, calculate
the circuit input voltage attenuation at the frequency of | Zi | minimum and at twice that frequency.
4. Determine the capacitance C of a tank circuit necessary to have a
resonant frequency νC = 20 kHz if L = 10mH, R = 10kΩ, and
r = 10Ω. Use the high frequency approximation. Then, calculate Q,
τ, ν0 , of the circuit.
5. Estimate the time constant τ of the ring-down in figure 2.8. Supposing that R = 10kΩ, estimate r from figure 2.7.
6. Calculate the maximum frequency of the EM field isolated by a Faraday cage with a dimension d = 10mm (hint: consult the proper appendix).
2.4.2 Procedure
AF
T
1. Build a LCR series circuit with a resonant frequency of around 20kHz,
using inductance, capacitance, and resistance values calculated in
the pre-lab problems. Then, do the following steps:
(a) Using the oscilloscope and knowing the expected magnitude
and phase values at the resonant frequency νC , find νC and compare it with the theoretical value computed using the components measured values.
DR
(b) Verify the circuit transfer function HC (ν) using the data acquisition system and the proper software.
(c) Estimate the quality factor Q of the circuit from the transfer
function measurement and compare it with the theoretical value.
CHAPTER 2. RESONANT CIRCUITS
(d) Explain why the input voltage Vi changes in amplitude if we
change frequency.
(e) Considering the harmonic distortion of the function generator,
explain why the frequency spectrum of the input signal changes
quite drastically when we approach the resonance νC .
(f) Download the simulation file from the ph5/105 website for the
LCR series circuit, input the proper components values, run the
AC response simulation, and find the discrepancies beteween
your measurement and the simulation.
(g) Modify the simulated circuit to qualitatively account for the
eventual notch measured between 100 kHz and 1 MHz (hint:
use the inductor model specified in the appendix considering
the extra capacitor only).
2. Build a LCR parallel circuit with a resonant frequency around 20kHz,
using inductance, capacitance, and resistance values calculated in
the pre-lab problems. Then, do the following steps:
(a) Using the oscilloscope and knowing the expected magnitude
and phase values at the resonant frequency νC , find νC and compare it with the theoretical value computed using the components measured values.
(b) Verify the circuit transfer function HC (ν) using the data acquisition system and the proper software.
AF
T
(c) Estimate the quality factor Q of the circuit from the step response.
(d) Download the simulation file from the ph5/105 website for the
LCR parallel circuit, input the proper components values, run
the AC response simulation, and find the discrepancies beteween your measurement and the simulation.
(e) Modify the simulated circuit to qualitatively account for the
eventual notch measured between 100 kHz and 1MHz (hint:
use the capacitor model specified in the appendix considering
the extra inductors only).
DR
54
2.4. LABORATORY EXPERIMENT
55
3. Check the effect of the Faraday cage ( a metallic coffee can) using 10x
probe connected to the oscilloscope. Add a 1m long wire to increase
the antenna effect.
Note the differences when the antenna is approached to the fluorescent lights, and when you touch the antenna.
Keeping the cage in the same position and without touching the
cage, explain what you observe and coarsely estimate the amplitude
and frequency content of the picked-up signal in the following conditions:
(a) Antenna outside the cage,
(b) Antenna inside the cage,
DR
AF
T
(c) Antenna inside the cage with ground probe connected to the
cage.
AF
T
CHAPTER 2. RESONANT CIRCUITS
DR
56
Chapter 3
The Operational Amplifier
3.1 Introduction
Operational amplifiers are one of the most extensively used analog integrated circuits especially because of their ability to approximate reasonably well the ideal behavior. For this reason, real operational amplifiers
can be quite often modeled as ideal or quasi-ideal. Moreover, the versatility this device, which hides a large internal complexity 1 , makes the
operational amplifier suitable for many different applications.
AF
T
In the first section, the mathematical formulation makes the ideal operational amplifier concept quite awkward, especially after a first reading.
Using the ideal device properties in conjunction with the so called feedback network, which links the output of the amplifier inputs, will clarify
the definition of the ideal operation amplifier.
The subsequent section is dedicated to the explanation of some basic
operational amplifier circuits. Finally, section 3.5 introduces a more realistic model of operational amplifier together with some of the particular
behavior of this electronic device.
1A
57
DR
modern operational amplifier made of a cascade of stages, each one designed
mainly to match the ideal characteristics, can have around 50 components both active
and passive. See the Analog Devices Web site, for example.
CHAPTER 3. THE OPERATIONAL AMPLIFIER
58
3.2 The Ideal Operational Amplifier
The ideal operational amplifier (Op-Amp) is a linear amplifier with two
differential inputs v+ , v− and one output vo (see figure 3.1) and with the
following characteristics:
• v 0 = A v ( v + − v − ),
Av > 0, (linearity)
• input resistance Ri → ∞,
• output resistance Ro → 0,
• voltage gain Av → ∞,
• frequency response constant for any frequency.
Aside the welcome property of linearity and infinite frequency response,
the need of all the other characteristics can be justified as follows. Infinite input resistance Ri means essentially that the Op-Amp inputs do not
produce perturbations to any circuit to which they are connected to. Zero
output resistance Ro perfectly isolates the Op-Amp from any perturbation.
Infinite input impedance and zero output impedance implies also no dissipation of energy. The condition of infinite voltage gain Av is necessary if
we want a device able to deliver any gain, once a network which connects
the output to the input is added to the Op-Amp. In general, this kind of
network is called feedback network.
v−
vi = v +− v−
v+
−
A >0
vo= Avv i
vi = v +− v−
+
v+
AF
T
−
+
v−
+
−
vo= Avv i
DR
Figure 3.1: Op-Amp symbol, some definitions of variables and ideal model with
an voltage controlled source.
3.2. THE IDEAL OPERATIONAL AMPLIFIER
59
3.2.1 Ideal Op-Amp Fundamental Equation (Golden Rules)
The consequence of the following conditions
• Av → ∞,
• vo < ∞ if vi = v+ − v− < ∞,
is the following very useful and important formula
v+ − v− = 0,
at all times.
i+ − i− = 0,
at all times,
(3.1)
Equation 3.1 will be called the first Op-Amp golden rule.
The consequence of the following condition
is
• Ri → ∞,
(3.2)
Equation 3.2 will be called the second Op-Amp golden rule.
These two rules are fundamental for the solution of any circuit involving Op-Amps. We will see in the next sections the importance of this equations once a feedback network is connected to the Op-Amp.
3.2.2 Op-Amp Input Output “Logic”
Rf
R
+
Vs
−
I
I
−
+
Vo
DR
A
AF
T
It is worthwhile here to notice the behavior of Op-Amp output as function
of the two inputs. From the definition of Op-Amp we have that a signal
sent to the negative input V− is amplified and changed in sign. A signal
sent to the positive input V+ is just amplified. Two signals sent each to one
input are indeed subtracted and amplified.
Figure 3.2: Op-Amp with a feedback network.
60
CHAPTER 3. THE OPERATIONAL AMPLIFIER
3.2.3 Op-Amp with a Feedback Network
Let’s consider the circuit in figure 3.2, where a feedback resistance R f is
connected to the negative input. The current through the resistors R and
R f is the same because the ideal Op-Amp input does not drive any current
(Ri = ∞). Furthermore, since Vi = V+ − V− = 0 and with the use Ohm’s
law and the KVL, it follows that
I=
Vs
Vo
=− .
R
Rf
(3.3)
The output voltage Vo and voltage gain A will be
Vo = AVs ,
A=−
Rf
.
R
The gain of the Op-Amp depends just on the resistances ratio R f and
R.
3.2.4 The Virtual Ground
AF
T
Let’s re-analyze the circuit in figure 3.2. Considering that the golden rule
imposes Vi = V+ − V− = 0, and the negative input is grounded, the node
A must always be at zero voltage. This is equivalent to having A virtually
grounded. The adjective virtual is necessary because even if A is at the
potential of the ground there is no current flowing through A ( Ri = ∞) as
in a real ground. In other words, the virtual ground happens to be because
the Op-Amp does its best to keep Vi = 0.
3.3 Commonly Used Op-Amp Circuits
DR
In the study of the several common Op-Amp configurations, we will use
the approximation of an ideal circuit. A more realistic model is often necessary to understand some behaviors of real circuits. For an initial design,
and where the the ideal Op-Amp characteristics are well approximated,
the ideal model is quite often sufficient.
3.3. COMMONLY USED OP-AMP CIRCUITS
61
Rf
R
−
Vo
+
+
Vs
−
Figure 3.3: Non-inverting configurations of the Op-Amp.
3.3.1 Non-Inverting Amplifier
Let’s consider the non-inverting configuration of the Op-Amp in figure
3.3. Because of Vi = 0, we will have
Vs − V− = Vs − RI = 0.
Considering that the output voltage Vo is
Vo = ( R f + R) I,
Vs =
R
Vo .
R + Rf
The output voltage Vo and voltage gain A will be
Vo = AVs ,
A = 1+
Rf
.
R
AF
T
we can use the expression of I to obtain
DR
Considering that in this configuration Vs is directly connected to V+ and
V− is not a virtual ground, the input impedance of the amplifier is Ri + R,
where Ri is the real input impedance of the Op-Amp.
CHAPTER 3. THE OPERATIONAL AMPLIFIER
62
3.3.2 Inverting Amplifier
This circuit has been already discussed in section 3.2.3. For completeness,
the solution and some comments are here reported
Rf
.
R
It is worthwhile to notice that because V− = 0, the circuit input impedance
is just R. Having values of R typically of few kΩ, the inverting configuration doesn’t preserve the high impedance characteristic of an Op-Amp. A
connection of the circuit input to a network can potentially create appreciable perturbations.
Vo = AVs ,
A=−
3.3.3 Differential Input Stage
Rf
V2
I1
R1
V1
−
I1
Vo
+
R2
R0
AF
T
Figure 3.4: Differential input configuration of the Op-Amp.
Let’s now solve the differential input circuit of the Op-Amp in figure
3.4.
Writing the voltage drop across R1 and R f , we obtain the linear system
V− − V1 = R1 I,
Vo − V− = R f I.
DR
Solving the system with respect to Vo , we get
Rf
Rf
Vo = 1 +
V− −
V1 .
R1
R1
3.3. COMMONLY USED OP-AMP CIRCUITS
vi (t)
−
63
vo (t)
G
+
Figure 3.5: Voltage follower or unity gain buffer.
Using the voltage divider equation to obtain V+ and because V+ − V− =
0, we have
R0
V− = V+ =
V2 ,
R2 + R0
and finally, we get
Vo =
Rf
R1 + R f
R0
V2 −
V1 .
R1 R2 + R0
R1
A way to to obtain the same voltage gain for V2 and V1 is to impose
R0 = R f and R1 = R2 = R. The output voltage becomes
Vo = A(V2 − V1 ),
A=
Rf
.
R
AF
T
This differential configuration is not very convenient because it does
not preserve the high input impedance of the Op-Amp. In fact , considering that the Op-Amp input impedance is very high, we have that the
resistance seen from V2 is R2 + R0 . Usually, the sum of those resistors is at
least one order of magnitude smaller than the Op-Amp input impedance.
If we need to build a variable gain differential amplifier, we will need to
change more than one resistor value. Matching the resistances values can
become an issue when thermal drifts become important.
More practical and stable configurations called instrumentation amplifiers are available “off the shelf”.
3.3.4 Voltage Follower (Unity Gain “Buffer”)
DR
The circuit sketched in figure 3.5 is called voltage follower or unity gain
buffer. The feedback line with no load gives
vo (t) = v− .
CHAPTER 3. THE OPERATIONAL AMPLIFIER
64
Moreover, because of the golden rule we will have
v− = v+ ,
which implies
v o = vi .
The output voltage vo (t) follows the input voltage vi (t) with unitary
gain.
Considering that the high impedance input and the low impedance
output values of Op-Amps are close to the state of the art in the electronic
design2 , the voltage follower can be used as an isolation stage (buffer)
between two circuits.
3.3.5 Integrator Amplifier
Rf
Cf
R
−
+
vo (t)
AF
T
vi (t)
Figure 3.6: Active integration stage using an Op-Amp.
2 Devices
DR
expressly made to work as input unity gain buffer, and output unity gain
buffer are also available. Analog Devices SSM2141 and SSM2142 are complementary
buffers devices which can drive long delay lines for example.
3.3. COMMONLY USED OP-AMP CIRCUITS
65
Let’s consider the circuit in figure 3.6 without the resistance R f . The
voltage drop vo across the capacitor C f is
vo (t) = −
1
Cf
Z t
−∞
(3.4)
i (τ )dτ
and the current flowing through the resistance R is
i (t) =
vi ( t )
.
R
Placing the expression of i (t) obtained from the previous equation into
eq.(3.4), we will obtain
vo (t) = −
1
τ
Z t
−∞
vi (t′ )dt′ ,
τ = RC f .
DR
3.3.6 Differentiator Amplifier
AF
T
Real Op-Amps or signals connected to the input have often (always )
a DC offset. This offset is indeed integrated and after a given time will
saturate the amplifier output. This saturation is essentially a manifestation of the instability of the circuit at low frequency. Moreover, the initial
charge of the capacitor is undefined, making the initial output state unpredictable.
A common way to avoid these problems is to introduce the resistance
R f in parallel with the capacitor C f which reduces the amplifier DC gain.
An intuitive way to understand the effect of this feedback resistance is that
it does not allow the capacitor to be charged "ad libitum". The choice of
the R f is not so trivial if we want to preserve the characteristic of good
integrator. Using the simple phasor analysis it easy to prove that the good
integrator condition is ω ≫ 1/C f R f .
If the DC current must be integrated, we can place a switch in parallel
with the capacitor to be opened when the integration is started. In this
way we will have the capacitor state completely defined.
Let’s now consider the circuit in figure 3.7 without the feedback capacitor
C f . Applying a similar analysis to that used in the integrator amplifier we
CHAPTER 3. THE OPERATIONAL AMPLIFIER
66
Cf
Rf
i(t) C
−
vi (t)
+
vo (t)
Figure 3.7: Differentiator stage using an Op-Amp.
will have
dvi
,
dt
vo (t) = − R f i.
i (t) = C
and indeed
dvi
,
τ = R f C.
dt
This configuration without C f doesn’t work well with real Op-Amps,
because of stability problems. In fact, the introduction of the capacitor
compromises the internal compensation of the Op-Amp. Placing a capacitor C f in the feedback network restores the compensation making the overall circuit stable. The choice of C f is not trivial if we want to preserve the
circuit differentiator characteristics.
Section 3.5.2 explains in more details the effect of this configuration on
the compensation of a real Op-Amp.
AF
T
vo (t) = − τ
3.4 Negative Feedback Amplifiers
DR
Let’s consider an amplifier with a negative feedback network as show in
figure 7.1. Considering the voltage at the summation point output is
V ′ = Vi − β(ω )Vo ,
3.4. NEGATIVE FEEDBACK AMPLIFIERS
Vi
V´
+
67
Vo
A(ω)
−
βVo
β(ω)
Figure 3.8: Amplifier with negative feedback.
and the amplifier gain is A(ω ), the output voltage will be
Vo = AV ′ = A(Vi − βVo ) .
Collecting Vo , we will have
Vo =
A
Vi ,
1 + βA
and the so called closed loop transfer function, ACL , will finally be
ACL (ω ) =
A
.
1 + βA
(3.5)
We can clearly see that if the denominator goes to zero for a given frequency ω ∗ , we are in trouble, ACL (ω ∗ ) diverges, and the amplifier saturates. The trick to avoid this situation, is to study the following equation
(3.6)
AF
T
AOL (ω ) = β(ω ) A(ω ) = −1,
where | AOL | = 1
DR
where AOL is the feedback amplifier open loop transfer function. If the phase
where the magnitude of AOL is equal to one is different from 1800 plus
multiples of 3600 , the denominator never goes to zero and the saturation
is avoided. However, this is not enough because we can have just an oscillation with no saturation if the AOL phase is too close to 1800 . The rule of
thumb is to have a so called phase margin of about 600 from 1800 . Finally,
we can formulate the criterion for the stability:
⇒ −120 < arg( AOL ) < 120 .
CHAPTER 3. THE OPERATIONAL AMPLIFIER
68
Another important result of the theory of feedback amplifier is the following straightforward result
if
βA ≫ 1
⇒
ACL (ω ) ≃
1
.
β
Where the open loop transfer function Aβ is greater than one, the feedback amplifier response does not depend on the response A(ω ) of the amplifier with no feedback. It is worthwhile to notice that the ideal amplifier
(A → ∞) has ACL = 1/β for all angular frequencies. A close loop ideal
amplifier does not have undesired instabilities, but just the ones that can
be introduced by the feedback network.
3.4.1 Input Impedance Feedback Amplifier
The input impedace of a feedback amplifier is
Zi = Zi′ (1 + βA) ,
where Zi′ is the impedance of the amplifier without feedback network. For
frequency values such that
β ( ω ) A (ω ) ≫ 1
⇒
Zi ≃ Zi′ βA .
In conclusion, the impedance of a feedback amplifier scales with the open
loop gain. In other words, where the loop gain is much greater than one
the input impedance increases by the loop gain.
3.4.2 Output Impedance Feedback Amplifier
Zo =
Zo′
1 + βA
For frequency values such that
⇒
Zo ≃
Zo′
.
βA
DR
β (ω ) A (ω ) ≫ 1
AF
T
The output impedace of a feedback amplifier is
Where the loop gain is much greater than one the output impedance
decreases by the loop gain.
3.4. NEGATIVE FEEDBACK AMPLIFIERS
69
3.4.3 Simple Feedback Network
Just to familiarize with the concept of feedback amplifier let’s analyze
some very simple circuit like the Op-Amp non inveriting and inverting
amplifiers.
3.4.3.1 Non-Inverting Configuration
Zf
Z−
V−
−
Vi
Vo
+
Figure 3.9: Non-inverting configuration Op-Amp with generic impedance.
Considering the Op-Amp Non-Inverting configuration as shown in figure 3.9, and the voltage divider equation we have
V− =
Z−
Vo ,
Z f + Z−
(3.7)
and the feedback network transfer function is
β(ω ) =
V−
Z−
=
.
V0
Z f + Z−
ACL ≃
Zf
1
= 1+
.
β
Z−
3.4.3.2 Inverting Configuration
β=
Vi
V0
DR
In this case the feedback network transfer function β is
AF
T
The approximate gain of the feedback amplifier is as expected
CHAPTER 3. THE OPERATIONAL AMPLIFIER
70
Zf
Vi
Z−
V−
−
Vo
+
Figure 3.10: Inverting configuration Op-Amp with generic impedance.
Considering the inverting configuration stage as shown in figure 3.10,
because of the virtual ground we have
Z−
Vi = ZI
⇒ β(ω ) = −
,
−Vo = Z f I
Zf
and the gain of the feedback amplifier is simply
ACL ≃
Zf
1
=−
.
β
Z−
3.5 The Real Op-Amp
AF
T
Lets consider in this section a more realistic model of the Op-Amp by including a finite input impedance Ri , non zero output impedance Ro , finite and frequency dependent A (ω ), input bias currents ib+ , bb− and input
voltage offset vb . Using ideal components, the equivalent circuit of the real
Op-Amp is shown in figure 3.11.
3.5.1 Bias Currents and Voltage and Current Offsets
DR
Imbalances inside of the Op-Amp, mainly due to differences in the electronics components, produce undesirable bias currents and a voltage offset at the inputs. Input voltage and current offsets can be modeled by
introducing ideal generators as shown in figure 3.11. Current Offset is
defined as the difference in the magnitude of the bias currents, i.e.
ios = |ib+ | − |ib− |
3.5. THE REAL OP-AMP
71
A way to characterize the voltage offset is to use the voltage follower
configuration (see section 3.3.4) with the input Vi connected to the ground.
The voltage offset will be directly the output voltage Vo .
Current input biases can be studied connecting a resistor between the
one input and ground and measuring the voltage drop across the resistor.
vb
v−
−
vi = v +− v−
v+
vo
+
−
ib−
Ri
+
Ro
+
ib+
vo= Av(v +−v−−vb )
Av (ω)
−
Figure 3.11: Equivalent circuit for an Op-Amp using ideal components. Voltage
offsets and current biases are taken into account using ideal voltage and current
generators.
3.5.2 Compensated Op-Amp Transfer Function
AF
T
Practical Op-Amps are often designed to have a frequency response dominated by a single pole, i.e. the transfer function with no feedback from
DC to a given frequency is well approximated by just a simple low-pass
filter. In this case, the Op-Amp transfer function with no feedback can be
written as
A0
A(ω ) =
,
(3.8)
1 + j ωω0
DR
where A0 is the DC gain and ω0 is the angular frequency of the dominant
pole (the cut-off angular frequency of the low pass filter). This behavior
is obtained by introducing a compensating circuit (quite often a capacitor)
in the architecture of the Op-Amp.
This choice comes from the stability requirement that we mentioned
in the previous section. In fact, an amplifier with a transfer function with
CHAPTER 3. THE OPERATIONAL AMPLIFIER
72
100
Differential Open Loop Gain (dB)
80
60
40
20
0
1
10
2
10
3
4
10
10
Frequency (Hz)
5
10
6
10
7
10
Figure 3.12: Differential gain of the AD711 Op-Amp (cut-off frequency ν0 =
18Hz, unity gain frequency ν1 = 4MHz, and DC gain a0 = 110dB). Dominant
single pole behavior is valid up to about 4MHz, where the slope becomes steeper
than 1/ω.
AF
T
a dominant pole cannot lose more than 900 making quite easy the design
of a stable feedback network. For example, feedback networks with just
resistors, (ideal resistors don’t loose phase) will not generate oscillations.
Typical values for pole frequencies are between 5Hz and 100Hz. Figure
3.12 shows the differential transfer function of the Op-Amp AD711 with
no feedback network.
DR
Let’s study more in details the compensated Op-Amp response with a
feedback.
Considering the frequency response of a compensated feedback amplifier the closed loop transfer function with a feeback network β(ω ) will
3.5. THE REAL OP-AMP
73
be
ACL (ω ) =
A0
1 + j ωω0
!,
1+
βA0
1 + j ωω0
!
,
A0
,
1 + βA + j ωω0
A0
ω
=
1+j
.
1 + A0 β
ω0 (1 + βA0 )
=
3.5.2.1 Compensated Op-Amp with Constant Frequency Response Feedback
In the particular case that β(ω ) is constant and β = β 0 ≥ 1, the previous
equation becomes
A0
,
1 + β 0 A + j ωω0
A0
ω
=
1+j
.
1 + A0 β
ω0 (1 + βA0 )
ACL (ω ) =
and therefore
A1
,
ACL (ω ) =
1 + j ωω1
(
A0
A1 =
1+ A 0 β 0
ω1 = ω0 (1 + β 0 A 0 )
AF
T
In this case, the feedback Op-Amp response ACL is the same as of the
open loop transfer function AOL but with a smaller DC gain (about 1/β 0 )
and higher cut off angular frequency ω1 of about ω0 β 0 A0 .
3.5.2.2 Compensated Op-Amp in the Differentiator Configuration
Let’s find the frequency response of the "real" Op-Amp differentiator circuit of figure 3.7. For the basic differentiator (no capacitor C f ) the feedback
network transfer function is
1
ω
Z
=−
= j 1,
Zf
jωRC
ω
ω1 =
1
.
RC
DR
β=−
Considering eq. (3.5), we have that the gain of the differentiator is
CHAPTER 3. THE OPERATIONAL AMPLIFIER
74
ACL (ω ) =
1
A(ω )
=
,
1 + jA(ω ) ω1 /ω
1/A + j ω1 /ω
and finally
ACL (ω ) =
A0
1 + j A0 ωω1 +
ω
ω0
= − A0 ω0
ω2
jω
.
− jωω0 + A0 ω0 ω1
Resonance occurs when the denominator goes to zero, i.e for
s
p
ω02
ω0
±j
+ A0 ω0 ω1 ≃ ± j A0 ω0 ω1 ,
ω0 ≪ ω1 , A ≥ 1 .
ω∗ = j
2
4
3.5.3 The Common Mode Rejection Ratio (CMRR)
We want characterize the rejection of an Op-Amp output as a differential
amplifier, of signals sent to both inputs. For an ideal Op-Amp we expect
to obtain Vo = 0 for all frequencies, i.e. a perfect rejection. To define a convenient parameter which measures the rejection it is necessary to define
the following ones, the common mode gain
AC (ω ) =
Vo
,
V+ − V−
V+ = V− = Vs sin(ωt),
and the differential mode gain
Vo
,
V+ − V−
V+ = Vs sin(ωt),
V− = 0.
AF
T
A D (ω ) =
The Common Mode Rejection Ratio (CMRR) is defined as the modulus of
the ratio of the differential gain A D over the common mode gain AC , i.e.
CMRR(ω ) =
A D (ω )
AC (ω )
DR
Ideally, the CMRR should be infinity for all frequencies.
For values below ~90, CMRR can be measured using the Op-Amp differential configuration (see figure 3.4) by measuring AC and A D as a function of the frequency. To minimize possible large systematic errors, it is
3.5. THE REAL OP-AMP
75
necessary to have the same gain for the to inputs V1 and V2 . This can be
achieved by placing a trimmer in the voltage divider mesh of the differential configuration circuit. Adjusting the trimmer we can minimize Vo for a
single frequency and study the CMRR for a given bandwidth.
Figure 3.13shows the CMRR as a function of frequency of a typical
Op-Amp. A typical value for CMRR is 90dB.
90
CMRR (dB)
80
70
60
50
40
30
1
10
100
1000
Frequency (Hz)
10000
100000
Figure 3.13: CMRR as a function of frequency of a typical Op-Amp.
AF
T
3.5.4 The Gain Bandwidth Product (GBWP)
The gain bandwidth product, GBWP, is a common way to characterize the
gain with respect to the available bandwidth of amplification. Using the
previously defined notation, the GBWP is simply defined as
DR
GBWP = A0 ω0 .
The larger the GBWP the better is the Op-Amp, and the closer the OpAmp is to the ideal operational amplifier.
CHAPTER 3. THE OPERATIONAL AMPLIFIER
76
Input/Output Voltage (V)
16
14
12
10
8
Input
Output
6
4
2
0
0
10
20
30
40
50
60
70
time (us)
Figure 3.14: Slew rate illustration. The output vo takes 40µs to reach the desired
voltage. The device is said to be slew rate limited.
3.5.5 The Slew Rate (SR)
The slew rate or maximum slew rate SR of an Op-Amp is defined as the
maximum rate of the output voltage vo per unit time
∆vo (vi )
SR = max
.
∆t
AF
T
The slew rate can be easily observed sending a square wave (see figure
3.14) to the Op-Amp input vi , and looking at the raising and falling slope
of the output signal vo . If the slopes do not change while changing the
input amplitude, then the Op-Amp is slew rate limited.
A similar procedure can be applied using a sinusoidal signal as input.
In this case, if we increase the input amplitude too much, the output will
become distorted and will look somehow closer to a triangular wave than
a sinusoid. Considering a sinusoid of frequency ω0 and amplitude Vo
d
Vo sin ω0 t = ω0 Vo max {cos ω0 t} = ω0 V0 .
SR = max
dt
DR
For an undistorted signal with amplitude V and maximum frequency
ω, we must have
SR ≫ ωV .
3.5. THE REAL OP-AMP
77
Usually, a frequency compensated Op-Amp has an integrating stage
somewhere, and therefore the slew rate is often proportional to the inverse
of the capacitance of the compensating network.
Typical good slew rate values are of the order of few V/µs.
The slew rate parameter essentially measures the ability of an Op-Amp
to follow voltage changes for large voltage inputs. The inability of following the voltage input usually comes from limitations on the current inside
the amplifier compesation circuit. In fact, on a capacitor compensated amplifier the bottle-neck id the current to charge or discharge the capacitor.
3.5.6 Ideal versus Real and Practical Considerations
The following table summarizes the main characteristic of an ideal OpAmp together with those of typical real Op-Amp. In some cases we can
find Op-Amps excelling some of the mentioned characteristics , quite often
at the expense of other characteristics.
Ideal Op-Amp
Typical Op-Amp
Open-Loop DC Gain Av
∞
> 104
Open-Loop Bandwidth
∞
∼ 10Hz (dominant pole)
Common Mode Rejection
Ratio CMRR
∞
> 70dB
Input Resistance Ri
∞
> 10MΩ
Output Resistance Ro
0
< 100Ω
Input Current δI±
0
< 0.5µA
Input Offset Voltage δV±
0
< 10mV
Input Offset Current δIi
0
DR
AF
T
Property
< 200pA
78
CHAPTER 3. THE OPERATIONAL AMPLIFIER
What are the conditions that dictate the range of the feedback impedances
R f ? Apart from special cases, the feedback current I should be only a small
fraction of the maximum output current Io , i.e. I = 1%Io . A typical OpAmp has a maximum output of 10mA at 10V, i.e.
Rf =
V
10V
=
= 100kΩ
I
10 · 1%mA
Anyway, typical feedback resistors should be in the range of R f = 50 −
1MΩ.
Small difference on the differential stage of the Op-Amp produces a
DC offset δV at the input, which can produce large DC output if the gain
is extremely high. For example, if we have
AF
T
G ≥ 104 , ⇒ Vo ≥ 10V.
DR
δV = 10mV,
3.6. PROBLEMS PREPARATORY TO THE LABORATORY
79
3.6 Problems Preparatory to the Laboratory
1. Supposing that the open-loop gain of an Op-Amp is a simple low
pass filter (first order) with DC gain 144dB and cut-off frequency
f 0 = 10Hz, sketch in a bode diagram, the magnitude of the frequency
response of a non-inverting stage with gain G = 10 at 10Hz.
2. Prove that the good integrator condition for the circuit in figure 3.6
is ω ≫ 1/R f C f . (Hint: calculate the response of the circuit and
compare it with the ideal response of the ideal inverting integrator,
i.e. A(ω ) = −1/( jωRC). Calculate the DC gain of the integrator
transfer function.
3. Using an integrator stage with a feedback resistor R f , and time constant τ = RC, compute the values for τ and R f needed to integrate
a sinusoidal wave with frequency f > 1kHz and with 10% of losses
in the integration. Choose a value of R ≫ Rs , where Rs = 50Ω is the
input impedance of the used function generator.
4. Show that the slope of an integrated square wave is the inverse of
of the time constant τ = RC f of the Integrator shown in figure 3.6.
Which characteristics of the square wave are needed to fulfill the requirement to not saturate the integrator output ?
5. Consider a differential stage having the following resistances values:
R1 = R2 = 50 kΩ, R f = R0 = 100 kΩ . Calculate the following
quantities
3.7 Laboratory Procedure
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T
(a) the two input impedances Z1 and Z2 ,
(b) the output impedance Zo ,
(c) Considering that the Op-Amp max output current and voltage
are respectively Imax =10mA and Vmax = 10V, calculate the
smallest load Rmin it can drive .
DR
Read carefully the entire procedure before starting the experiment, and
note on your log book all the unpredicted behavior you experience in the
circuits response.
80
CHAPTER 3. THE OPERATIONAL AMPLIFIER
Consult the data-sheet to properly map the µ741 and AD711 Op-Amp
pin-out.
Op-Amp output high frequency noise can be reduced by adding 100nF
capacitors closest as possible to the ±15V power supplies input of the OpAmp.
Before powering your circuit up, cross-check the power supply connections.
It is always a good practice to turn on the power supplies at the same
time to avoid potential damages of the Op-Amps.
1. Build a non-inverting staga amplifier with a µ741 Op-Amp with a
gain of 100 and do the following:
(a) measure the transfer function of the amplifier,
(b) download the simulation file from the ph5/105 website for the
circuit, input the proper components values,
(c) run the AC response simulation with the ideal compensated
Op-Amp and find the discrepancies beteween your measurement and the simulation,
(d) replace the ideal Op-Amp with the model of the Op-Amp you
are using and compare the frequency response with the simulation.
2. Using the previous circuit,
(a) estimate the slew rate of the Op-Amp,
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T
(b) compare the measured slew rate to the simulation using the OpAmp model,
(c) replace the AD711 and remeasure the slew rate.
3. Build an integration stage with a time constant τ ∼ 100µs using an
AD711 Op-Amp . Include a feedback resistor R f to avoid saturation
at the output and do the following steps:
DR
(a) measure the impulse response,
(b) measure the frequency response,
3.7. LABORATORY PROCEDURE
81
(c) estimate the integrator time constant τ using a square wave,
(d) download the simulation file from the ph5/105 website for the
circuit, input the proper components values,
(e) run the AC response simulation with the ideal compensated
Op-Amp and find the discrepancies beteween your measurement and the simulation.
DR
AF
T
(f) run the transient analysis and compare it with the measured
impulse response.
AF
T
CHAPTER 3. THE OPERATIONAL AMPLIFIER
DR
82
Bibliography
[1] The Art of Electronics, Paul Horowitz and Winfield Hill, Cambridge
University Press.
[2] Microelectronics Jacob Millman & Arvin Grabel, McGraw-Hill Electrical Engineering Series
[3] Analog Devices web site www.analog.com AD625: Programmable
Gain Instrumentation Amplifier Data Sheet (Rev. D, 6/00).
83
DR
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T
[4] An Introduction to Operational Amplifiers with Linear IC, Second Edition, Luces M. Faulkenberry, edited by John Wiley and Son.
BIBLIOGRAPHY
DR
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T
84
Chapter 4
Diodes and Transistors
4.1 Introduction
In this chapter we will analyze two new electronic devices, the semiconductor diode and the bipolar junction transistor (BJT). For a better understanding of their behavior and characteristics, we will also introduce some
basic applications.
Unfortunately, there will be no time to study the quite complex physics
of semiconductors, and especially the conduction mechanism, which substantially differs from that of metals. The interested student should look
for a course and books on solid state physics.
It is important to notice that to quickly grab how the BJT device works,
it is fundamental to acquire a clear understanding of the semiconductor
diode’s behavior.
AF
T
4.2 The Semiconductor Junction (Diode)
The semiconductor junction or semiconductor diode is a device which shows
non-linear behavior due to its peculiar conduction mechanism.
In fact, if ID and VD are the current and the voltage difference across
the junction, we will have
− 1 ),
DR
ID (VD ) = I0 (e
− qVD
ηk B T
(4.1)
where I0 is the reverse saturation current, k B = 1.3807 · 10−23J/K, the
85
CHAPTER 4. DIODES AND TRANSISTORS
86
ID( µA)
−100
I0
Vb
−50
1
0
Von
VD (V)
0.5
Figure 4.1: Diode characteristic (continuous curve), simplified diode characteristic (dashed curve). Note the different scales in first and third quadrant of the
diode characteristic plot.
Boltzmann constant, T the absolute temperature , q = −1.60219 · 10−19 C,
the electron charge, and η a dimensionless parameter which depends on
the diode type. Considering that the ambient temperature is T ≃ 300K,
we will have k B T ≃ 4.14 · 10−21 J ≃ 0.026eV. For silicon diodes the reverse
saturation current I0 is of the order of few tenths of nano-amperes.
Instead of following Ohm’s law, the semiconductor junction follows an
exponential law. Deviations from this law are negligible depending on the
current magnitude and the diode characteristics.
1 The thermally generated carriers accelerated by the
AF
T
Figure 4.1 shows standard symbols for a semiconductor diode and the
I-V characteristic. The break-down voltage Vb reported in the same figure
is the reverse voltage which essentially short circuits the junction (typically
between -100V and -50V). This behavior is not accounted in equation (4.1),
and is generated by the so called avalanche multiplication mechanism and
the Zener mechanism1 .
DR
electric field have enough energy
to disrupt the electrons bond of the crystal atoms producing new carriers (electron-holes
pairs). The new and accelerated pairs generate new carriers producing an avalanche of
carriers, and indeed a break-down current.
A sufficient strong electric field can also disrupt electrons bonds creating an electronhole reverse current. This effect is called Zener Breakdown mechanism.
4.2. THE SEMICONDUCTOR JUNCTION (DIODE)
A
ID
A
D0
VD
ID
K
87
A
D0
VD
ID
K
D0
VD
K
Figure 4.2: diode standard symbols. Starting from left, diode symbol , Zener
diode symbol, and Schottky diode symbol. Diode terminals A, and K are respectively called anode and cathode.
A simplified model of the junction diode is that of a perfect switch, i.e.
ID (V ) =
∞ V ≥ Von
,
0 V < Von
where Von is the diode turn-on voltage or cut-in voltage, which depends
on the junction type and on the current magnitude. For current up to
ID ∼ 100 mA, silicon diodes have Von ≃ 0.6V, and germanium diodes
have Von ≃ 0.3V .
For voltages greater than Von , the diode is a short circuit (current is not
limited by the diode) and is said to be forward biased. For smaller values it
is an open circuit (current across the diode is zero ) and is reverse biased.
4.2.1 Zener Diodes
DR

