WIP Introduction to Analog Electronics
Ole Herman Schumacher Elgesem
February 29, 2016
Contents
1 Formulae
1.1 Definitions . . . . . . . . . . .
1.2 Resistors . . . . . . . . . . . .
1.3 Capacitors and Inductors . .
1.4 RC . . . . . . . . . . . . . . .
1.5 Bipolar Junction Transistors .
1.6 Operational Amplifiers . . . .
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4
4
4
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5
5
2 Basic Concepts
2.1 Water analogy . . . . . . . . . . . . . . .
2.2 Charge . . . . . . . . . . . . . . . . . . . .
2.3 Current . . . . . . . . . . . . . . . . . . .
2.3.a Measuring current . . . . . . . . .
2.4 Voltage . . . . . . . . . . . . . . . . . . .
2.4.a Measuring Voltage . . . . . . . . .
2.5 Resistance . . . . . . . . . . . . . . . . . .
2.5.a Ohms law . . . . . . . . . . . . . .
2.5.b Measuring resistance . . . . . . . .
2.6 Voltage source . . . . . . . . . . . . . . .
2.7 Current source . . . . . . . . . . . . . . .
2.8 Alternating and Direct Current(AC/DC)
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7
7
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8
3 Power
3.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.a Watts Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Impedance
4.1 Resistance . . . . . . . . . . . . . . . . . .
4.2 Reactance . . . . . . . . . . . . . . . . . .
4.3 Complex impedance . . . . . . . . . . . .
4.4 Conductance . . . . . . . . . . . . . . . .
4.5 Susceptance . . . . . . . . . . . . . . . . .
4.6 Admittance . . . . . . . . . . . . . . . . .
4.7 Updated Ohm’s law . . . . . . . . . . . .
4.8 Impedance and Admittance table . . . . .
4.9 Impedance vector - magnitude and phase
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9
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11
11
11
5 Circuits
5.1 Fundamentals . . . . . .
5.2 Series circuits . . . . . .
5.3 Parallel circuits . . . . .
5.4 Kirchhoffs Voltage Law
5.5 Kirchhoffs Current Law
5.6 Voltage divider . . . . .
5.6.a Load . . . . . . .
5.7 Current divider . . . . .
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11
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12
12
12
12
12
6 Periodic signals
6.1 Sinusoid . . . . . . . . . . . . . .
6.1.a Amplitude . . . . . . . . .
6.1.b Frequency & Period . . .
6.1.c Phase shift . . . . . . . .
6.2 Square wave . . . . . . . . . . . .
6.2.a Pulse width & Duty Cycle
6.2.b Rise & fall time . . . . . .
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13
13
13
13
13
13
13
13
7 Capacitors
7.1 Physical properties . . . . . . . .
7.2 Stored capacitor charge . . . . .
7.3 Time constant . . . . . . . . . . .
7.4 Capacitive reactance . . . . . . .
7.5 Voltage & Current . . . . . . . .
7.6 Parallell capacitors . . . . . . . .
7.7 Series capacitors . . . . . . . . .
7.8 Practical capacitors . . . . . . . .
7.9 RC circuits . . . . . . . . . . . .
7.10 Capacitive voltage divider . . . .
7.11 Power consumption in capacitors
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13
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13
13
13
13
8 Inductors
8.1 Physical properties
8.2 Induced voltage . .
8.3 Inductive reactance
8.4 Practical Inductors
8.5 RL circuits . . . .
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14
14
14
14
14
14
9 Filters and frequency Response
9.1 DC characteristics . . . . . . . . . .
9.2 Time domain . . . . . . . . . . . . .
9.3 Frequency domain . . . . . . . . . .
9.4 Harmonics . . . . . . . . . . . . . . .
9.5 Fourier series and Fourier transform
9.6 Low-Pass filter . . . . . . . . . . . .
9.7 High-Pass filter . . . . . . . . . . . .
9.8 Band-pass filter . . . . . . . . . . . .
9.9 Band-stop filter . . . . . . . . . . . .
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10 Resources / Further Reading
14
2
Introduction
Background...