V < Vb
 −∞
0 0 ≤ V < Von ,
ID (V ) =

+∞
V ≥ Von
AF
T
Zener diodes are particular semiconductor diodes with adequate power dissipation to operate in the break-down voltage region. They have a well
defined Vb , with values ranging from about few volts to several hundreds
volts. Zener diode symbol is shown in Figure 4.2. Approximating the
characteristics with a piecewise linear relationship, we have
Often, the break-down curve is virtually vertical so that the previous
approximation of the reverse biased region is quite good.
CHAPTER 4. DIODES AND TRANSISTORS
88
4.2.2 Schottky Diodes
A junction made of a semiconductor and a metal can behave like a semiconductor diode[2]. For example, Lightly doped silicon and aluminum
can form a semiconductor junction. Such kind of devices, called Schottky
barrier diodes (or simply Schottky diodes), still follow the diodes characteristics (4.1) with usually a lower turn-on voltage Von and a larger in magnitude reverse saturation current Is . The symbol for Schottky diode device
is shown in Figure 4.2.
4.3 Diode Dynamic Impedance
For linear devices the current is proportional to the applied voltage and
for a given frequency the impedance (V/I ) is constant. With non-linear
circuits this is not true anymore, but we can generalize the impedance
concept introducing the dynamic impedance
Rd =
dV
dI
Let’s apply this definition to the diode. Starting from the I-V characteristic equation and neglecting the reverse saturation current, after some
algebra we obtain
ID
kB T
ln
.
VD = −η
q
I0
Rd = − η
kB T 1
q ID
AF
T
Taking the derivative on both sides we obtain
As we can see, the dynamic impedance of the diode depends on the
current ID .
Considering a silicon diode with a typical value of η = 2 at room temperature ( T = 300 K ), we will have
5.2 · 10−2
,
ID
ID = 1mA
⇒ Rd ≃ 52 Ω .
DR
R d ( ID ) ≃
For small variations of the current around 1mA, we can assume that
the impedance of a forward biased diode with η = 2 is ∼ 50 Ω . The
4.4. PRACTICAL CIRCUITS
89
ID
D
+
Vs
RL
VL
−
Figure 4.3: Half-wave rectifier circuit.
dynamic impedance concept will be quite useful for studying the bipolar
junction transistor.
4.4 Practical Circuits
To better understand the behavior of a semiconductor junction, let’s analyze a few typical applications of semiconductor diodes. Some other applications in connection to other components will be studied in the following
chapters.
4.4.1 Rectifiers, AC to DC Conversion
4.4.1.1 Half-Wave Rectifier
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T
The purpose of a rectifier circuit is to convert alternating current into a
unidirectional current. This can be achieved using semiconductor diodes.
The typical alternating current to direct current converter is a rectifier connected to an active low pass filter with a so called regulator circuit, which
smooths the rectifier output and minimizes ripples. The simplest regulator is a capacitor placed in parallel with the rectifier output. Regulators
can be easily found in literature (see [1]).
DR
The simplest rectifier circuit is the so called half-wave rectifier shown in
Figure 4.3.
Using the diode ideal characteristic, it is quite straightforward to predict the voltage difference across the the resistor R L . In fact, when the
sinusoidal signal is positive, it will forward bias the diode and we will
CHAPTER 4. DIODES AND TRANSISTORS
90
V
V0
0.7V
t
VL
Vs
Figure 4.4: Voltage difference across the load connected to the half-wave rectifier
output.
have a voltage drop across the resistor VL = RI. For the negative half
cycle, because the diode is reverse biased VL must be zero.
Considering the diode threshold voltage V0 , and the diode resistance
R f during the positive half cycle we will have
VL =
RL
(Vs − V0 ),
R f + RL
and if
RL ≫ R f
⇒
VL ≃ (Vs − V0 ).
VL =
RL
Vs ,
Rr + R L
and if
R L ≪ Rr
⇒
VL ≃
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T
During the negative half cycle we will have
RL
Vs ≃ 0.
Rr
DR
The main disadvantage of this circuit is the very poor efficiency (less
than 50% of current is rectified). In fact, instead of rectifying the entire
signal the circuit chops the negative half cycle out (see Figure 4.4).
4.4. PRACTICAL CIRCUITS
91
A
D1
D3
vs (t)
C
D
RL
D4
D2
B
Figure 4.5: Full-wave rectifier bridge circuit or simply bridge rectifier.
4.4.1.2 Full-Wave Rectifier Bridge
The Full-Wave rectifier bridge (see Figure 4.5), a more efficient way of rectifying an AC current, uses four arranged diodes in the so called bridge
configuration. To understand the circuit “logic”, let’s consider the two
possible states of the nodes A and B shown in Figure 4.5.
• When the node A is positive (B negative) the diodes D2 , and D3 are
forward biased (i.e. the diodes are a “short circuit”) and D1 , and
D4 are reverse biased (i.e. the diodes are an “open circuit”). The current flows through the resistor R L and the node C is positive.
• When the node A is negative (B positive), the diodes D1 , and D4 are
forward biased (short circuit) and D2 , and D3 are reverse biased. The
current flows through the resistor R L and the node C is still positive.
4.4.2 Voltage Limiter (Diode Clamp)
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T
Using the full-wave rectifier we will indeed have the negative half cycle
rectified as shown in Figure4.6.
DR
Diodes can be used to limit the voltage applied to an input as shown in
Figure 4.7. Let’s consider the diode D1 connected to Vmax . If Vi exceeds
Vmax + Von the diode is not reverse biased anymore and starts conducting,
i.e the circuit limits the input voltage Vi to Vmax + Von . Analogously, D2
limits the minimum input voltage Vi to Vmin + Vo . The resistor is necessary
to limit the current flowing through the diodes. In fact, without the resistor
CHAPTER 4. DIODES AND TRANSISTORS
92
V
2V0
VL
t
Vs
Figure 4.6: Voltage difference across the load connected to the full-wave rectifier
output
Vmax
Vi
R
D1
Device
to
Protect
D2
Vmin
Figure 4.7: Diode clamps circuit.
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T
if we exceed one of the voltage limits an excessive current can destroy the
forward biased diode junction. The worst scenario is when the broken
diode becomes an open circuit and then the device to protect becomes
completely unprotected.
4.5 The Bipolar Junction Transistor (BJT)
DR
The bipolar junction transistor is essentially a device formed by two semiconductor junctions which share one semiconductor layer (see Figure 4.8).
The common layer is called the base and the two others are the collector
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
Collector
n
93
Base Emitter
p
n
E
C
B
Figure 4.8: Qualitative physical model of a npn junction.
and emitter. We will have then the emitter-base and the collector-base junctions.
There are two types of BJT: the npn and the pnp transistor. In the pnp
transistor the collector and the emitter are p-type and the base is n-type.
The npn transistor has a p-type base , and n-type collector and emitter.
Standard symbols for both types are shown in Figure 4.9.
Because the two junctions have two possible states (forward or reverse
biased), the BJT can have four possible operating modes as shown in the
following table
Forward-Active:
Bias
Emitter-Base
Forward
Reverse
Forward
Reverse
Bias
Collector-Base
Reverse
Reverse
Forward
Forward
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T
Operating
Mode
Forward-Active
Cutoff
Saturation
Reverse-Active
The BJT approximates a current-controlled source of current as explained
in section 4.5.2.
Cutoff:
DR
Both junctions are reverse biased. Neglecting the reverse saturation current, no current flows through the junctions. This mode, together with the
saturation mode, is used to implement the switch device (see section4.5.5).
CHAPTER 4. DIODES AND TRANSISTORS
94
IC
IB
C
IC
IB
+
pnp VCE
B
+
npn
B
−
IE
E
C
VCE
−
IE
E
Figure 4.9: Standard circuit symbols for npn and pnp transistors.
Saturation:
Both junctions are forward biased, and the current IC flows from the collector through the emitter.
Reverse-Active:
The BJT still approximates a current-controlled source of current, but the
amplification factor is usually less than that of the forward-active mode.
4.5.1 The Collector Emitter Characteristic
Saturation Region
AF
T
Figure 4.10 shows collector emitter characteristic curves family of a typical
npn transistor. Each curve corresponds to a given value of the base current
IB , with the base emitter junction forward biased.
The curves have three regions which are called, the saturation, forwardactive, and breakdown regions. The break-down region starts for VCE values
larger than those shown in the plots .
DR
The saturation region is where the collector emitter voltage difference VCE
slightly changes as a function of the collector current IC . For the 2N2222
this regions is where VCE is between 0V to about 0.3V.
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
95
Ib=200µA
35
30
Ib=150µA
Collector Current IC [mA]
25
20
Ib=100µA
15
10
Ib=50µA
5
Ib=20µA
Ib=10µA
0
0
0.5
1
Collector Emitter Voltage VCE [V]
1.5
Figure 4.10: Collector Emitter voltage characteristics for the 2N2222 npn transistor. The value above each curve is the corresponding base current IB .
Forward-Active Region
Break-Down Region
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T
The collector current IC slightly changes as a function of the collector emitter voltage VCE . Normally, this region is quite larger than the saturation
region. For the 2N2222 it is where VCE is between 0.3V to about 50V.
This is the region where the VCE doesn’t change and IC rapidly increases.
In this case, the conduction in the junction is produced by the avalanche
mechanism. For the 2N2222 this region starts from VCE > 60V.
4.5.2 The BJT as a Current-Controlled Current Source (CCCS)
DR
As stated before, the bipolar junction transistor is a device that approximates a current-controlled source of current CCCS (see Figure 4.11). In
other words, because its current output io is proportional to the current
CHAPTER 4. DIODES AND TRANSISTORS
96
ii
io
io
A i (1)
i
vi
A ii
vo
A i (2)
i
R
A i (3)
i
vo
Figure 4.11: Ideal current controlled source (diamond symbol), i.e. the current
output io is proportional to the current input ii , and is independent of the load R.
In other words, if we change the load R and vo consequently, io stays constant.
input ii we can linearly control io by changing ii , i.e.
i o ( t ) = β F i i ( t ).
4.5.3 BJT Simplified DC Model
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T
if | β F | > 1 then the BJT is a current amplifier.
As shown in Figure 4.11, once ii is set io must be constant independently of the load R placed at the output. If the voltage across the output
vo changes we don’t expect to see any changes on io . The curve height simply depends on the current input ii .
This approximation is valid for the so called small signal model and
the low frequency model. Non linearities arise for large signals and at
high frequency the response cannot be flat.
It is clear from the VCE characteristic that, if we want to use the BJT as
CCCS , we have to bias it with a DC voltage to work in the forward-active
region.
DR
The simplified DC model of the BJT for the forward-active mode is shown
in Figure 4.12 with two different arrangements. The first mimics the topology of the BJT symbol, and the second the topology of the CCCS in Figure
4.11. This model is good enough to properly bias the transistor to work as
an amplifier.
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
C
iC
iB
+
B
iC
B
βF i B
iB
97
C
=
−
VBE
+
βF i B
−
VBE
E
E
Figure 4.12: Simplified DC model for the bipolar junction transistor working in
forward-active mode. The two drawings are just two different arrangement of
the same circuit.
The current controlled current source represents the VCE characteristic
in the forward-active region. The battery in the base emitter circuit represents the voltage across the base-emitter forward biased junction (it could
be replaced with a diode). A typical value is VBE = 0.7V.
4.5.4 The BJT as an Amplifier
+VCC
vi
+
C
RC
IB
B
vo
+
−
IC
C
VCC
+
−
VBE
+
βF i B
−
E
VCC
+
AF
T
RB
−
forward-active DC model (right).
DR
Figure 4.13: The BJT as an AC basic amplifier (left ), and same circuit using the
Left circuit of Figure 4.13 shows the basic configuration of a BJT as a
simple current amplifier. Resistors RB and RC are chosen to properly bias
CHAPTER 4. DIODES AND TRANSISTORS
98
and limit the currents across the junctions. The capacitance at the input
is necessary to prevent the DC bias from reaching the device connected to
the amplifier input. Let’s better analyze how to properly bias the transistor
junctions.
4.5.4.1 BJT Amplifier Bias
To obtain the largest voltage dynamic range, and considering the VCE characteristic, and neglecting the saturation region, we must have
VCC ≃ 2 VCE .
(4.2)
Plugging the forward-active DC model into the amplifier circuit as
shown in Figure 4.13, we will have2
VCC = VCE + RC IC ,
Considering equation4.2 and the previous equation, the collector resistor
value will be
VCC − VCE
V
RC =
= CE .
IC
β F IB
For the base resistor we will have
VCC = VBE + RB IB ,
RB =
2VCE − VBE
.
IB
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T
and finally
This circuit is not very useful because the junctions bias and the gain
depend on β F , which is quite often not well know and can easily vary
by a factor of two for the same transistor. Moreover, β F is quite sensitive to
temperature fluctuations. Anyway, this circuit is pedagogically interesting
because of it simplicity.
2 The
DR
repeated index is a common convention used to distinguish between the voltage
of the transistor’s connections and the source voltages applied to the transistor connections. In this case between the collector voltage VC and the source voltage VCC .
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
99
Numerical Example
A typical BJT a transistor has VCE between 1V and 10V. Considering the
following parameters


β F = 100


625Ω

 RC ≃
VCE = 5V
⇒
R
≃ 116.25kΩ .
 VBE = 0.7V
 B

VCC ≃
10V

IB = 80µA
4.5.4.2 BJT Amplifier Gain, Input and Output Impedance (Low Frequency Model)
IB
C
B
+
VCC
−
IC
RC
RB
Vi
hie
hfe I B
Vo
VCC
+
−
E
Figure 4.14: BJT basic amplifier using the low frequency model (gray box). The
parameters h f e = β F and hie are provided by the manufacturer
Ri ≃ RB || hie ,
DR
AF
T
Because the emitter-base junction is forward biased, the input impedance
seen from the points B and E is quite low. This consideration with the fact
that the BJT approximates a CCCS is sufficient enough to define a model
for the BJT transistor response for the low frequency region. Figure 4.14
shows the model applied to the basic amplifier. The resistance hie is indeed
the dynamic input impedance of the forward biased emitter-base junction.
From Figure 4.14 we can easily calculate the amplifier voltage gain,
which is
R I
R
| Av | = C C = h f e C
(β F = h f e )
hie IB
hie
Considering that the ideal current source is an open circuit and the
ideal voltage source is a short circuit, we will have
Ro ≃ RC .
CHAPTER 4. DIODES AND TRANSISTORS
100
Thermal fluctuations can substantially change the response of the BJT. A
way to avoid such kind of behavior is to add a feedback network. Essentially, a feedback network samples the output and sends it back to the
input with negative sign minimizing the output fluctuations. For example,
if the amplifier gain increases because of a temperature increase, the feedback signal will increase as well reducing the input signal by the amount
necessary to keep the gain constant. Feedback networks can create instabilities due to phase delays in the loop (the feedback signal can change
sign). It is indeed necessary to satisfy stability criterion to avoid oscillations. A detailed explanation of feedback theory can be found in [2] and
[3].
4.5.5 BJT as Switch
+VCC
RC
vo
+
vi
vi
+
t0
t1
t
−
RB
−
vo
t0
t1
t
AF
T
Figure 4.15: BJT as a switch
Figure 4.15 shows a npn BJT configured as a switch. In this case, the
function of the two resistors RB and RC are just to limit the current flowing
through the transistor junctions.
The input voltage vi control the output state of the switch. For sake
of simplicity let’s neglect the reverse currents components to study the
circuit.
DR
• If vi = 0, The emitter-base junction is reverse biased and no current
flows through the circuit. This implies that vo ≃ VCC and (BJT in
cutoff state).
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
101
• If vi = V and supposing that this voltage forward bias both junctions
we will have vo ≃ 0 (BJT in saturation).
Let’s consider now the BJT reverse currents.
• If vi = 0, we will have iC = ICO and vo = VCC − ICO RC . Because
ICO ∼ 1nA ICO RC is negligible and v0 = VCC .
• If vi = V, then v0 is essentially the voltage drop VBE of the forward
biased base-emitter junction vo = 0.7V.
4.5.6 BJT as Diode
Q
Figure 4.16: BJT as diode.
4.5.7 Current Mirror
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T
Figure 4.16 shows the typical configuration used to make a BJT working as simple diode. The emitter-base junction acts as a simple semiconductor diode. Short circuiting the collector-base ensures that the collectorbase junction is always reverse biased.
Let’s consider the left circuit shown in Figure 4.17 where VCC forward bias
the emitter-base junction. From the KVL obtain
IR =
VCC − VBE
.
R
DR
If VCC and R are kept constant (VBE = 0.7, typically ) then IR is constant
as well. Applying the KCL to node we obtain
IR = IC + IB
CHAPTER 4. DIODES AND TRANSISTORS
102
VCC
VCC
IR
R
IR
R
IC2
IC
IC1
IB
Q1
Q1
Q2
IB1
IB2
Figure 4.17: BJT as diode.
and considering that
Ic = βIB
we will finally have
IR =
1
1+
β
IC .
⇒ IC ≃ IR .
The collector current IC is indeed constant if VCC and R are kept constant.
Let’s now consider the circuit on the right-hand side of Figure 4.17.
Because of the KVL we will have
VBE1 = VBE2
IC1 = IC2 .
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T
Supposing that the two transistors Q1 and Q2 are perfectly identical
and because they have the same VBE we must have
We will have indeed that the output IC2 will work as a constant current
source.
Let’s analyze the stability of the circuit for a change on the transistor
parameter β. From the KVL and KCL we have
VCC − VBE
R
= IC + 2IB
IR
DR
IR =
4.5. THE BIPOLAR JUNCTION TRANSISTOR (BJT)
103
After some simple algebra and considering that IC = βIB we will have
IC =
β VCC − VBE
β+2
R
Studying the fluctuation of the transistor we will have
∆IC
∆β
≃2 2 .
IC
β
In other words, the stability of this current source due to the fluctuations of the transistor properties are expected to be remarkably good. In
fact, if we suppose to have β = 100 and change of 100% in β then
∆I
β = 100
⇒ C ≃ 0.02
∆β = 100
IC
DR
AF
T
For their simplicity, current mirror are extensively used in ICs design
where a constant current source is needed.
AF
T
CHAPTER 4. DIODES AND TRANSISTORS
DR
104
Bibliography
[1] ??Find a reference to Regulators??
[2] Microelectronics, Jacob Millman & Arvin Grabel, McGraw-Hill Electrical Engineering Series.
105
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[3] The art of Electronics Second Edition, Paul Horowitz & Winfield Hill,
Cambridge University Press
BIBLIOGRAPHY
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106
Chapter 5
Basic Op-Amp Applications
5.1 Introduction
In this chapter we will briefly describe some quite useful circuit based on
Op-Amp, bipolar junction transistors, and diodes.
5.1.1 Inverting Summing Stage
Rf
V1
R
I1
V2
R
A
I
Vn
−
Vo
+
AF
T
I2
I
R
In
Figure 5.1: Inverting summing stage using an Op-Amp. The analysis of the
DR
circuit is quite easy considering that the node A is a virtual ground.
Figure 5.1 shows the typical configuration of an inverting summing
stage using an Op-Amp. Using the virtual ground rule for node A and
107
CHAPTER 5. BASIC OP-AMP APPLICATIONS
108
Ohm’s law we have
In =
Vn
,
R
N
I=
∑ In .
n =1
Considering that the output voltage V0 is
Vo = − R f I,
we will have
N
Vo = A
∑ Vn ,
A=−
n =1
Rf
.
R
5.1.2 Basic Instrumentation Amplifier
Instrumentation amplifiers (In-Amp) are designed to have the following
characteristics: differential input, very high input impedance, very low
output impedance, variable gain, high CMRR, and good thermal stability.
Because of those characteristics they are suitable but not restricted to be
used as input stages of electronics instruments.
Vi−
R2
+
G
−
R1
−
Vo
+
R1
Vi+
G
+
Input Stage
R2
AF
T
−
Differential
Stage
Figure 5.2: Basic instrumentation amplifier circuit.
DR
Figure 5.2 shows a very basic In-Amp circuit made out of three OpAmps. In this configuration, the two buffers improve the input impedance
of the In-Amp, but some of the problems of the differential amplifier are
5.1. INTRODUCTION
109
still present in this circuit, such as common variable gain, and gain thermal
stability.
A straightforward improvement is to introduce a variable gain on both
amplifiers of the input stage as shown in Figure 5.3(a), but unfortunately,
it is quite hard to keep the impedance of the two Op-Amps well matched,
and at the same time vary their gain and maintaining a very high CMRR.
A clever solution to this problem is shown in Figure 5.3(b). Because of the
virtual ground, this configuration is not very different from the previous
one but it has the advantage of requiring one resistor to set the gain. In
fact, if R3 = R4 then the gain of the Op-Amps A1 and A2 can be set at the
same time adjusting just RG .
For an exhaustive documentation on instrumentation amplifiers consult [2].
Vi−
Vi−
+
+
A1
A1
−
−
R1
R3
R4
R2
Vi+
R3
RG
−
R4
−
A2
+
(a)
Vi+
A2
+
(b)
AF
T
Figure 5.3: Improved input stages of the basic instrumentation amplifier.
5.1.3 Voltage to Current Converter (Transconductance Amplifier)
DR
A voltage to current converter is an amplifier that produces a current proportional to the input voltage. The constant of proportionality is usually
called transconductance. Figure 5.4 shows a Transconductance Op-Amp,
which is nothing but a non inverting Op-Amp scheme.
CHAPTER 5. BASIC OP-AMP APPLICATIONS
110
if
Zf
R
−
+
Rf
vs (t)
Zf
Amperometer
Figure 5.4: Basic transconductance amplifier circuit.
The current flowing through the impedance Z f is proportional to the
voltage vs . In fact, assuming the Op-Amp has infinite input impedance,
we will have
vs (t)
i f (t) =
.
R
Placing an amperometer in series with a resistor with large resistance
as a feedback impedance, we will have a high resistance voltmeter. In
other words, the induced perturbation of such circuit will be very small
because of the very high impedance of the operational amplifier.
5.1.4 Current to Voltage Converter (Transresistance Amplifier)
v o ( t ) = − R f i s ( t ).
AF
T
A current to voltage converter is an amplifier that produces a voltage proportional to the input current. The constant of proportionality is called
transimpedance or transresistance, and its units are Ω. Figure 5.5 show a
basic configuration for a transimpedance Op-Amp. Due to the virtual
ground the current through the shunt resistance is zero, thus the output
voltage is the voltage difference across the feedback resistor R f , i.e.
DR
Photo-multipliers photo-tubes and photodiodes drivers are a typical
application for transresistance Op-amps. In fact, quite often the photocurrent produced by those devices need to be amplified and converted
into a voltage before being further manipulated.
5.2. LOGARITHMIC CIRCUITS
111
is
Rf
−
Rs
is
v o=−is R f
+
Figure 5.5: Basic transimpedance Op-Amp.
5.2 Logarithmic Circuits
By combining summing circuits with logarithmic and anti-logarithmic amplifiers we can build analog multipliers and dividers. The circuits presented here implements those non-linear operations just using the exponential current response of the semiconductor junctions. Because of that,
they lack on temperature stability and accuracy. In fact, the diode reverse
saturation current introduces an offset at the circuits output producing a
systematic error. The temperature dependence of the diode exponential response makes the circuit gain to drift with the temperature. Nevertheless
these circuits have a pedagogical interest and are the basis for more sophisticated solutions. For improved logarithmic circuits consult [1] chapter 7,
and [1] section 16-13 and[3].
Q
vi
R
−
vo
vi
R
−
+
vo
+
(b)
DR
(a)
AF
T
D
Figure 5.6: Elementary logarithmic amplifiers using a diode or an npn BJT.
112
CHAPTER 5. BASIC OP-AMP APPLICATIONS
5.2.1 Logarithmic Amplifier
Figure 5.6 (a) shows an elementary logarithmic amplifier implementation
whose output is proportional to the logarithm of the input. Let’s analyze
this non-linear amplifier in more detail.
The Op-Amp is mounted as an inverting amplifier, and therefore if vi
is positive, then vo must be negative and the diode is in conduction. The
diode characteristics is
i = Is e−qvo /k B T − 1 ≃ Is e−qvo /k B T
Is ≪ 1,
where q < 0 is the electron charge. Considering that
i=
vi
,
R
after some algebra we finally get
vo =
kB T
[ln (vi ) − ln ( RIs )] .
−q
5.2.2 Anti-Logarithmic Amplifier
AF
T
The constant term ln ( RIs ) is a systematic error that can be measured
and subtracted at the output. It is worth to notice that vi must be positive
to have the circuit working properly. An easy way to check the circuit is to
send a triangular wave to the input and plot vo versus vi .
Because the BJT collector current Ic versus VBE is also an exponential
curve, we can replace the diode with an npn BJT as shown in Figure 5.6.
The advantage of using a transistor as feedback path is that it should provide a larger input dynamic range.
If the circuit with the BJT oscillates at high frequency, a small capacitor
in parallel to the transistor should stop the oscillation.
DR
Figure 5.7 (a) shows an elementary anti-logarithmic amplifier, i.e. the output is proportional to the inverse of logarithm of the input. The current
flowing through the diode is
i ≃ Is e−qvi /k B T
Is ≪ 1.
5.2. LOGARITHMIC CIRCUITS
113
R
R
D
vi
vi
vo
−
Q
vo
−
+
+
(b)
(a)
Figure 5.7: Elementary anti-logarithmic amplifiers using a diode or an pnp BJT.
Considering that
vo = − Ri ,
thus
vo ≃ − RIs e−qvi /k B T .
If the input vi is negative, we have to reverse the diode’s connection. In
case of the circuit of Figure 5.7 (b) we need to replace the pnp BJT with an
npn BJT. Same remarks of the logarithmic amplifier about the BJT, applies
to this circuit.
Logarithmic
Amplifier
v2
Logarithmic
Amplifier
R
R
R
−
+
AF
T
v1
Anti−Log
Amplifier
vo
anti-logarithmic amplifiers
DR
Figure 5.8: Elementary analog multiplier implementation using logarithmic and
CHAPTER 5. BASIC OP-AMP APPLICATIONS
114
5.2.3 Analog Multiplier
Figure 5.8 shows an elementary analog multiplier based on a two log one
anti-log and one adder circuits. Fore more details about the circuit see [1]
section 7-4 and [1] section 16-13.
R
v1
v2
Logarithmic
Amplifier
Logarithmic
Amplifier
R
−
R
+
Anti−Log
Amplifier
vo
R0
Figure 5.9: Elementary analog divider implementation using logarithmic and
anti-logarithmic amplifiers.
5.2.4 Analog Divider
AF
T
Figure 5.8 shows an elementary logarithmic amplifier based on a two log
one anti-log and one adder circuits. Fore more details about the circuit see
[1] section 7-5 and [1] section 16-13.
5.3 Multiple-Feedback Band-Pass Filter
DR
Figure 5.10 shows the so called multiple-feedback bandpass, a quite good
scheme for large pass-band filters, i.e. moderate quality factors around 10.
Here is the recipe to get it working. Select the following parameter
which define the filter characteristics, i.e the center angular frequency ω0
the quality factor Q or the the pass-band interval (ω1 , ω2 ) , and the passband gain A pb
5.3. MULTIPLE-FEEDBACK BAND-PASS FILTER
115
C1
Vi
R3
C2
R1
−
G
Vo
+
R2
R0
Figure 5.10: Multiple-feedback band-pass filter.
ω0 =
√
ω2 ω1
ω0
Q =
ω2 − ω1
A pb < 2Q2
Set the same value C for the two capacitors and compute the resistance
values
Q
ω0 CA pb
Q
=
ω0 C (2Q2 − A pb )
2Q
=
ω0 C
R2
R3
Verify that
R3
< 2Q2
2R1
DR
A pb =
AF
T
R1 =
See [1] sections 8-4.2, and 8-5.3 for more details.
116
CHAPTER 5. BASIC OP-AMP APPLICATIONS
5.4 Peak and Peak-to-Peak Detectors
The peak detector circuit is shown in Figure 5.11. The basic ideal is to
implement an integrator circuit with a memory.
To understand the circuit letŽs first short circuit Do and remove R.
Then the Op-amp A0 is just a unitary gain voltage follower that charges
the capacitor C up to the peak voltage. The function of D0 and of A1 (high
input impedance) is to prevent the fast discharge of the capacitor.
Because of D0 the voltage across the capacitor is not the max voltage at
the input, and this will create a systematic error at the output vo . Placing
a feedback from vo to vi will fix the problem. In fact, because v+ must be
equal to v− , A0 will compensate for the difference.
Introducing the resistance (R ≃ 100kΩ) in the feedback will provide
some isolation for vo when vi is lower than vC .
The Op-Amp A0 should have a high slew rate (~20 V/µs) to avoid the
maximum voltage being limited by the Op-Amp slew rate.
The capacitor doesn’t have to limit the Op-Amp A0 slew rate S, i. e.
iC
dv
≪
=S
C
dt
DR
5.5 Zero Crossing Detector
AF
T
It is worthwhile to notice that if D0 and D1 are reversed the circuit
becomes a negative peak detector.
The technology of the hold capacitor C is important in this application.
The best choice to reduce leakage is probably polypropylene, and after
that polystyrene or Mylar.
Using a positive and a negative peak detector as the input of a differential amplifier stage we can build a peak-to-peak detector (for more
details see [1] section 9-1). Some things to check: holding time (a given %
drop from the maximum) versus capacitor technology, systematic errors ,
settling time required for the output to stabilize.
When vi is positive and because it is connected to the negative input then
vo becomes negative and the diode D1 is forward biased and conducting.
5.6. ANALOG COMPARATOR
117
R
D1
−
D0
−
vi
A
A
vC
vo
+
+
C0
RESET
Figure 5.11: Peak detector circuit.
5.6 Analog Comparator
An analog comparator or simply comparator is a circuit with two inputs vi ,
vre f and one output vo which fulfills the following characteristic:
vo =
V1 , vi > vre f
V2 , vi ≤ vre f
AF
T
An Op-Amp with no feedback behaves like a comparator. In fact, if we
apply a voltage vi > vre f , then V+ − V− = vi − vre f > 0. Because of the
high gain, the Op-Amp will set vo to its maximum value +Vsat which is a
value close to the positive voltage of the power supply. If vi < vre f , then
vo = −Vsat . The magnitude of the saturation voltage are typically about
1V less than the supplies voltages.
Depending on which input we use as voltage reference vre f , the Opamp can be an inverting or a non inverting analog comparator.
DR
It is worthwhile to notice that the analog comparator circuit is also a 1
bit analog digital converter , which converts voltages to the two levels V1
and V2 .
CHAPTER 5. BASIC OP-AMP APPLICATIONS
118
Vi
−
V1
Vo
+
R+
Rf
(utp)
V+
t
(ltp)
V+
+Vsat
t
−Vsat
Figure 5.12: Schmitt Trigger and its qualitative response to a signal that swings
up and down between and through the saturation voltages ±Vsat .
AF
T
5.7 Regenerative Comparator (The Schmitt Trigger)
1 Otto
DR
The Regenerative comparator or Schmitt Trigger1 shown in Figure 5.12 is a
comparator circuit with hysteresis.
It is important to notice that the circuit has a positive feedback. With
positive feedback, the gain becomes larger than the open loop gain making
the comparator to swing faster to one of the saturation levels.
Herbert Schmitt (1913-1998) American scientist considered the inventor of this
device, that appeared in an article in 1938 with the name of "thermionic trigger"[3].
5.8. PHASE SHIFTER
119
Considering the current flowing through R+ and R f ,we have
I=
V+ − Vo
V1 − V+
,
=
R+
Rf
⇒
V+ =
V1 R f + Vo R+
.
R f + R+
The output Vo can have two values, ±Vsat . Consequently, V+ will assume just two trip points values
(utp)
V+
=
V1 R f + Vsat R+
R f + R+
(ltp)
V+
(utp)
=
V1 R f − Vsat R+
R f + R+
(ltp)
When Vi < V+ , Vo is high, and when Vi < V+
To set V+ = 0 it requires that
V1 = −
, Vo is low.
R+
Vo
Rf
This circuit is usually used to drive an analog to digital converter (ADC).
In fact, jittering of the input signal due to noise which prevents from keeping the output constant, will be eliminated by the hysteresis of the Schmitt
trigger (values between the trip points will not affect the output).
See [1] section 11 for more detailed explanations.
Example1: (Vsat = 15V)
5.8 Phase Shifter
AF
T
Supposing we want to have the trip points to be V+ = ±1.5V, if we set
V1 = 0 then R f = 9R+ .
DR
A phase shift circuit shown in Figure 5.13, produces a sinusoidal signal
at the output Vo which is equal to the sinusoidal input Vi with a defined
phase shift . The basic idea of this clever circuit is to subtract using an
Op-Amp two equal sinusoids one of which is lagging behind because is
low pass filtered (the difference of these two same frequency sinusoid is a
phase shifted attenuated sinusoid). The Op-Amp provides also the unitary
gain.
CHAPTER 5. BASIC OP-AMP APPLICATIONS
120
R0
R0
−
Vi
Vo
+
C
R
Figure 5.13: Phase shifter circuit.
Using the same method applied to compute the differential Op-Amp
stage transfer function, we obtain
H (ω ) =
1 − jωRC
,
1 + jωRC
The amplitude and phase of the transfer function are therefore
H (ω ) = 1
for any frequency,
ϕ(ω ) = arctan (−ωRC) − arctan (ωRC) = −2 arctan (ωRC) .
Example
AF
T
The phase shift is double the one of a simple low pass filter and therefore can go from 0◦ down to −180◦ . Exchanging the potentiometer and
the capacitor changes the phase lag to a phase lead.
Supposing that we want a phase shift of −90◦ for a 1 kHz sinusoid with
capacitor with a reasonable capacitance value C = 10 nF, then
− tan
Cω
2
=
10−8
1
= 15.915 kΩ.
· 2π · 103
DR
R=
ϕ
The value of R0 can be in the range of few kilohoms to tens kilohoms.
5.9. GENERALIZED IMPEDANCE CONVERTER (GIC)
121
5.9 Generalized Impedance Converter (GIC)
A
Z1
+
A
−
Z2
Z
Z3
+
−
Z4
Z5
Figure 5.14: Generalized Impedance Converter (GIC).
The fundamental equation of the GIC, which can be found after tedious
calculation, is
Z Z3 Z5
Z= 1
Z2 Z4
• C to L converter and vice versa,
• impedance scaler
• negative resistance
DR
• LC parallel to LC series and vice versa,
AF
T
By a careful choice of passive components, one can implement types
of impedance with values impossible to attain with standard passive components. This holds true where the Op-Amps behave very closely to ideal
Op-Amps and if one lead of Z is grounded. For example, we can make:
CHAPTER 5. BASIC OP-AMP APPLICATIONS
122
This circuit is also a gyrator, i.e. a two voltage controlled current sources,
where the currents have opposite direction. For more details consult [7].
5.9.1 Capacitor to Inductor Converter
If we choose, for example,
Z1 = Z2 = R1 ,
Z3 = R3 ,
Z4 =
1
,
jωC
Z5 = R5 ,
then the GIC becomes
Z = jωR3 R5 C .
DR
AF
T
The circuit will behave as an inductor with inductance L = R3 R5 C. If
we chose large resistance values R3 , R5 and reasonably large capacitance
values C, we can implement very large inductance values practically impossible to attain using standard inductors.
5.10. PROBLEMS PREPARATORY TO THE LABORATORY
123
5.10 Problems Preparatory to the Laboratory
1. Considering the following circuit, determine the voltage output Vo
for the following input voltages Vi = −2V, 1V, 1.5V, 3V
+10V
Vi
−
Vo
G
+1.5V
+
−10V
2. Consider the Schmitt trigger of Figure 5.12.
(a) If Vo = −15V and V+ = 0V, compute V1 .
(b) If Vo = +15V, and V1 = 15V, compute V+ .
3. Design a Schmitt trigger with two diode clamps and one resistor connected to the output.
(a) Limit the output Vo from 0 to 5V.
(b) Compute the resistance value R necessary to limit the diode current to 10mA.
4. What is the practical maximum and minimum output voltage of the
logarithmic amplifier in Figure 5.6?
DR
AF
T
5. Chose and study at least two circuits to study and design, one from
this chapter and one form the next one on active filters .
New circuits different than those ones proposed in this chapter are
also welcome. For a good source of new circuits based on Op-Amps
see [1] , [4], and [2].
AF
T
CHAPTER 5. BASIC OP-AMP APPLICATIONS
DR
124
Bibliography
[1] Luces M. Faulkenberry, An introduction to Operational Amplifiers
with Linear IC Applications, Second Edition.
[2] Charles Kitchin and Lew Counts, A Designer’s Guide to
Instrumentation Amplifiers (2nd Edition), Analog Devices
(http://www.analog.com/en/DCcList/0,3090,759%255F%255F42,00.html)
[3] Theory and Applications of Logarithmic Amplifiers, National
Semiconductors, AN-311 ( http://www.national.com/an/AN/AN311.pdf ).
[4] Horowitz and Hill, The Art of Electronics, Second Edition
[5] Microelectronics Jacob Millman & Arvin Grabel, McGraw-Hill Electrical Engineering Series
[6] A thermionic trigger, Otto H Schmitt 1938 J. Sci. Instrum. 15 24-26.
125
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[7] Sedra Smith, Microelectronics Circuits, third edition, Oxford University Press.
BIBLIOGRAPHY
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126
Chapter 6
Active Filters
Introduction
An electronic circuit that modifies the frequency spectrum of an arbitrary
signal is called filter. A filter that modifies the spectrum producing amplification is said to be an active filter. Vis à vis its definition, it is convenient
to study the filter characteristics in terms of the frequency response of its
associated two port network
H (ω ) =
Vo (ω )
,
Vi (ω )
where Vi and Vo are respectively the input voltage and the output voltage
of the network, and ω the angular frequency. Depending on the design,
active filters have some important advantages:
AF
T
• they can provide gain,
• they can provide isolation because of the typical characteristic impedances
of amplifiers,
• they can be cascaded because of the typical characteristic impedances
of amplifiers,
Here some disadvantages:
127
DR
• they can avoid the use of inductors greatly simplifying the design of
the filters.
CHAPTER 6. ACTIVE FILTERS
128
• they are limited by the amplifiers’ band-with, and noise,
• they need power supplies,
• they dissipate more heat than a passive circuit.
Let’s make some simple definitions useful to classify different types of
filters.
6.1 Classification of Ideal Filters
Based on their magnitude response | H (ω )|, Some basic ideal filters can be
classified as follows:
|H(ω)|
1
|H(ω)|
1
0
ω0
ω
0
ω0
Low-pass
ω
High-pass
|H(ω)|
|H(ω)|
1
1
0
0
ω1
ω
Band-pass
|H(ω)|
1
0
Notch
ω
DR
ω0
ω0
ω1
Stop-band/band-reject
ω
AF
T
ω0
Practical filters approximate more or less the ideal definitions.
6.2. FILTERS AS RATIONAL FUNCTIONS
129
DA1
A1
DA3
A3
DA2
A2
ω1 ω2 ω3 ω4
ω 5 ω 6 ω 7ω 8
ω
Figure 6.1: Graphical definition of the filter performance specifications, and hypothetical filter response (red curve) that satisfy the specification.
Usually, the filter requirements are specified defining the band frequencies with their gains (attenuation or amplification) gain ripples, and slope
transitions in terms of power of the frequency. Figure 6.1 shows a quite
general graphical definition of the design parameters of a filter with an
hypothetical design. For a complete specification one should also define
the requirement for phase response.
6.2 Filters as Rational Functions
Let’s consider filters whose transfer function can be expressed as rational
function or standard form
β 0 + β 1 jω + β 2 ( jω )2 + . . . + β M ( jω ) M
For the filter not to diverge
M≥N
⇒
.
AF
T
H (ω ) =
α0 + α1 jω + α2 ( jω )2 + . . . + α N ( jω ) N
| H (ω )| < ∞ for any value of ω
DR
Writing the transfer function as a polynomial factorization we obtain
( ω − z1 ) n 1 ( ω − z2 ) n 2 . . . ( ω − z N ) n N
H (ω ) = k
( ω − p 1 ) m1 ( ω − p 2 ) m2 . . . ( ω − p M ) m M
CHAPTER 6. ACTIVE FILTERS
130
Denominator roots p1 , p2 , . . . , pn are called poles, and numerator roots
z1 , z2 , . . . , zm are called zeros. The integers n1 , n2, . . . , n N , and m1 , m2, . . . , m N
are therefore the multiplicity of poles and zeros.
Poles and zeros values determine the shape of the filter, and apart from
zero frequency, one could say that poles provide attenuation and zeros
amplification.
The transition from transmission to attenuation, and vice versa, in the
filter magnitude | H (ω )| is characterized by an asymptote slope which determine the so called filter order.
For example, considering the RC low pass filter with ω0 = 1/RC, we
have one pole p1 = jω0
H (ω ) =
ω0
⇒ first oder low pass filter with cut-off freq. ω0
ω0 + jω
For the RC high pass filter with ω0 = 1/RC, we have one pole p1 = jω0
and one zero z1 = 0
H (ω ) =
ω
⇒ first oder high pass filter with cut-off freq. ω0
ω0 + jω
AF
T
In the next sub-sections, we will analyze into more details filters with
the following transfer function
ω1
+ ω12
Q1
H (ω ) = H0
ω
−ω2 + jω 0 + ω02
Q
DR
−ω2 + jω
6.2. FILTERS AS RATIONAL FUNCTIONS
131
6.2.1 Second Order Low-Pass Filter
Figure below shows the second order low pass filter bode plot, with resonant frequency ωres , and characteristic frequency ω0
Hmax
Filter
ωres
Magnitude [dB]
10
ω0
0
−10
−20
−30
−40
2
10
ωres ω0 103
10
ωres ω0 103
Frequency [rad/s]
10
4
Phase [Deg]
0
−50
φres
−90
−100
−150
2
10
4
ω02
H0
ω
−ω2 + jω 0 + ω02
Q
Resonance
ω0
r
1−
1
2Q2
Maximum
H0 r
Q
1
1−
4Q2
DC
Gain
High Freq.
Gain
H0
0
DR
Transfer Function
AF
T
The second order low-pass filter written in standard form
CHAPTER 6. ACTIVE FILTERS
132
6.2.2 Second Order High-Pass Filter
Magnitude [dB]
Hmax
10
Filter
ωres
ω0
0
−10
−20
−30
2
3
10
4
ω0 ωres10
10
ω0 ωres103
Frequency [rad/s]
10
Phase [Deg]
150
100
90
φres
50
0
2
10
4
H0
ω2
ω
−ω2 + jω 0 + ω02
Q
Resonance
ω0
r
1
1−
2Q2
Maximum
H0 r
Q
1
1−
4Q
DR
Transfer Function
AF
T
The second order high-pass filter written in standard form
DC High Freq.
Gain
Gain
0
H0
6.2. FILTERS AS RATIONAL FUNCTIONS
133
6.2.3 Band-Pass Filter
Hmax5
Filter
ωres
Magnitude [dB]
0
ω0
−5
−10
−15
−20
−25
−30
2
3
10
4
ω0ωres 10
10
ω0ωres 103
10
Phase [Deg]
50
φres
0
0
−50
2
10
4
Frequency [rad/s]
AF
T
The band-pass filter written in standard form is
Transfer Function
Resonance
Maximum
DC Gain
High Freq.
Gain
ω0
Q
H0
ω
−ω2 + jω 0 + ω02
Q
ω0
H0
0
0
DR
jω
CHAPTER 6. ACTIVE FILTERS
134
For example, depending on the output we consider, the already studied LRC series circuit is a low-pass, a band-pass, or a high-pass filter with
the transfer function described above. When we will study difference filters topologies we will reduce their transfer function into one of the standard form above.
6.3 Common Circuit Filters Topologies
This is a brief and not exhaustive at all list of filter topologies that use
resistors, capacitors, and operational amplifiers to implement the filters
types described above:
• Infinite gain, multiple feedback (IGMF)
• State Variable (SV)
DR
• Switched Capacitor Filters (SC)
AF
T
• Generalized Sallen-Key (GSK)
Cascading these implementation allows to increase the filter order.
6.4. INFINITE GAIN MULTIPLE FEEDBACK CONFIGURATION (IGMF)135
6.4 Infinite Gain Multiple Feedback Configuration (IGMF)
I4
Vi
Y4
Y5
I3
I1
A VA
Y1
Y3
B
V−
V+
I2
Y2
−
Vo
+
Figure 6.2: Infinite Gain Multiple Feedback Filter.
Let’s consider the circuit in Figure 6.2 with generic admittances Y1 , Y2 , Y3 , Y4 ,
and Y5 . Applying the KCL to node A and considering the circuit virtual
ground (V− = 0), we have
VA Y3 + (Vo − VA ) Y4 + (Vi − VA ) Y1 + VA Y2 = 0 .
(6.1)
Again, applying KCL to node B and for the virtual ground we have
Vo Y5 + VA Y3 = 0
⇒
VA = −
Y5
Vo
Y3
AF
T
Replacing the last expression into eq. 6.1 and after some algebra we
obtain the generic transfer function for the circuit
Vo
Y1 Y3
.
=−
Vi
Y5 (Y1 + Y2 + Y3 + Y4 ) + Y3 Y4
DR
Choosing the proper type of admittances we can construct different
types of active filters, low-pass band-pass, and high-pass. It is worthwhile
noticing that IGMF configuration allows to implement low-pass, bandpass, and high-pass filter with capacitors, resistor and no inductors. This
simplifies considerably the design of the filters.
CHAPTER 6. ACTIVE FILTERS
136
6.4.1 Low-pass Filter
C5
R4
Vi
R1
R3
−
Vo
G
+
C2
Figure 6.3: Low-pass filter configuration of the infinite gain multiple feedback
filter.
A possible choice to implement a low-pass filter as shown in Figure 6.3 is
Y1 =
1
,
R1
Y2 = jωC2 ,
Y3 =
1
,
R3
1
,
Y5 = jωC5 ,
R4
and the transfer function of the circuit becomes
1/R1 R3
Vo
.
=−
Vi
jωC5 (1/R1 + jωC2 + 1/R3 + 1/R4 ) + 1/R3 R4
Y4 =
Vo
=
Vi
−
−ω2
1
R1 R3 C2 C5
AF
T
Rearranging the expression to obtain a rational fraction in ω we finally
obtain
1
1
+ jω (1/R1 + 1/R3 + 1/R4 ) +
C2
R3 R4 C2 C5
.
DR
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.1 we find that the frequency
ω0 , the quality factor Q, and the DC gain H0 are respectively
s
C2
R4
1
,
Q = ω0
.
,
H0 = −
ω0 =
R3 R4 C2 C5
R1
(1/R1 + 1/R3 + 1/R4 )
6.4. INFINITE GAIN MULTIPLE FEEDBACK CONFIGURATION (IGMF)137
6.4.2 High-pass Filter
C4
Vi
C1
R5
C3
−
Vo
G
+
R2
Figure 6.4: High-pass filter configuration of the infinite gain multiple feedback
filter.
A possible choice to implement a high-pass filter as shown in Figure 6.4is
Y1 = jωC1 ,
Y2 =
1
,
R2
Y4 = jωC4 ,
Y5 =
1
,
R5
Y3 = jωC3 ,
and the transfer function of the circuit becomes
Vo
jωC1 jωC3
.
= −1
1
/R5 ( jωC1 + /R2 + jωC3 + jωC4 ) + jωC3 jωC4
Vi
Rearranging the expression to obtain a rational fraction in ω we obtain
−ω2 (−C1/C4 )
.
1
1
+
−ω2 + jω (C1 + C3 + C4 )
R5 C3 C4
R2 R5 C3 C4
AF
T
Vo
=
Vi
DR
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.2 we find that the frequency
ω0 , the quality factor Q, High frequency gain H∞ are respectively
s
1
R5 C3 C4
C1
,
Q = ω0
,
H∞ = − .
ω0 =
R2 R5 C3 C4
C4
(C1 + C3 + C4 )
CHAPTER 6. ACTIVE FILTERS
138
6.4.3 Band-pass Filter
C4
Vi
R1
R5
C3
−
Vo
+
R2
Figure 6.5: Band-pass filter configuration of the infinite gain multiple feedback
filter.
A possible choice to implement a Band-pass filter is shown in Figure 6.5.
The admittances are
1
1
,
Y3 = jωC3 ,
,
Y2 =
Y1 =
R1
R2
1
,
R5
and the transfer function of the circuit becomes
Vo
jωC3 /R1
= −1
.
1
1
Vi
/R5 ( /R1 + /R2 + jωC3 + jωC4 ) + jωC3 jωC4
Y4 = jωC4 ,
Y5 =
DR
AF
T
Rearranging the expression to get a rational fraction in ω we finally
obtain
C3 + C4
jω
Vo
R5
R5 C3 C4
=−
.
C3 + C4
R1 + R2
Vi
R1
2
−ω + jω
+
C3 C4 R5
R1 R2 R5 C3 C4
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.3 we find that the resonance
frequency, and the quality factor are respectively
s
R1 + R2
R5 C3 C4
R5
ω0 =
,
Q = ω0
,
H0 = −
R1 R2 R5 C3 C4
C3 + C4
R1
6.5. GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK)
139
6.5 Generalized Sallen-Key Filter Topology (GSK)
I4
Vi
Y4
I3
I1
Y1
A VA
B
V+
I3 Y3
V−
Y2
C
I5
Y5
+
Vo
−
Y6
I6
Figure 6.6: Generalized Sallen-Key Topology.
Let’s consider the circuit in Figure 6.6 with generic admittances Y1 , Y2 , Y3 , Y4 , Y5 ,
and Y6 . Applying the KCL to node A , we have
(Vi − VA ) Y1 + (Vo − VA ) Y4 + (V+ − VA ) Y2 = 0 .
(6.2)
Applying KCL to node B
(V+ − VA ) Y2 + V+ Y3 = 0
⇒
VA =
Y2 + Y3
V+ .
Y2
Applying KCL to node C
⇒
V− = V+ =
Y6
V0 .
Y6 + Y5
AF
T
(V0 − V− ) Y6 − V− Y5 = 0
DR
Replacing the expression found for VA , and V+ into eq. (6.2) and after
quite some boring algebra, we obtain
Y1 Y2 Y6
Y5
Vo
= 1+
.
Vi
Y6 Y1 Y6 (Y2 + Y3 ) + Y3 Y6 (Y2 + Y4 ) − Y2 Y4 Y5
Let’s analyze some admittances’ configuration of the this filter topology.
CHAPTER 6. ACTIVE FILTERS
140
6.5.1 GSK Second Order Low-pass Filter
C4
R1
Vi
R2
Vo
+
−
C3
R6
R5
Figure 6.7: Low-pass filter configuration of the Generalized Sallen-Key filter.
A possible choice to implement a low-pass filter as shown in Figure 6.7 is
Y1 =
1
,
R1
Y4 = jωC4 ,
Y2 =
1
,
R2
Y5 =
1
,
R5
Y3 = jωC3 ,
Y6 =
1
,
R6
and the transfer function of the circuit becomes
1+
R6
R5
1
R1 R2 C3 C4
.
1
1
1
1 R6
2
−ω + jω
+
+
−
R1 C4 R2 C4 R2 C3 R5
R1 R2 C3 C4
AF
T
Vo
=
Vi
DR
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.1we find that the frequency
square ω02 , the quality factor Q, and the DC gain H0 are respectively
1
R1 R2 R5 C3 C4
R6
2
ω0 =
.
,
Q = ω0
,
H0 = 1 +
R1 C4 R2 C3
R5 ( R1 + R2 ) C3 − R1 R6 C4
R5
6.5. GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK)
141
6.5.2 Simple Case
If R1 = R2 = R, C3 = C4 = C, and R5 = R6 = 0, then
ω02
Vo
,
=
Vi
−ω2 + jωω0 + ω02
ω02 =
1
R2 C 2
,
,Q = 1,
which is the transfer function of a second order low-pass filter with low
quality factor.
6.5.3 GSK Second Order High-pass Filter
R4
Vi
C2
C1
Vo
+
−
R3
R6
R5
AF
T
Figure 6.8: High-pass filter configuration of the Generalized Sallen-Key filter.
To implement a low-pass filter as shown in Figure 6.8 one needs to choose
the admittances as follows
Y4 =
1
,
R4
Y2 = jωC2 ,
Y5 =
1
,
R5
Y3 =
1
,
R3
Y6 =
1
,
R6
DR
Y1 = jωC1 ,
and the transfer function of the circuit becomes
CHAPTER 6. ACTIVE FILTERS
142
Vo
=
Vi
R6
1+
R5
−ω² + jω
−ω2
1
1 R6
1
+
−
R3 C2 R3 C1 R4 C1 R5
1
+
R3 R4 C1 C2
.
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.2 we find that the frequency
square ω02 , the quality factor Q, and the DC gain H0 are respectively
1
R3 R4 R5 C1 C2
R6
2
ω0 =
.
,
Q = ω0
,
H0 = 1 +
R3 C1 R4 C2
R5 (C1 + C2 ) R3 − C1 R6 R4
R5
6.5.4 Simple Case
If R1 = R2 = R, C3 = C4 = C, and R5 = R6 = 0, then
Vo
−ω2
,
=
Vi
−ω2 + jωω0 + ω02
ω02 =
1
,
R2 C 2
,Q = 1,
which is the transfer function of a second order high-pass filter with low
quality factor.
6.5.5 GSK Band-pass Filter
R4
R1
C2
AF
T
Vi
Vo
+
−
R3
C3
R6
DR
R5
Figure 6.9: Band-pass filter configuration of the Generalized Sallen-Key filter.
6.5. GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK)
143
To implement a band-pass filter as shown in Figure 6.9 one needs to choose
the admittances as follows
Y1 =
1
,
R1
Y4 =
1
,
R4
Y2 = jωC2 ,
Y5 =
Y3 =
1
,
R5
1
+ jωC3 ,
R3
Y6 =
1
,
R6
and the transfer function of the circuit becomes
Vo
=
Vi
1+
R6
R5
jω
R1 C3
.
R
1
R1 + R4
1
1
C
+
C
6
2
3
2
+
+
+
−
−ω + jω
C2 C3 R1
C3 R3 C2 R4 C3 R4 R5
C2 C3 R1 R3 R4
Comparing the denominator of the previous equation with the denominator of the transfer function in section 6.2.3 we find that the frequency
square ω02 , the quality factor Q, and the DC gain H0 are respectively
C2 C3 R1 R3 R4 R5
,
(C2 + C3 ) R3 R4 R5 + C2 R1 R4 R5 + C3 R1 R3 R5 − C2 R1 R3 R6
ω02
R1 + R4
1
=
,H0 =
C2 C3 R1 R3 R4
R1 C3
R6
1+
R5
6.5.6 Simple Case
Q
.
ω0
AF
T
Q = ω0
If R1 = R3 = R4 = R, C2 = C3 = C, and R5 = R6 = 0, then
√
2
,
ω0 =
RC
,Q =
√
2
,
3
DR
ω0
Vo
Q
,
=
ω
Vi
−ω2 + jω 0 + ω02
Q
jω
which is the transfer function of a second order high-pass filter with low
quality factor.
CHAPTER 6. ACTIVE FILTERS
144
6.6 State Variable Filter Topology (SV)
The state variable filter provides a low pass, a band pass, and a high pass
filter outputs. At the same time, it allows to change the gain, the cut-off
frequencies, and the quality factor independently, but it requires 4 OpAmps.
Vi
R1
R2
R3
−
G
C1
R4
VHP
+
VBP
−
C2
R5
−
G
VLP
G
+
+
R6
R7
Figure 6.10: State variable filter circuit.
6.7 Practical Considerations
6.7.1 Component Values
AF
T
TBF
How do we select the values of capacitance and resistance? Here are some
considerations that should help the filter design:
DR
• reducing the resistance values reduces the thermal noise and therefore the filter noise,
• reducing resistance values minimizes the op-amp voltage offsets,
6.7. PRACTICAL CONSIDERATIONS
145
• increasing the resistance reduce the current load on the op-amps,
• increasing the resistances usually allows to decrease the capacitance
and therefore it make easier to find capacitors because of the small
capacitance values needed,
• reducing the capacitance minimizes the capacitance fluctuations due
to temperature,
• increasing the capacitance allows to reduce resistance values and
therefore the thermal noise.
As we can clearly see, some of the consideration cannot be used at the
same time. Based on the design requirements one can decide which of the
consideration above are more important to finally meet the design requirements.
Rules of Thumb
Particularly critical design often overrule these following rules:
• Capacitor with capacitance less of ~100 pF should be avoided,
• Try to use resistor with resistance between few kilo-ohms to few hundreds of kilo-ohms.
6.7.2 Components technology
Capacitors
Resistor
AF
T
The use of low loss dielectric is very important to obtain good results.
If possible one should use plastic film capacitors or C0G/NPO ceramic
capacitors, 1% tolerance for temperature stability.
DR
Low thermal noise resistors such as metal film resistors 1% tolerance for
temperature stability should be used.
AF
T
CHAPTER 6. ACTIVE FILTERS
DR
146
Bibliography
147
DR
AF
T
[1] Hank Zumbahlen, State Variable Filters, Mini Tutorial MT-223, Analog Devices
BIBLIOGRAPHY
DR
AF
T
148
Chapter 7
Basics on Oscillators
7.1 Introduction
AF
T
Waveform generators are circuits which provide a periodic signal with
constant frequency, phase, and amplitude. The quality of these devices
are measured by the frequency stability, amplitude stability, and absence
of distortion. The last characteristics is essentially cleanness of the spectrum signal. For example, the spectrum of a perfect sinusoidal oscillator
must be a delta of Dirac at the oscillating frequency. Practically, sinusoidal
oscillators has a sharp narrow peak at the oscillation frequency, and other
less taller peaks at different frequencies, mainly at multiples of the oscillation frequency (harmonics ).
In this chapter we will study the criterion to sustain a sinusoidal oscillation with a positive feedback amplifier, the so-called Barkhausen criterion, and some simple circuit to produce different waveforms.
Direct Digital Synthesis[1], a more versatile and effective technique to
produce arbitrary waveforms, is out of the scope of these simple notes.
7.2 Barkhausen Criterion
DR
Let’s consider an ideal amplifier with a positive feedback network as show
in figure 7.1. Considering that the summation point output is
Vi + β(ω )Vo ,
149
CHAPTER 7. BASICS ON OSCILLATORS
150
Vi + β Vo
Vi
Vo
A(ω)
βVo
β(ω)
Figure 7.1: Amplifier with positive feedback
and the amplifier gain is A(ω ), the output voltage will be
Vo = A(Vi + βVo ),
Collecting Vo we will finally have
Vo =
A
Vi .
1 − βA
For
| β(ω ) A(ω )| = 1,
arg [ β(ω ) A(ω )] = 0, 360, ...
ℜ[ βA] = 1,
AF
T
the output Vo diverges. If the previous condition is satisfied for the
angular frequency ω0 , any excitation at the frequency ω0 will make the
output to oscillate at the frequency ω0 with infinite amplitude. If Vi goes
to zero as fast as 1 − βA then the output will theoretically oscillate at the
frequency ω0 with amplitude A.
The previous condition which can be rewritten as
ℑ[ βA] = 0
(7.1)
DR
is the so called Barkhausen criterion for the oscillation.
The term βA is called the open loop gain or simply loop gain since
that is exactly the gain of the loop in the feedback amplifier network when
the loop is open at the summing point.
In the discussion of the oscillator circuits, we will assume that the amplifier is able to deliver the required positive or negative gain without
adding any additional phase. In the general case, this is clearly a crude
approximation, but it is used here just to simplify the study of the circuits.
7.2. BARKHAUSEN CRITERION
151
7.2.1 Gain Stability
Oscillators with exactly unitary open loop gain at a given frequency and
input Vi equal to zero at any time are just a mere mathematical abstraction.
In real circuits, there will always be some noise at ω0 and the gain cannot
be kept absolutely stable. For example, external perturbations, drifts due
to temperature, and components aging would make these two conditions
impossible to keep.
Practically, it is necessary to have a loop gain βA somewhat larger than
unity to start and sustain the oscillation. This can lead to a slow drift of
the oscillation amplitude, and in the worst case, the oscillation can even
saturate or stop.
It is worthwhile to notice that large values of the amplifier gain A, that
produce saturation at the output, can be used to generate squares or pulse
waves. Moreover, cascading a proper filtering stage, one can select just
one frequency and make a quite amplitude stable sinusoidal generator.
7.2.2 Automatic Gain Control (AGC)
7.2.3 Oscillation Kick-start
AF
T
To properly sustain the oscillation in case of temperature drifts, we need
to add to the positive feedback path another feedback loop this time negative to stabilize the gain. This path often called Automatic Gain Control
(AGC) circuit can be done using temperature sensitive components. For
example, semiconductor diodes, transistors, or even incandescent bulbs,
whose resistivity increases or decreases with temperature can be used in
the AGC.
DR
We don’t have to provide an initial kick to start the oscillation. This is
true, because every time we turn on a circuit on or we toggle a switch, a
step like perturbation propagates through the circuit providing an initial
excitation at the right frequency. Moreover, the probability to have a small
signal perturbation ( due to the omnipresent noise ) at the right frequency
is usually quite high.
CHAPTER 7. BASICS ON OSCILLATORS
152
VDD
RD
C
C
C
Vo
Vo
Vi
Q0
gmVi
R
R
R
rd
RD
RS
Figure 7.2: Phase shift oscillator using a JFET as amplification stage (left
gray rectangle) and a phase shift network (right gray rectangle). The circuit on the left represents the low frequency model of the JFET amplifier.
7.2.4 Frequency Stability
7.3 Phase Shift Oscillator
AF
T
The frequency stability of an oscillator is a quite complex topic of study.
Here we can simply say that it depends mainly on the ability of the circuit to maintain the loop gain phase constant to 0◦ or to multiples of 360◦ .
Phase fluctuations will therefore introduce noise in the oscillator frequency.
DR
The phase shift oscillator exemplifies the concepts set forth above. Referring to figure 7.2, we can distinguish the JFET amplifier stage and the
positive feedback network made of three cascaded RC phase shifting filters.
Supposing that the amplifier load ZL is negligible, i.e. | ZL | ≫ RD ||rd
then, the amplifier will just change sign (180◦ ) to any signal injected in the
gate. The network feedback will provide additional phase shift to satisfy
the Barkhausen criterion at a given angular frequency ω0 .
7.4. THE WIEN-BRIDGE OSCILLATOR
153
It can be proved that
β(ω ) =
Vi
=
Vo
1−
5
(ωτ )2
+j
1
1
(ωτ )3
6
ωτ
−
τ = RC ,
(7.2)
The amplifier gain, supposed to be constant is A = − gm RD , where gm is
the JFET amplifier gain.
Imposing the condition ℑ[ βA] = 0, we get
1 1
ω0 = √ .
6τ
Replacing the previous expression in the open loop gain Aβ and using
the second condition ℜ [ βA] = 1, we get
gm RD = 29
To sustain the oscillation, the amplifier must have a gain of at least
29/RD .
7.4 The Wien-Bridge Oscillator
The Wien Bridge Oscillator show in figure 7.3, uses a differential amplifier
to provide positive and negative feedback to satisfy the two condition of
oscillation.
Referring to figure 7.3 , setting YC = 1/ ( jωC) , and thanks to the voltage divider equation we can write
R + YC +
and
β(ω ) =
RYC
YC + R
Vo =
1
(YC + R)
RYC
2
1
V+
=
Vo
3 + j ωτ −
Vo =
+1
1
ωτ
1
YC
R
AF
T
V+ =
RYC
YC + R
+
R
YC
+3
Vo
τ = RC.
ωτ −
1
= 0,
ωτ
⇒
DR
The oscillation will happen where the phase shift is zero, i.e. for
ω0 =
1
.
τ
CHAPTER 7. BASICS ON OSCILLATORS
154
Rf
Vo
R−
−
V+
Vo
A
R
Rf
C
+
V−
R
V+
R
C
R−
C
R
C
Figure 7.3: Wien Bridge oscillator, and components rearrangement to
show the bridge topology.
The angular oscillation frequency ω0 depends on the inverse of the resistance R and the capacitance C.
Because the attenuation at the resonant frequency is
V+
1
= .
Vo
3
Rf
Vo‘
= 1+
.
V+
R−
AF
T
the negative feedback must have a theoretical gain of A(ω0 ) = 3. The
resistances R− and R f must be given by the usual equation
DR
The oscillation frequency can be continuously tuned using coupled
variable resistors.
To minimize distortions due to the Op-amp saturation when the gain
is larger than one, it is required to provide a circuit with variable gain.
Essentially, we need an overall gain larger than one for small signal to
sustain the oscillation and gain of about 1 or less for large signal to avoid
distortion. The negative feedback path shown in figure 7.4 does the job.
7.5. LC OSCILLATOR
155
D1
D0
Rf
Rf
Figure 7.4: Automatic gain control circuit for the Wien bridge oscillator
negative feedback.
For large signals one of the diodes becomes forward biased reducing the
feedback resistance and the Op-Amp gain. For smaller signal the gain is
not affected by the diodes.
Practically, Wien Bridge oscillators are used in the kilohertz region with
a variable range up to ~10 times ω0 .
7.5 LC Oscillator
A quite general form of oscillator circuits is depicted in figure 7.5. In this
case it is not straightforward to separate the oscillating feedback network
and the amplifier itself. Let’s suppose that the amplifier is ideal but has a
non zero output resistance Ro . Referring to figure 7.5 we have
β=
Vi
.
V0∗
Vo =
or
Z
V0∗ ,
Z + Ro
Z = Z2 || ( Z1 + Z3 ) ,
DR
and
AF
T
Applying the voltage divider equation twice we have the two equations
Z1
Vo
Vi =
Z1 + Z3
Z
1
1
=
.
∗
Vo
Z + Ro V0
CHAPTER 7. BASICS ON OSCILLATORS
156
Vi
Ro
Vo*
+
A
Vo
Vo
−
Z3
+
A(V+− V− )
Z2
Z3
Vi
Z2
−
I=0
Z1
Z1
Figure 7.5: LC Oscillator circuit using an ideal Op-Amp with non zero output
impedance Ro and its equivalent ideal circuit. Note that the feedback loop is
connected to the negative input of the amplifier, and therefore to get a positive
loop feedback the feedback network has to flip the signal phase by 180◦ .
After some algebra we finally get
β=
Z1 Z2
.
Ro ( Z1 + Z2 + Z3 ) + Z2 (Z1 + Z3 )
(7.3)
Let’s consider the case of the LC tunable oscillators, i.e. the impedances
are purely reactive (real part equal to zero)
Zi = jXi ,
Xi > 0
for i = 1, 2, 3
β=
For β to be real
AF
T
Then the previous eq. (7.3) becomes
− X1 X2
.
jRo ( X1 + X2 + X3 ) − X2 (X1 + X3 )
X1 + X2 + X3 = 0 ,
and
X1
,
X1 + X3
DR
β ( ω0 ) =
where ω0 is the oscillation frequency. Using the two previous equation we
finally get
7.6. CRYSTAL OSCILLATOR
157
X
β ( ω0 ) = − 1
X2
⇒ AOL
X
= −A − 1
X2
.
Since AOL must be positive and A > 0, then X1 and X2 must have same
sign. For example they have be both capacitors or inductors. From the
condition of imaginary part equal to zero we find that if X1 and X2 are capacitors, then X3 must be an inductor, and vice versa. Here is the oscillator
circuit name depending on the choice of the reactance:
• Colpitts Oscillator: X1 and X2 capacitive reactances and X3 an inductive reactance ( X1,2 = −1/(ωC1,2 ), X3 = ωL3 ).
The oscillator angular frequency and the gain in this case are
v
u
1
C2
u
,
ω0 = t β ( ω0 ) =
C1 C2
C1
L
3
C1 + C2
• Hartley oscillator: X1 and X2 inductive reactances and X3 a capacitive reactance ( X1,2 = ωL1,2 , X3 = −1/(ωC3 )).
The oscillator angular frequency and the gain in this case will be
s
1
L
ω0 =
,
β ( ω0 ) = 1
C3 ( L1 + L2 )
L2
7.6 Crystal Oscillator
AF
T
Using a BJT amplifier we can usually obtain higher oscillating frequency
than using standard operational amplifiers. In this case the high frequency
hybrid-π model[2] must be used to properly model the transistor behavior. Moreover, the BJT amplifier low input impedance makes the design
more complicated.
1 Piezoelectricity
DR
Crystal oscillators are based on the property of piezoelectricity1 exhibited
by some crystals and ceramic materials. Piezoelectric materials change
was discovered by Jacques and Pierre Curie in the 1880’s during experiments on quartz.
CHAPTER 7. BASICS ON OSCILLATORS
158
C3
L3
C2
C1
L2
L1
Figure 7.6: Colpitts (left) and Hartley (right) feedback circuits β(ω ) for the
LC oscillator circuit of Figure 7.5.
2 Mechanical
AF
T
size when an electric field is applied between two of its faces. Conversely,
if we apply a mechanical stress, piezoelectric materials generate an electric
field. Some crystals have internal mechanical resonances with very high
quality factors (quartz can reach quality factors of 104 ) 2 and can be indeed
used to generate very stable oscillators.
Figure 7.7 shows the circuit symbol for a piezoelectric component and
the equivalent circuit modeled using ideal components.
Usually, to apply an electric field to a crystals is necessary to make a
conductive coating on two parallel faces, and this process creates a capacitor with an interposed dielectric. This explain the presence of the capacitor
of capacitance C p in the model. The LCR series circuit accounts for the particular mechanical resonance we want to use to build the oscillator.
To design a crystal oscillator it is important to study the reactance ( the
imaginary part of the impedance) whose qualitative behavior is shown in
figure 7.8. Where the reactance is essentially inductive and very close to
the resonance, the crystal behaves as a simple equivalent inductor. We can
indeed replace the inductor Ls of the LC oscillator of figure 7.5 with the
piezoelectric crystal to build a simple oscillator.
Crystal oscillators using a Colpitts configuration and a BJT in commonemitter or common-collector configuration, can work from few kHz up to
DR
resonance stability depends mainly on the fact that the resonance value
is determined by the crystal geometry. It the crystal size slightly depends on the temperature we can have very stable resonators. Active temperature stabilization can clearly
improve frequency stability.
7.6. CRYSTAL OSCILLATOR
159
Cs
Ls
Cp
Rs
Figure 7.7: Circuit symbol for a piezoelectric oscillator (or quartz oscillator) and the equivalent electronic circuit. The LCR series circuit accounts
for the sharp mechanical resonance The capacitor C p in parallel describes
the capacitance of the crystal for frequency far for the resonance.
X
X(ω)
Inductive
Half Plane
Capacitive
Half Plane
AF
T
ω
DR
Figure 7.8: Qualitative behavior of the crystal reactance versus frequency.
160
CHAPTER 7. BASICS ON OSCILLATORS
~100MHz.
7.7 Relaxation Oscillators
Relaxation oscillators include a wide class of non-linear systems many of
them found in different fields such as mechanics, biology, chemistry, electricity, to just mention a few.
Relaxation oscillators are characterized by the following properties:
• a non linear mechanism that provides a bistable state,
• a relaxation process that creates the transition from one stable state
to the other,
• a period of oscillation characterized by a relaxation phenomena, i.e.
by the time constant of the relaxation process,
7.7.1 Square Wave Generator
AF
T
The canonical example of relaxation oscillator is the seesaw with one bucket
on one end and a weight on the other with the bucket continuously filled
by a constant water flow. When the bucket is filled, it changes the equilibrium of the seesaw, and the system transitions to the new state of equilibrium. In the new state, the bucket is tilted enough to be emptied and
therefore the system transition back to the older state. The seesaw + waterflow is clearly the bistable nonlinear system, and the relaxation process is
the emptying of the bucket.
Balthasar van der Pol3 was one of the first to analyze a relaxation oscillator system.
3 Dutch
DR
A very simple relaxation oscillator is the RC charge discharge oscillator
show in figure 7.9. The ideal Op-Amp is configured as a Schmitt trigger
which provides the non-linear bistable states. The negative feedback provides the relaxation mechanism.
To qualitatively understand the circuit, let’s suppose that the Schmitt
trigger output rails down to the Op-Amp power supply voltage −Vss . The
physicist of the beginning of 20th century whose work covered several different fields such as applied mathematics, radio waves, and electrical engineering.
7.7. RELAXATION OSCILLATORS
C0
R
vC
161
+Vss
2 vC
+Vss
−
t
vo
G
−Vss
2
+
−Vss
+Vss
vo
t
R+
Rf
−Vss
Figure 7.9: The Op-Amp version of the RC Charge Discharge Oscillator.
capacitor will start to charge down and its voltage VC will swing down to
reach −Vss with a characteristic time constant τ = RC. Once VC ≤ −Vss /2
, the Schmitt trigger output will switch to +Vss and the the capacitor voltage VC will start swinging to +Vss with the same characteristic time constant τ. This cycle will keep repeating generating a square wave at the
Schmitt trigger output.
The Period of the oscillation can be computed considering the time for
exponential decay with time constant τ to go from Vss /2 to −Vss /2. After
some algebra one obtains
T = 2 log (3) τ
7.7.2 Triangular Wave Generator
AF
T
which shows in this case ( same resistors on the positive feedback loop)
that the period does not depend on the voltage limits but only on τ. In a
more general case when the positive feedback loop resistors R0 are different, T will depend also on the values of those resistors.
DR
A triangular waveform generator can be easily built by cascading a Schmitt
Trigger and an Op-amp integration stage as shown in Figure 7.10.
Let’s suppose that the Schmit Trigger output vst has railed up to +VSS .
A current Vss /R will start charging the capacitor C and as consequence,
the output vo which is connected to the capactor will decrease. Once vo is
lower than the Schmitt Trigger lower tripping voltage VLT , vst will transition to −Vss and the capacitor will start discharging and vo will increase.
CHAPTER 7. BASICS ON OSCILLATORS
162
When vo will reach the high trip voltage VHT then vst will go back to
+Vss and the cycle will repeat again over and over, generating a triangular
wave.
Let’s now figure out the period of the triangular waveform by finding
out the time to charge and discharge the capacitor. If vst is at +Vss , then vo
will go from VHT down to VLT with a slope Vss /RC. From Figure 7.10 we
can easily se that
VLT − VHT
Vss
,
=−
T1
RC
and
T1 = RC
VHT − VLT
Vss
Analogously
Vss
VHT − VLT
=
,
T2
RC
and
T2 = RC
VHT − VLT
Vss
Replacing the tripping voltages expression
VHT = −VLT =
R+
Vss
Rf
we finally obtain the triangular wave period T
R+
−
Vss
Rf
= 4RC
R+
.
Rf
AF
T
Vss
!
DR
T = T1 + T2 = 2RC
R+
Vss −
Rf
7.7. RELAXATION OSCILLATORS
163
vo
V HT
C
+Vss
−
t
v st
G
R
v−
V LT
G
+
−Vss
R+
vo
−
+
T1
T2
vst
+Vss
vo
t
Rf
−Vss
DR
AF
T
Figure 7.10: Triangular wave form generator made using a Schmitt Trigger
and an Op-amp integration stage.
AF
T
CHAPTER 7. BASICS ON OSCILLATORS
DR
164
Bibliography
[1] http://www.analog.com/UploadedFiles/Tutorials/450968421DDS_Tutorial_rev122-99.pdf
165
DR
AF
T
[2] Microelectronics, Jacob Millman, and Arvin Grabel , Mac-Graw Hill
BIBLIOGRAPHY
DR
AF
T
166
Chapter 8
Phase Locked Loop (PLL)
A phase locked loop PLL1 is a circuit with a feedback network that synchronizes an oscillator, the reference oscillator (REF), to another oscillator,
the controlled oscillator (CO), so that they will oscillate (be locked together)
at the same frequency.
vi
Input
vph
Phase
Detector
vlp
Filter
Voltage
Controlled
Oscillator
vo
Output
Phase Locked
Loop
AF
T
Figure 8.1: Phase-Locked loop block diagram.
1 It
DR
To implement a PLL we need a circuit that generates a signal proportional to the phase difference between the REF and the CO. This continuously changing signal is then used to correct the frequency of the CO to be
the same of the REF. In fact, if the phase difference is constant or zero the
two oscillators must have the same frequency. In other words, keeping the
phase difference constant makes the oscillator frequencies the same.
seems that the first phase locked loop was proposed by the French scientist De
Belleseize in 1932.
167
CHAPTER 8. PHASE LOCKED LOOP (PLL)
168
v2
v1
vmix
R
vlp
C
Figure 8.2: Mixer Phase-Detector and low-pass filter. The two cascaded
circuits produce the error signal to be sent to the VCO.
Another way to see this is that the time variation of phase is proportional to the frequency difference between the oscillators, and therefore the
phase difference is the signal we need to correct the change of frequency
between the two oscillators.
Let’s look the at Figure 8.1 containing the block diagram of a basic PLL.
The reference signal from REF is sent to the PLL input, goes into the phase
detector, which gives a signal proportional to the phase difference between
the CO and the reference. This signal has high frequency noise and in general needs to be low pass filtered and compensated. Then, the filtered signal goes to the voltage controlled oscillator (VCO). The VCO, the core of the
PLL, has a circuit to control the frequency by changing its voltage input.
This input is therefore driven with a voltage with the proper sign to zero
the phase difference between the reference frequency and the CO.
TBD
8.2 Phase Detector
AF
T
8.1 Phase Locked Loop Basic Analysis
DR
Phase detectors convert the phase difference between two signals into a
signal proportional to the phase difference.
Phase detectors can be classified into two types. Type I phase detectors
are designed to be driven by analog signals, whereas Type II are driven by
digital signal and in particular by the transitions/edges of such signals.
8.2. PHASE DETECTOR
169
8.2.1 Analog Mixer Phase Detector
The analog mixer is a device that ideally multiplies two arbitrary signals.
If the two signals are simple sinusoids with the same frequency and different phase, the output can be decomposed into two components. If the
two input signals are
v1 = V1 sin (ωt) ,
v2 = V2 sin (ωt + δφ0 ) ,
then after some algebra, the multiplied signal vmix (the mixer output) will
be
1
1
vmix = v1 v2 = V1 V2 sin (δφ0 ) + V1 V2 sin (2ωt + δφ0 ) .
2
2
The output of the mixer is therefore a signal proportional to the phase
difference of the two sinusoidal signals plus another component proportional to a sinusoid at twice the frequency of v1 or v2 . Applying an appropriate low pass filter, we can remove the two omega component and
finally get our wished phase difference detector.
8.2.2 Logic Gate Phase Detector
DR
AF
T
Another basic type I phase detector, a XOR logic gate with a low-pass filter, is shown in Figure 8.3 together with the time plots of the circuit voltages, and the phase detector characteristic. As shown in the time plots, the
XOR pulses output v xor have a duration of the time difference ∆T between
the two same period T input square waves v1 and v2 . Those pulses are
then integrated by the low pass filter whose ouput vl p will be on average
proportional to ∆T and therefore to the phase difference ∆φ between the
square waves v1 and v2 .
Let’s find the phase detector characteristic of the XOR PD. If there is no
time difference and the capacitor is not charged then vl p is zero. As soon a
time difference appears, the capacitor will charge and vl p will increase. If
we keep increasing ∆T, vl p will reach a maximum when v1 and v2 are off
by T/2 because v xor is always on. Then, if we keep increasing even further ∆T, the capacitor will start discharging and vl p will decrease and will
eventually reach zero. By continuing to increase ∆T, this cicle will repeat
producing a triagular plot as shown in the phase detector characteristic of
Figure 8.3.
CHAPTER 8. PHASE LOCKED LOOP (PLL)
170
v1
v1
vxor
R
t
vlp
v2
v2
XOR
t
C
vxor
t
vlp
t
vlp
Average
0
π
2π
3π ∆φ
Figure 8.3: Phase-Detector and low-pass filter. The two cascaded circuits
produce the error signal proportional to the phase difference which is then
sent to the VCO.
AF
T
8.3 Voltage Controlled Oscillator (VCO)
As we already said before, a voltage controlled oscillator is an oscillator
whose frequency can be controlled by changing the circuit voltage input.
The circuit shown in Figure 8.4 shows how to implement such a circuit using an analog multiplier and the triangular waveform generator explained
in chapter7.
DR
Looking at the VCO circuit, we see that effect of the analog multiplier
placed at the output of the Schmitt Trigger is simply to change the gain of
the output vst by a factor vi /VR and therefore the wave slopes by the same
factor. As a consequence, the wave period will change. Considering this
8.4. VARACTORS OR VARYCAP
171
C
vi
+Vss
R
v st
−
G
v−
G
+
−Vss
R+
vo
−
+
Rf
Figure 8.4: Voltage Controlled Oscillator (VCO) circuit built using the triangular waveform generator explained in chapter7.
scaling factor, the frequency of the triangular waveform is then
ν ( vi ) =
1 R f vi
.
4RC R+ VR
(8.1)
8.4 Varactors or Varycap
DR
AF
T
A varactor diode or varycap is a voltage controlled capacitance. It is essentially a reverse biased p-n junction whose capacitance increases if the reverse bias decreases.
Intuitively, a reverse biased p-n junction is a capacitor with the depletion region acting as an insulator. Increasing the reverse bias the p-n depletion region increases and therefore the capacitance decreases. The major
difference between a varactor and a diode is that the varactor is optimized
to be a variable capacitance (as much as the technology allows) controlled
with a bias.
Typical values are from tens to hundreds of picofarads. Because the
small variation of capacitance available they cannot be effectively used at
low frequency.
Varactors commonly available are the Motorola’s MVAM115, and the
Phillips BB112, BB212, BB204.
172
CHAPTER 8. PHASE LOCKED LOOP (PLL)
8.5 CMOS 4046 PLL Circuit
The CMOS 4046 PLL is an integrated circuit which implements a VCO
and two PDs and some extra circuits to simplify the construction of a PLL
circuit.
The VCO frequency range is set with the components R1 , R2 , and C1 .
Resistor R1 and capacitor C1 values set the maximum frequency f max of
the VCO. Resistor R2 and Capacitor C1 set an optional frequency offset
f min . The values limitations are:
• 5kΩ ≤ R1 ≤ 1MΩ
• R2 ≤ 1MΩ
• C1 ≥ 100pF, 5V ≤ VDD < 10V
• C1 ≥ 50pF, 10V ≤ VDD < 20V
VCO input has a very high input impedance which allows us to use a wide
range of values for the capacitor C and the resistor R for the low-pass filter
circuit.
8.6 Pre-lab Problems
• Derive equ. 8.1 using the analysis done in chapter 7.
AF
T
• Considering that VR = 10 V, and a max VCO input voltage vi = 5 V,
determine the values of R+ , R f , R and C to set the VCO frequency
between 0 Hz and 10 kHz.
• Sketch a complete PLL circuit using the VCO circuit reported in section 8.3 and the mixer phase detector reported in section 8.2.1. Use
the square wave output of the VCO to close the phase locked loop.
DR
• Considering that the mixer has differential inputs, modify your previous sketch by adding a circuit to the VCO input that allows the
oscillator to oscillate at 5 kHz frequency (free runnning frequency)
when the the PLL input is disconnected. Use the VCO components
values computed in the previous exercise.
8.6. PRE-LAB PROBLEMS
vi
173
VDD
16
14
8
vlp
2
3
vo
XOR
C
4
R
PD
6
C1
7
vVCO
9
VCO
R1
11
R2
12
SIMPLIFIED
CD4046B
DR
AF
T
Figure 8.5: Simplified circuitry of the CMOS 4046 with components to set
the VCO frequency range and the low-pass circuit compensating circuit.
CHAPTER 8. PHASE LOCKED LOOP (PLL)
174
fVCO
fmax
fc
fmin
vmin
VDD
2
vmax
VDD
vVCO
Figure 8.6: VCO characteristic of the CMOS 4046 .
DR
AF
T
• Determine the value of the mixer low-pass filter components R, C,
for a cut-off frequency of 1 kHz.
Bibliography
[1] William F. Egan, Phase-Lock Basics, Wiley-Interscience, 1998.
-
Third
Edition,Wiley-
AF
T
Techniques
175
DR
[2] Floyd Gardner, Phaselock
Interscience, 2005.
BIBLIOGRAPHY
DR
AF
T
176
Chapter 9
Final Project, Some Useful
Circuits
This appendix is a collection of practical consideration, suggestions, and
useful simple circuits to be used with transducers that can help you building your final project.
9.1 Solderless Breadboard
The figure below shows the circuit breadboard contacts topology and suggested power supply connections.
W
X
A
B
C
D
E
TRENCH
10
15
20
25
30
35
40
45
50
F
G
H
I
J
Z
Y
60
−15V
GND
+15V
DR
IC
55
AF
T
5
Here some rules and guidance to follow when assembling a circuit using a solderless breadboard
177
178
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
• Wires gauge AWG 24 must be used.
• Do not force thick leads into the board holes connections. Leads fit
must be somehow loose to allow the spring loaded contact to work
properly.
• Green jacket wires should be used to connect components to ground
(GND), red jacket wires for +15 V, and black jacket wires for −15 V.
A different color should be used for feedback connections.
• Component’s leads should be cut short to place the components as
close as possible to the board.
• ICs must be placed on and along the trench otherwise opposite on
side IC’s pins will be short circuited.
• Always double check power supply connections to avoid damaging
or more likely destroying ICs components.
This type of pragmatism will minimize the number of surprises you will
have building your circuit and also will help you debug your circuit.
9.2 Decoupling/Bypassing Capacitor
Decoupling or bypassing capacitors are capacitor placed between the ICs
power supply inputs and ground to filter out (decouple) noise coming
from the power supply lines.
We can somehow distinguish two type of noise:
AF
T
• transient noise due to fluctuation of the voltages generated by sudden increase of current demanded by the ICs,
• broad band or high frequency noise either random or constant with
a given frequency spectrum.
DR
Large capacitors acting as a current reservoir are needed to compensate
for voltage transients, and small capacitor are required for high frequency
noise filtering.
A typical value for high frequency noise decoupling capacitors is 100 nF.
These capacitors should be placed very close to the ICs power supply inputs to avoid the inductive effects of long electric connections. The faster
9.3. VARIABLE RESISTORS
179
the IC circuitry is the more important the decoupling becomes especially
for high frequency noise. For transient noise, much larger capacitor with
values, between 1 µF to 10 µF, are necessary to provide some kind of "current reservoir". Apart from special cases, they can be placed somewhere
in the circuit board. Usually, decent off the shelf power supplies already
include those capacitors.
A very extensive source of information about decoupling capacitor is
found in the bibliography.
9.3 Variable Resistors
It is good practice to place potentiometer to set amplifier gains, match resistor pairs, or in circuits to null voltage offset or set the proper voltage
divider outputs. Variable resistors come in so many flavors that it is time
consuming to choose the right components. For the project purpose, it is
convenient to use at least 10 turn trimmers potentiometers with up right
wiper screw, which are compact and fit reasonably well onto the solderless
breadboard. As shown in the figure below, trimmers should be placed in
series with a resistor every time one needs to limit the current to a maximum . The figure also shows where usually the wiper (B) is located.
Vi
Rx
Vo
Wiper
Wiper
DR
9.4 Surface Mount Devices
AF
T
R
Due the absence of leads, Surface Mount Devices (SMD) cannot be easily used with solderless prototyping boards. Adapters, which permits to
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
180
solder SMDs onto a small printed circuit boards with legs, can be used to
properly wire these type components on the solderless board.
There is a plethora of SMD standards such as Small Outline Integrated
Circuit (SOIC), mini-SOIC, Small Outline Package (SOP), Shrink SOP (SSOP)
Thin SSOP (TSSOP), et. cetera. Each standard differs by ICs dimensions
(especially the footprint) and distance between leads. Some of them have
so small packages that it is practically impossible to solder ICs without
expressely made tooling.
Soldering SOIC components on PCBs is somehow a quite simple operation once done two or three times. SOIC leads are about 1.3 mm apart
and therefore small an narrow solder iron tips allow to solder each lead
at a time. Smaller packaging are quite difficult to solder and because of
their small size, it is easy to damage the ICs if overheated. Consult the
data-sheet before soldering SMDs on adapters to check how long the component can withstand the maximum temperature. SOIC adapters with 8
pin connection are available in the laboratory.
9.5 Switches
Switching circuits to drive high power load are shown in the figure below
Vcc
Load
R
D0
BJT
1k−3.3kΩ
Load
R
AF
T
D0
D1
D1
Vcc
N−MOSFET
15kΩ
DR
The diode D0 at the input is necessary in case the switch input can go
lower than zero volts. The diode in parallel with the load is necessary if the
load is inductive to damp over-currents generated when the load switch
status. Typical inductive load are relays whose large inductance values
come from the relay’s solenoid.
9.6. VOICE COIL LOUD-SPEAKERS DRIVER
181
9.6 Voice Coil Loud-speakers Driver
Voice coil loud-speakers are transducers with a quite low input impedance,
usually of 4 Ω or 8 Ω and few to several watts of power consumption. They
therefore need quite some current to be properly driven. For our purposes
(unless the project is to make a very powerful speaker) a 50 mW to 100 mW
speaker for the mid-range acoustic frequencies is more than sufficient.
Usually, because Op-Amps can provide no more than few milliamperes,
they cannot drive a speaker directly. A current booster such as the pushpull amplifier with BJTs able to dissipate about 0.5 W, which can increase
the current by a factor 10 to 100 easily, can drive a small speaker in the
mid-range acoustic frequencies. It is fundamental to limit the current to
ensure that the transistor power rating are not exceeded. The circuit below includes the resistor necessary to limit the current flowing through
the BJTs which needs to be calculated based on the power supply voltage
VCC and the rms dissipation Prms the transistors can sustain. Resistors R
power dissipation must be also computed .
Vcc
vi
R
R0
R1
v+
Q1
−
G
vo
vA
+
Q1
R
RL
DR
−Vcc
AF
T
v−
Heat-sinks should be placed on the BJTs to ensure that the amplifier
works reliably.
182
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
9.6.1 Off the shelf Audio Amplifiers
ICs power audio amplifier suitable to power small speakers (~1 Wspeakers)
are for example the LM380 and the LM386. Heat sink should be used with
those off the shelf audio anplifiers.
9.7 Microphones
A simple condenser (capacitor) microphone can be used as a transducer
to convert audio waves into a current. A condenser microphone exploits
the change in capacitance caused by sound waves modulating the distance
of the capacitor plates. This change corresponds to a charge variation of
the capacitor which generates a current proportional to the sound wave
intensity. A voltage bias is required to charge the capacitor otherwise there
would not be a capacitance to be modulated by the sound waves. Usually
one capacitorŽs plate is a stiff conductor and the other is a light small
conducting membrane under tension which can absorb the sound wave
energy and vibrate.
Electret microphones are condenser microphones with a so called electret material as dielectric which has an intrinsic electric field analogously
to permanent magnet and its magnetic field. The electric field generates
a capacitance without the need of a bias voltage as in a conventional condenser microphone. A n-channel JFET connected as show in the figure
below "hide" the capacitive load of the electret microphone.
VDD
R
+
AF
T
VDD
R
Vo
Vo
+
Mic.
Q0
Mic.
−
+
−
C
DR
V0
−
9.7. MICROPHONES
183
The following basic preamplifier circuit can be used to amplify the
weak microphone signal to further conditioning.
Cf
15 V
VCC
100pF
Rf
R1
Mic.
AOM−4544P−2
+
6.8kΩ
C
100kΩ
+
−
−
G
10µ F
+
R2
3.3kΩ
H ( jω ) = −
R f jωRC
1
,
R 1 + jωRC 1 + jωR f C f
R=
AF
T
The voltage divider resistors R1 and R2 set the positive bias voltage for
the condenser microphone to the required level specified by the transducer
characteristics. It also sets the maximum current that the transducer will
drain. The capacitor C bypass the DC component which is too large to
be amplified. The capacitor value should be such that the high pass cutoff frequency is about 20 Hz. Electrolytic capacitor can be used because
of the microphone positive bias and therefore even large values such as
10 µF can be easily used. The capacitor C f adds a pole at high frequency
to filter out high frequency oscillations and noise. The low-pass cut-off
frequency should be at about 15 kHz. Between the poles, the inverting
amplifier gain is set as usual by the ratio R f /R and should probably be
quite large, between 50 to 100 depending on the microphone used. The
amplifier transfer function is
R1 R2
.
R1 + R2
• Part #: AOM-4544P-2-R
• Directivity: Omnidirectional
DR
The electret condenser microphone available in the lab has following
characteristics
184
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
• Sensitivity: -44dB
• Frequency Response: from 20 Hz to 20 kHz,
• Max supply voltage Max: 10 V
• Max drain current : 500 µA
• Impedance: R (source impedance of microphone and bias, see schematic)
• Pin-out:
(+)
(−)
9.8 Phototransistors
AF
T
It is worthwhile to notice that a speaker can be used as a microphone
as well. In fact, vibrations induced by the sound waves on the speaker
membrane will move the solenoid respect to the permanent magnet. The
relative velocity between solenoid and magnet will induce a current in the
solenoid proportional to the sound wave intensity. Dynamic microphones
have essentially the same topology of a speaker and do not require a bias
voltage as condenser microphones do.
DR
A common type of phototransistor is the photo-BJT whose base region is
enlarged compared to standard BJT and has the base exposed such that
light can be absorbed to generate a photocurrent. Photo-BJTs can either
have two leads , Collector and Emitter leads or all the three BJT leads.
The circuit below is the simplest configuration that can be used to have
a photo-BJT working as switch or an amplifier driven by a light source.
The operation mode depends on the value of the load R L that sets the
current ICC . For the circuit to work as a switch or as an amplifier, we need
9.9. PHOTODIODES
185
to either keep the transistor working on the saturated or the active mode
when light is impinging on the transistor, and therefore
VCC = R L IC
Switch
⇒
(max )
⇒
VCC > R L IC
Amplifier
VCC
RC
Vo
RL
Usually, because of their faster response and lower noise, photodiodes
work better as transducer to convert light power into voltage. With modulated light, photodiodes should be therefore preferred.
9.9 Photodiodes
Rs
A
Io
ID
Vo
Iph
VD
D
A
Io
Cj
Rp
DR
Pi
K
AF
T
Photodiodes are essentially semiconductor junctions which generate a current (photocurrent) when exposed to light. They are therefore used to measure light intensity. The equivalent photodiode circuit is show below.
Vo
K
186
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
Note the opposite direction of the photocurrent I ph and the forward
biased diode current ID .
• I ph : photocurrent proportional to the amount of incident light of
power P on the photodiode active surface, i.e.
I ph = −α (λ) P ,
where α depends on the light wavelength λ.
• ID : diode PN junction current
− qVD /K B T
ID = Is e
−1 .
The saturation current is usually called dark current, i.e. current with
no incident light.
• R p : shunt resistance (in parallel.)
• Rs : series resistance.
• Cj : capacitance due to the PN junction. The smaller the N doped
active region surface, the smaller is the capacitance.
The I-V characteristic of the photodiode for different values of incident
power is qualitatively shown in the figure below. Note how the input dynamic range where the response is linear increases when the photodiode
is reverse biased.
0
AF
T
ID
VD
Pi
2P
i
3P
i
Vo
DR
Vo
9.9. PHOTODIODES
187
Fast and low noise photodiodes are hard to build and it is even harder
in case of high power application. Ultra fast photodiodes can have an
active area of about less than 100 µm2 and bandwidth of 25 GHz. Typical
semiconductor substrate for visible light are Silicon (Si) and for infrared
light Indium-Gallium-Arsenide (InGaAs).
9.9.1 Photovoltaic Mode
In this mode, the photodiode works thanks to the photovoltaic effect. Some
of the electron-hole pairs generated by incident photons do not give up
their energy as in a metal or are complectely extracted from the solid in
the photoelectric effect. They just stay inside the semiconductor with energy in the conduction band of the semiconductor. These pairs then build
up generating a charge distribution and therefore an electric field that finally produces the photocurrent.
As shown in the figure below, the photocurrent is measured as a voltage drop across a load or is transformed into a voltage with a transimpedance
amplifier.
Rf
Vo
−
D
+
Major advantages of this configuration:
• Smaller dark current ⇒ low current noise.
• linear output because smaller dark current.
• accurate voltage response.
Major disadvantage:
Vo
AF
T
R >> R p
DR
• slow response. The Cj in parallel to the current generator low pass
filter the signal.
188
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
9.9.2 Photocurrent Mode
In this mode, the generated photocurrent is collected by the revers bias
voltage applied across the photodiode. As shown in the figure below,
apart from having the photodiode reverse biased with −VDD the circuits
are the same as the photovoltaic mode circuits.
Cf
−VDD
Rf
Vo
−
R
D
+
−VDD
Major advantages of this configuration:
• Fast response. The reverse voltage bias −VDD reduces the junction
depletion region and therefore the junction capacitance.
Major disadvantages:
• larger dark current than the unbiased photodiode.
Consult [3, 4] for more detailed information.
9.10 Ultrasonic Transceivers
TBD
DR
9.11 Velocity Sensors
AF
T
• small OpAmp current bias amplified by the transimpedance.
TBD
9.12. POSITION SENSORS
189
9.12 Position Sensors
DR
AF
T
TBD
AF
T
CHAPTER 9. FINAL PROJECT, SOME USEFUL CIRCUITS
DR
190
Bibliography
[1] Semiconductor Packaging Information by Type, Texas Instruments,
http://www.ti.com/sc/docs/psheets/type/type.html
[2] Decoupling Techniques,
MT-101 Tutorial,
Analog Devices,
www.analog.com/static/imported-files/tutorials/MT-101.pdf
[3] Photodiode Technical Information - Hamamatsu Photonics,
http://sales.hamamatsu.com/assets/applications/SSD/photodiode_technical_information.pd
191
DR
AF
T
[4] Characteristics and use of infrared detectors, Hamamatsu Photonics,
http://sales.hamamatsu.com/assets/applications/SSD/infrared_kird9001e04.pdf
BIBLIOGRAPHY
DR
AF
T
192
Appendix A
Decibels
A.1 Definition of Bel and Decibel
The bel1 is defined as the logarithm in base ten of a power P normalized
to a reference power Pr , i.e.
X B = log10
P
.
Pr
(A.1)
The decibel is 10 bels
X dB = 10 X B = 10 log10
P
.
Pr
(A.2)
AF
T
Bels and decibels are dimensionless. The bel is not a unit, but a formula
to conveniently scale homogeneous quantities thanks to the properties of
the logarithmic function.
Bels are not commonly used because if we consider integer values of
bels they do not provide a good resolution. We will just consider decibels
for the remainder of this notes.
Considering that
V2
P=
= | Z| I 2 ,
|Z|
1 The
DR
and supposing that we use the same reference impedance magnitude | Zr |
units scale "bell" was name after Alexandre Graham Bell, scientist and inventor.
193
APPENDIX A. DECIBELS
194
for P, and Pr ,we can rewrite equ. (A.2) as
X dB = 20 log10
I
V
= 20 log10 ,
Vr
Ir
where Vr , and Ir are respectively the voltage and the current across the
reference impedance | Zr |. In other words, we have to measure the voltage
or the currents across equal impedances, to get the decibels.
A.2 Generalization of the Use of Decibel
For practical purposes, the decibel is also used to report the ratio of any
homogeneous quantities such as the voltage output Vo over the voltage
input Vi of a two port network, or in general, the ratio of any kind of
homogeneous quantities x1 , x2
X dB = 20 log10
x1
.
x2
In this case there is no normalization respect to a reference load Rr or
power Pr .
A.3 Useful Table and Properties
The next table is quite useful to easily translate decibels into magnitude
0 1
1 1.1
2
3
4
5 6
1.2 1.4 1.6 1.8 2
7
8
9
10
2.2 2.5 2.8 3.2
AF
T
[dB]
Magnitude
For convenience, letŽs rewrite some useful properties of the logarithm
function
=
=
=
=
log x + log y,
log x − log y,
n log x,
logb x/ logb a.
DR
log( x y)
log( x/y)
log x n
loga x
A.4. STANDARD POWER REFERENCES
195
A.4 Standard Power References
Decibels comes in many flavors (different reference powers) depending on
the application, radio frequency, microwaves, optics, acoustics, et cetera.
For example the following definition is quite often used
X dBm( Rr ) = 10 log10
V 2 /R
1mW
The value of Rr depends on the application field
DR
AF
T
Radio Frequency
TV Frequencies
Audio Frequencies
Rr [ Ω ]
50
75
600
AF
T
APPENDIX A. DECIBELS
DR
196
Appendix B
Resistor Color Code
Nominal values of resistances are coded using colors bands around the
resistors (see figure below). The bands identify digits and the exponent
in base ten for the resistance value and the tolerance as explained in the
following table:
Band
1
Number
3 Bands Digit
4 Bands Digit
5 Bands Digit
2
3
Digit
Digit
Digit
4
(Tolerance band)
Exponent
Always 20%
Exponent
Tolerance
Exponent
Tolerance
5
Tolerance after
1000 hours
ABC
D
R = AB · 10C ,
AF
T
3 Band resistors have no band for the tolerance because it is assumed to
be 20% of the nominal values.The fifth band is not an industry standard,
but quite often it means the tolerance after 1000 hours of continuous use.
∆R = R · D
197
DR
The bands are counted from left to right. The following table reports
the coding of the values using colors and a mnemonic sentence to remember the color code table.
APPENDIX B. RESISTOR COLOR CODE
198
Mnemonic
Sentence
Big
Bart
Rides
Over
Your
Grave
Blasting
Violent
Guns
Wildly.
Go
Shoot (him?)
Color
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
Gold
Silver
Exponent Tolerance Tolerance (%)
(%)
5th Band
0
20
1
1
1%
2
2
0.1%
3
0.01%
4
0.001%
5
6
7
8
9
-1
5
-2
10
For example, the nominal resistance of a 4 band resistor having the
sequence brown, black, orange and gold is
Rnom. = 10 kΩ
∆Rnom. = 5%10 kΩ
⇒
Rnom. = (10.0 ± 0.5) kΩ
DR
AF
T
Resistor size (volume) is related to the power dissipation capability. Typical used values are 1/8W, 1/4W, 1/2W, 1W.
Appendix C
The Cathode Ray Tube
Oscilloscope
Every time we need to analyze or measure an electronic signal in the time
domain we will probably use some type of oscilloscope. The oscilloscope
is therefore one of the most useful tools used in a laboratory. Practically, it
is an indispensable instrument for measuring, designing, manufacturing,
or repairing electronic equipment.
Quite often, one can still find old Cathode Ray Tube (CRT) oscilloscopes even in modern laboratory, mainly because of the inadequacy of
state of the art digital oscilloscope scopes to represent very fast signals. It
is therefore worthwhile to study this device and understand how a CRT
works and also its limitations.
AF
T
C.1 The Cathode Ray Tube Oscilloscope
• the cathode ray tube (CRT),
• the trigger,
DR
The cathode ray tube oscilloscope is essentially an analog1 instrument that is
able to measure time varying electric signals. It is made of the following
functional parts (see figure C.1):
1 Hybrid instruments combining the characteristics of digital and analog oscilloscopes,
with a CRT, are also commercially available.
199
200
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
• the horizontal input,
• the vertical input,
• time base generator.
Let’s study in more detail each component of the oscilloscope.
C.1.1 The Cathode Ray Tube
The CRT is a vacuum envelope hosting a device called an electron gun ,
capable of producing an electron beam, whose transverse position can be
modulated by two electric signals (see figures C.1 and C.7).
When the electron gun cathode is heated by wire resistance because
of the Joule effect it emits electrons . The increasing voltage differences
between a set of shaped anodes and the cathode accelerates electrons to a
terminal velocity v0 creating the so called electron beam.
The beam then goes through two orthogonally mounted pairs of metallic plates. Applying voltage difference to those plates Vx and Vy , the beam
is deflected along two orthogonal directions ( x and y ) perpendicular to
its direction z. The deflected electrons will hit a plane screen perpendicular to the beam and coated with florescent layer. The electrons interaction
with this layer generates photons, making the beam position visible on the
screen.
C.1.2 The Horizontal and Vertical Inputs
DR
C.1.3 The Time base Generator
AF
T
The vertical and horizontal plates are independently driven by a variable
gain amplifier to adapt the signals v x (t), and vy (t) to the screen range. A
DC offset can be added to each input to position the signals on the screen.
These two channels used to drive the signals to the plates signals are called
horizontal and vertical inputs of the oscilloscope.
In this configuration the oscilloscope is an x-y plotter.
If we apply a sawtooth signal Vx (t) = αt to the horizontal input, the horizontal screen axis will be proportional to time t. In this case a signal vy (t)
C.1. THE CATHODE RAY TUBE OSCILLOSCOPE
201
CATODE RAY TUBE (CRT)
Line(60Hz)
External
Ramp
Generator
Internal
Level
s/div
TRIGGER
Preamplifier
TIME BASE GENERATOR
Amplifier
V/div
INPUT CHANNEL
AF
T
Source
DR
Figure C.1: Sketch of the functional parts of the analog oscilloscope,
preamplifier, amplifier, trigger, time base generator, and CRT.
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
202
applied to the vertical input, will depict on the oscilloscope screen the signal time evolution.
The internal ramp signal is generated by the instrument with an amplification stage that allows changes in the gain factor α and the interval
of time shown on the screen. This amplification stage and the ramp generator are called the time base generator.
In this configuration, the horizontal input is used as a second independent vertical input, allowing the plot of the time evolution of two signals.
Visualization of signal time evolution is the most common use of an
oscilloscope.
vy (t)
Signal
t
Trigger
t
Sawtooth
signal
T
T
t
C.1.4 The Trigger
AF
T
Figure C.2: Periodic Signal triggering. The second trigger is ignored because the ramp is still of the sawtooth signal is still increasing to its maximum αT.
DR
To study a periodic signal v(t) with the oscilloscope, it is necessary to synchronize the horizontal ramp Vx = αt with the signal to obtain a steady
plot of the periodic signal. The trigger is the electronic circuit which provides this function. Let’s qualitatively explain its behavior.
The trigger circuit compares v(t) with a constant value and produces
a pulse every time the two values are equal and the signal has a given
C.2. OSCILLOSCOPE INPUT IMPEDANCE
203
C
Input
DC
AC
−
High Voltage
Amplifier
GND
+
Cs
Rs
Deflection
Plates
Pre−Amplifier
Figure C.3: Oscilloscope input impedance representation using ideal components (gray box). Input channel coupling is also shown.
slope. The first pulse triggers the start of the sawtooth signal of period2 T
, which will linearly increase until it reaches the value V = αT, and then
is reset to zero. During this time, the pulses are ignored and the signal
v(t) is plotted for a duration time T. After this time, the next pulse that
triggers the sawtooth signal will happen for the same previous value and
slope sign of v(t), and the same portion of the signal will be re-plotted on
the screen.
C.2 Oscilloscope Input Impedance
2 In
DR
AF
T
A good approximation of the input impedance of the oscilloscope is shown
in the circuit of figure C.3. The different input coupling modes ( DC AC
GND ) are also represented in the circuit.
The amplifying stage is modeled using an ideal amplifier (infinite input impedance) with a resistor and a capacitor in parallel to the amplifier
input.
The switch allows to ground the amplifier input and indeed to vertically set the origin of the input signal (GND position), to directly couple
the input signal (DC position), or to mainly remove the DC component of
the input signal (AC position).
general, the sawtooth signal period T and the period of v(t) are not equal.
204
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
C.3 Oscilloscope Probe
An oscilloscope probe is a device specifically designed to minimize the capacitive load and maximize the resistive load added when the instrument
is connected to the circuit. The price to pay is an attenuation of the signal
that reaches the oscilloscope input3 .
Let’s analyze the behavior of a passive probe. Figure C.4 shows the
equivalent circuit of a passive probe and of the input stage of an oscilloscope. The capacitance of the probe cable can be considered included in
Cs .
Vi
Cp
Coaxial
Cable
Probe Tip
GND Clip
Cp
Rp
Vs
Cs
Rs
Rp
Probe
A
Rs
B
Vs
Cs
Oscilloscope Input Stage
Figure C.4: Oscilloscope input stage and passive probe schematics.The
equivalent circuit made of ideal components for the probe shielded cable
is not shown.
Considering the voltage divider equation, we have
1
1
= jωCs + ,
Zs
Rs
and then
Zs =
3 Active
Rs
,
jωτs + 1
1
1
= jωC p +
,
Zp
Rp
Zp =
Rp
.
jωτp + 1
DR
where
Vs
Zs
,
=
Vi
Z p + Zs
(C.1)
AF
T
H ( jω ) =
probes can partially avoid this problems by amplifying the signal.
C.3. OSCILLOSCOPE PROBE
205
Defining the following parameters
τp = C p R p ,
α=
Rs
,
Rs + R p
β=
Cp
,
Cs + C p
and after some tedious algebra, equation (C.1) becomes
H ( jω ) = α
1 + jωτp
,
1 + jω αβ τp
which is the transfer function from the probe input to the oscilloscope before the ideal amplification stage.
The DC and high frequency gain of the transfer function H ( jω ) are
respectively
H (0) = α,
H (∞) = β.
The numerator and denominator of H ( jω ) are respectively equal to
zero, (the zeros and poles of H ) when
ω = ωz = j
1
,
τp
ω = ωp = j
β 1
.
α τp
Figure C.5 shows the qualitative behavior of H for
α
β
> 1.
C.3.1 Probe Frequency Compensation
α
< 1 ⇒ over-compensation
β
α
= 1 ⇒ compensation
β
α
> 1 ⇒ under-compensation
β
AF
T
By tuning the variable capacitor C p of the probe, we can have three possible cases
DR
if α < β the transfer function attenuates more at frequencies above ωz ,
and the input signal Vi is distorted.
if α = β the transfer function is constant and the input signal Vi will be
undistorted and attenuated by a factor α.
206
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
Magnitude
α
Phase
β
ω1
Frequency
ω2
Figure C.5: Qualitative transfer function from the under compensated
probe input to the oscilloscope before the ideal amplification stage. As
usual, the oscilloscope input is described having an impedance Rs ||Cs .
α
= 1,
β
⇒
Cp
Rs
=
,
Rp
Cs
DR
and this condition implies that:
AF
T
if α > β the transfer function attenuates more at frequencies below ω p
and the input signal Vi is distorted.
The ideal case is indeed the compensated case, because we will have
increased the oscilloscope input impedance by a factor α without distorting the signal.
The probe compensation can be tuned using a signal able to show a
clear distortion when it is filtered. A square wave signal is very useful
in this case because it shows a quite different distortion if the probe is
under or over compensated. Figure C.6 sketches the expected square wave
distortion for the two uncompensated cases.
It is important to notice that if
• the voltage difference V1 across Rs is equal the voltage difference V2
across Cs , i.e V1 = V2
C.3. OSCILLOSCOPE PROBE
207
• the voltage difference V3 across R p is equal the voltage difference V4
across C p , i.e. V3 = V4
• and therefore V1 + V2 = V3 + V4 .
This means that no current is flowing through the branch AB when the
probe is compensated, and we can consider just the resistive branch of the
circuit to calculate Vs . Applying the voltage divider equation, we finally
get
Vs =
Rs
V
Rs + R i
The capacitance of the oscilloscope does not affect the oscilloscope input anymore, and the oscilloscope+probe input impedance Ri becomes
greater, i.e.
Ri = R s + R p .
V
t
AF
T
V
t
DR
Figure C.6: Compensation of a passive probe using a square wave. Left
figure shows an over compensated probe, where the low frequency content of the signal is attenuated. Right figure shows the under compensated
case, where the high frequency content is attenuated.
208
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
x
V0
Vy
θ
e
v0
Ey
h
z
d
D
Figure C.7: CRT tube schematics. The electron enters into the electric field
and makes a parabolic trajectory. After passing the electric field region it
will have a vertical offset and deflection angle θ.
C.4 Beam Trajectory
Let’s consider the electron motion through one pair of plates.
The electron terminal velocity v0 coming out from the gun can be easily
calculated considering that its initial potential energy is entirely converted
into kinetic energy, i.e
s
1 2
eV0
µv0 = eV0 , ⇒ v0 = 2
,
2
µ
AF
T
where µ is the electron mass, e the electron charge, and V0 the voltage
applied to the last anode.
If we apply a voltage Vy to the plates whose distance is h, the electrons
will feel a force Fy = eEy due to an electric field
Vy
.
h
The equation of dynamics of the electron inside the plates is
| Ey | =
ż = v0 ,
DR
µz̈ = 0, ⇒
µÿ = e| Ey |.
C.4. BEAM TRAJECTORY
209
Supposing that Vy is constant, the solution of the equation of motion
will be
s
eV0
2
z(t) =
t,
µ
y(t) =
1 eVy 2
t .
2 µh
Removing the dependency on the time t, we will obtain the electron
beam trajectory , i.e.
y=
1 Vy 2
z ,
4h V0
which is a parabolic trajectory.
Considering that the electron is transversely accelerated until z = d,
the total angular deflection θ will be
1 d Vy
∂y
=
.
tan θ =
∂z z=d
2 h V0
and displacement Y on the screen is
Y (Vy ) = y(z = d) + tan θD,
i.e.,
1d 1 d
( + D )Vy .
2 h V0 2
Y is indeed proportional to the voltage applied to the plates through a
rather complicated proportional factor.
The geometrical and electrical parameters of this proportional factor
play a fundamental role in the resolution of the instrument. In fact, the
smaller the distance h between the plates, or the smaller the gun voltage
drop V0 , the larger is the displacement Y. Moreover, Y increases quadratically with the electron beam distance d.
DR
C.4.1 CRT Frequency Limit
AF
T
Y (Vy ) =
The electron transit time through the plates determine the maximum frequency that a CRT can plot. In fact, if the transit time τ is much smaller
210
APPENDIX C. THE CATHODE RAY TUBE OSCILLOSCOPE
than the period T of the wave form V (t), we have
V (t) ≃ constant,
and the signal is not distorted.
The transit time is
d
τ=
=d
v0
r
τ ≪ T,
µ
.
2eV0
⇒
τ ≃ 1ns
DR
AF
T
Supposing that

V0 = 1kV



d = 20mm
µc2 ≃ 0.5MeV



e = 1eV
if
Appendix D
Electromagnetic Field Noise
D.1 Introduction
AF
T
Human and natural activities fill the surrounding space with electromagnetic fields (radiation) creating a very complex and unpredictable frequency
spectrum of radiation. For example, domestic appliances, bulbs, fluorescent lights, and power line grids mainly irradiate at 60Hz and harmonics
of 60Hz. Radios, televisions, wireless internet connections, and cellular
phones networks are other typical sources, which fill the radiation spectrum from the kilohertz to the gigahertz region. Light mainly produced by
the sun pervades the spectrum in the optical region. Radioactivity, gamma
ray burst (GRB) emitted by astrophysical sources are for eaxmple responsible for filling the high and very high region of the spectrum.
Portion of this so complex spectrum can be attenuated by the so called
electromagnetic shields but some others portions because of the energy
involved cannot be effectively even attenuated.
The so called radio frequency noise can be easily attenuated (shielded)
using a quite simple device known as the Faraday cage.
D.2 The Faraday Cage
211
DR
Gauss’s law states that a closed surface will prevent extern electrostatic
fields from reaching the space enclosed by the surface. If the electric field
is slowly varying i.e., its wavelength λ is large compared to the typical size
d of the enclosure), then the field on the surface can be considered static
212
APPENDIX D. ELECTROMAGNETIC FIELD NOISE
and Gauss’s law is then applicable. This enclosure is commonly called
Faraday cage.
Using this crude approximation we can state that all frequencies much
smaller than the following
c
ν∗ ∼
d
where c is the speed of light, will be effectively attenuated. For example if
d = 1 m then the Faraday cage will attenuate the external electromagnetic
fields with frequencies much smaller than ν∗ ∼ 300 MHz.
D.3 Practical Considerations
DR
AF
T
Normally, when we perform a measurement we cannot easily fit the lab in
a small Faraday cage. Anyway, most of the time it is sufficient to enclose
the physical system under measurement inside the cage . Then to perform
the measurement we will have to connect the instrument sitting outside
the cage to the system. The instruments leads acting like an antenna will
still pick-up some of the ambient electromagnetic radiation. This effect can
be amplified if we touch one of the leads increasing the antenna effect. A
way to minimize this effect is to connect Faraday cages together. Reasonably good instruments have a built in Faraday cage connected to ground.
Connecting the cages to ground will create a more or less single effective
cage which will attenuate the electromagnetic noise pick-up.
Appendix E
Common Emitter BJT Amplifier
The common emitter BJT amplifier is one of the most simple design that
allows to set the voltage amplification Av quite independently from the
BJT characteristics.
VCC
RC
IC
R1
Co
Vi
Ci
C
IB
Vo
Q VCE
B
E
IE
RE
AF
T
R2
Figure E.1: BJT Common emitter amplifier with coupling capacitors Ci ,
and Co .
213
DR
To properly set the BJT working point, we have to forward bias the
emitter base junction and reverse bias the collector base junction. But this
is not enough if we want to build an amplifier. The other requirement is to
set the voltage VCE where the VCE characteristic is flat and wide enough
APPENDIX E. COMMON EMITTER BJT AMPLIFIER
214
R1
+
−
VCC
IB
R2
B
VBE
C
IC
RC
+
+
VCC
hfe IB
−
IE
−
RE
E
Figure E.2: Common emitter equivalent circuit which simplifies the BJT
biasing understanding.
to accommodate the output signal excursion. In other words, we donŽt
want the output to swing into the BJT saturation region or into the break
down region.
The design parameters we have to set are are Av , IC ,VCE ,VCC , and essentially, the VCE characteristics contains all the information we need to
properly bias the BJT. As last remark, voltage gain and bias point are "intimately" related and cannot be completely independent.
E.1 Amplifier Design
The analysis of the circuit becomes quite easy if we observe from the VCE
characteristic that
IC ≫ IB .
(E.1)
Applying KVL to the output mesh, we will have
(E.2)
AF
T
VCC = RC IC + RE IE + VCC ≃ ( RC + RE ) IC + VCE
DR
If we want to optimize the dynamic range of the amplifier, and neglecting the saturation region, we will have to set according to the IC − VCE
characteristic
1
VCE ≃ VCC
2
Using this design condition and and voltage gain of this circuit we will
have
1
VCC
RE =
2 (1 + Av ) IC
E.2. RESUME
215
This equations together with the gain equation (E.8) set the values RC and
RE based only on the design parameters VCC ,IC , and Av . Let’s now find the
values of the voltage divider which forward bias the base-emitter junction.
If IB is negligible, then resistors R1 and R2 act as a simple voltage divider, i.e.
VB ≃
and for KVL
R2
VCC .
R1 + R2
(E.3)
VB = VBE + RE IE ≃ VBE + RE IC
where VBE must be the voltage drop of a forward polarized diode junction, typically between 6.0 V to 0.7 V.
Using the two expressions of VB and after some algebra, we get
VCC
R1 ≃ R2
−1 .
VBE + RE IC
E.2 Resume
Summarizing the results we have for IC ≫ IB
1
2 (1 + A v )
V
= α CC
IC
= Av RE
VCC
=
− 1 R2
VBE + αVCC
(E.4)
α =
RC
R1
E.3 Example

R


 1
R2
R


 C
RE
(E.6)
≃ 452 kΩ
≃ 40 kΩ
≃ 909 Ω
≃ 91 Ω
DR
Let’s set the following design values

Av =
10



VCC = 20 V
⇒
IC = 1 mA



R2 = 40 kΩ
(E.5)
(E.7)
AF
T
RE
APPENDIX E. COMMON EMITTER BJT AMPLIFIER
216
R1
Vi
IB
R2
B
C
IC
RC
Vo
hie
hfe IB
IB
RE
E
Figure E.3: Small signal circuit model for the common emitter BJT amplifier
E.4 Amplifier Gain and Sign
Using the equivalent small signal circuit model for the BJT and considering the impedance of the ideal voltage and current sources we can construct the circuit show in figure E.3. Then from that figure it is finally easy
to compute the voltage gain Av and the input and output impedance Ri
and Ro of the circuit.
In fact, considering that IB ≪ IC , RE ≫ hie , the input and the output
voltage are simply

 Vi = (hie + RE ) IE ≃ RE IC
R
, ⇒ Av ≃ C
(E.8)

RE
Vo = RC IC
The Common Emitter BJT amplifier is an inverting stage. In fact, considering that
⇒
VCC = Vo + ( RE + RC ) IC
AF
T
Vo = VCC − ( RE + RC ) IC
if IC increases, then Vo must decrease to keep VCC constant. If, we start
with an input current and voltage in phase they will end up being out of
phase by 180◦ degrees.
DR
E.5 Input and Output Impedance
The input impedance is the impedance seen from the inputs lead , and can
be easily computed considering that the ideal current source is an open
E.6. I/O COUPLING CAPACITORS
217
circuit, i.e.
Ri = R2 || R1 || (hie + Re ) .
The output impedance is then
Ro = RC .
E.6 I/O Coupling Capacitors
The coupling capacitors Ci will provide a way to send the input signal to
amplifier without perturbing the DC bias of the BJT. Similarly, placing the
capacitor Co to the output will allow to connect a load without perturbing
the DC bias of the BJT circuit.
Coupling capacitance should be selected to minimize the filtering effect
on the amplifier response. For example, Ci will create a RC high pass filter
with the R being the input resistance of the amplifier and Co will do the
same with the eventual resistance of the amplifier load.
VCC
RC
C
Vi
C
C
IB
Q VCE
B
E
IE
RE
DR
R2
Vo
AF
T
IC
R1
Figure E.4: BJT Common emitter amplifier with emitter bypassing capacitor.
218
APPENDIX E. COMMON EMITTER BJT AMPLIFIER
E.7 Emitter Bypass Capacitor
DR
AF
T
Adding a so called bypass capacitor CE in parallel with RE will not change
the DC bias of the BJT and will provide a the maximum possible voltage
gain at high frequency as seen in the simple BJT amplifier circuit.
Appendix F
Push-Pull BJT Amplifier
The BJT push-pull current amplifier is an amplifier topology which provides small standing currents, reasonable low distortion, and good efficiency. It is therefore quite useful to drive small loads such loudspeakers
which usually have a few Ohms input impedance.
Vcc
Q1
vo
+
Q1
vi
RL
−
−Vcc
AF
T
Figure F.1: Push-pull amplifier using a BJT complementary pair. The gray area
highlights the “pushing” part of the circuit with the transistor Q1 , which pushes
the current through the load R L . The other transistor Q̄1 is the circuit “pulling”
part that pulls the current through the load.
219
DR
The etymology of the term “push-pull” dates back to the first vacuum
tube amplifiers. It referred to the interpretation that one part of the circuit
“pushed” current through a transformer connected to the amplifier output
and the other complementary part “pulled” current from the transformer.
Different push-pull BJT configurations can be found in literature, but
APPENDIX F. PUSH-PULL BJT AMPLIFIER
220
Vo
VCC
v
vo
VBE
vi
VBE
−VBE
O
VBE
Vi
O
−VBE
t
−VCC
Figure F.2: Cross-over distortion. For − I0 < i < I0 the output current i is
zero because neither transistors Q1 and Q̄1 are in conduction.
the basic topology is the two complementary NPN PNP transistors connected as show in Figure F.1.
F.1 Push-Pull BJT Configuration
vo ≃ vi − VBE
AF
T
First, let’s consider just half of the BJT push-pull circuit, the one inside the
gray area shown in Figure F.1. When the sinusoidal input vi is positive
and greater than VBE but smaller than VCC , the NPN transistor Q1 turns
on, i.e. the base-emitter junction is forward biased and the collector-base
junction is reverse biased. The complementary transistor Q̄1 is in a cut off
state because the base-emitter junctions is reverse biased. Apart from an
offset VBE , the output vo follows the input
VCC > vo > VBE ,
vo ≃ vi + VBE
DR
whereVBE ≃ 0.65 V.
The previous analysis can be repeated for the negative portion of the
sinusoid just considering that transistor Q1 is now in its cut off mode, and
Q̄1 is on. We will have then
− VCC < vo < −VBE ,
F.2. DISSIPATION AND EFFICIENCY
221
Vcc
v−
vi
v+
Q1
−
G
vo
vA
+
Q1
RL
−Vcc
Figure F.3: Push-pull amplifier using an Op-Amp to minimize crossover
distortion.
F.2 Dissipation and Efficiency
AF
T
Figure F.2 shows the input and output signal of the circuit, and clearly the
distortion produced by of this type of circuit. This signal distortion, called
cross-over distortion, can be effectively minimized with some changes to the
circuit.
Figure F.3 a more complex and a little more costly but quite effective
way to minimize the cross-over distortion. In this case, the ideal OpAmp
eliminates the cross-over distortion thanks to the feedback from the circuit
output to the OpAmp negative input. In fact, because the OpAmp will
have to keep its input difference v+ − v− to zero it will have to compensate for all the difference between the push-pull circuit output and vi . The
correction will show up at the OpAmp output v A and therefore at vo eliminating the cross-over distortion. As usual, limitations of real OpAmps
and in particular bandwidth limitation make the correction more or less
effective.
DR
One of the advantages of a push-pull configuration is the low power consumption. In fact, considering the ideal case where the amplifier characteristics shows zero current when no signal is present, the amplifier does
not dissipate energy. In real circuits, there are always bias currents flowing
through but usually they are quite small making power consumption very
small compared to circuits with biased BJTs.
APPENDIX F. PUSH-PULL BJT AMPLIFIER
222
V(out)(2), df: 6.7Hz, Avgs: 12
V(in)(1), df: 6.7Hz, Avgs: 12
0
10
−1
Amplitude [ V/sqrt(Hz) ]
10
−2
10
−3
10
−4
10
1000
2000
3000
4000
5000
Frequency [Hz]
6000
7000
8000
Figure F.4: Simulated square-root of the power spectral density of the basic
push-pull circuit input and output. The simulation modeled the 2N2222
NPN and the 2N2907 PNP transistors, with a sinusoidal input amplitude
of 10 V and 1 kHz frequency.
F.3 Distortion
N
iC ≃
∑ Gk ibk .
k=0
Then, for a sinusoid
DR
i B = IB cos(ωt) ,
AF
T
We want to evaluate the type of distortion we will have due to the non
ideal BJTs base collector current characteristic ( iC versus i B ) .
Supposing that the characteristic can be written as a polynomial of the
form
and considering that the BJTs collector currents are 180◦ out of phase, we
will have that the collector currents of the two BJTs are
F.3. DISTORTION
223
N
i 1 ≃ IB
∑ Gk cosk (kωt)
k=0
N
=
α2k cos(2kωt) +
∑
k=1
N
∑ α2k+1 cos [(2k + 1)ωt] ,
k=0
N
i 2 ≃ IB
=
∑ Hk cosk (kωt + π )
k=0
N
N
k=1
k=0
∑ β2k cos(2kωt) − ∑ β2k+1 cos [(2k + 1)ωt] .
If the two transistors coefficients are the same (complementary BJTs
perfectly matched), then the currents difference cancels the even terms out
leaving just the odd terms
αk = β k ,
k = 1, 2, ..., N
⇒
i L = i1 − i2 = 2
N
∑ α2k+1 cos((2k + 1)ωt),
k=0
DR
AF
T
In this case, the amplifier output shows just the odd harmonics. In case
the two transistors are not perfectly matched we will also have the even
harmonics as well.
Figure F.4 shows the power spectra density output computed from a
simulation of the basic push-pull circuit of Figure F.1. The simulation used
the models of a 2N2222 NPN transistor, a 2N2907 PNP transistor and a
sinusoidal input with 10 V of amplitude and 1 kHz of frequency. Note that
the input is not an ideal sinusoid. As expected, the plots clearly show a
large difference between the input and the output for the odd harmonics.