3
1
1.1
Formulae
Definitions
Ohm, Ω
Siemens, S
Real(DC)
Resistance, R
Conductance, G
Imaginary(AC)
Capacitive and Inductive Reactance XC , XL
Susceptance, B
Total(Combined)
Impedance, Z
Admittance, Y
kg∗m2
s2
Energy(Joule, J):
E =F ∗d=
Charge(Coloumb, C):
Q=I ∗t=A∗s
2
Power(Watt, W):
P =
Current(Ampere, A):
I=
E
t
Q
t
kg∗m2
s3
=
=A
Voltage(Volt, V):
V =
P
I
=
kg∗m
A∗s3
Capacitance(Farad, F):
C=
Q
V
=
A2 s4
kg∗m2
Resistance(Ohm, Ω):
R=
V
I
=
kg∗m2
A2 ∗s3
Conductance(Siemens, S):
G=
1
R
=
A2 ∗s3
kg∗m2
Frequency(Hertz, Hz):
f=
1
T
Period(Second, s):
T =
1
f
=s
Inductance(Henry, H):
L=
N 2 µA
I
Induction voltage:
v(t) = L ∗
Voltage Gain:
A=
vout
vin
Current Gain:
β=
⇒
GT OT =
1.2
=
kg∗m2
A2 ∗s2
δi
δt
iout
iin
Resistors
Resistors in Series:
Resistors in Parallel:
Voltage Divider:
1.3
= s−1
RT OT = R1 + R2 + ... + Rn
GT OT = G1 + G2 + ... + Gn
⇒
RT OT =
1
G1
1
+ G1 +...+ G1n
1
R1
1
+ R1 +...+ R1n
1
C1
1
+ C1 +...+ C1n
1
L1
1
+ L1 +...+ L1n
2
2
x
Vx = Vs RR
T OT
Capacitors and Inductors
Capacitors in parallel:
Inductors in Series:
CT OT = C1 + C2 + ... + Cn
LT OT = L1 + L2 + ... + Ln
Capacative Reactance:
XC =
Capacitor Current:
IC =
1
2∗π∗f ∗C
δV
δt
Capacitors in Series:
Inductors in parallel:
CT OT =
LT OT =
2
2
Inductive Reactance:
XL = 2π ∗ f ∗ L
Phase angle:
θ = arctan( XRC )
∗C
t
Capacitor charge/discharge: v(t) = VF + (VI − VF ) ∗ e τ
1.4
RC
Impedance:
Time Constant: τ = RC
Z=
p
XC2 + R2
1RC: 63%
2RC: 86%
3RC: 95%
4
4RC: 98%
5RC: 99%
1.5
Bipolar Junction Transistors
Emitter Voltage:
VE = VB − 0.7v
Emitter Current:
IE = IC + IB
Collector Current:
IC = IB ∗ β
(Int.) Emitter Resistance:
rE =
Non-inverting amplifier:
A=
⇒
vout = vin
1.6
VT
IE
≈
26mV
IE
Operational Amplifiers
RF
RI
Inverting amplifier:
A=
Buffer/Voltage follower:
A=1
5
RF
RG N D
+1
(This page was intentionally left blank.)
6
2
2.1
Basic Concepts
Water analogy
Electronics and electricity is about the movement of electrons through different materials. The charge, voltage, current
and resistance terms can be conceptually hard to understand. An analogy is often used, and the author prefers the
water in pipes analogy. Electrons or more generally charge can be thought of as water in a system of pipes.
2.2
Charge
Every electron has the same charge, called the elementary charge:
e− = −1.602 ∗ 10−19 C
Electrons are defined to have negative charge. The absence of an electron, called a hole, is thought of as a charge carrier,
even though it isn’t a particle. Holes can move through the atomic structure of a material, much in the same way as
electrons. Consequently a hole has precisely the opposite charge of an electron:
e+ = −e− = 1.602 ∗ 10−19 C
As electrons are negatively charged per definition it can often be useful to think of the movement of the positive charge
carriers, the holes, instead. In the water analogy charge can be thought of as the water.
2.3
Current
Current, the flow of charge is defined as:
Q
(2.3.1)
t
Where Q is amount of charge and t is time. Thus current is the amount of charge passing a specific point in a unit of
time (per second). In the water analogy current is simply a current of water, the rate at which water flows through a
section of the pipe (liters / sec). The unit of current is Ampere:
I=
A=
C
s
Ampere is a base SI Unit, meaning that Coulomb and the other units used in electronics are defined in terms of Ampere:
C = 1A ∗ 1s
The Coulomb is the summed charge of one Ampere for one second(the second is also an SI base unit).
2.3.a
Measuring current
When measuring current it is important to connect your multimeter/amperemeter in series with the point you want to
measure. The current should flow through the meter. The meter should have very low impedance so it does not affect
the circuit.
2.4
Voltage
Voltage is a measure of how much work is needed to move charge between two places. Thus, voltage is a relative, not
absolute, unit. Voltage is a potential difference commonly from one point to a constant reference(ground). Definition:
V =
W
Q
(2.4.1)
The voltage difference between points A and B is defined by how much work(energy, Joules) is required to move 1C of
charge from A to B. In the water analogy voltage can be thought of as pressure in the pipes. It is easy to imagine the
work required to push water from a pipe with lower pressure to a pipe with higher pressure. Furthermore voltage can
create current. When completing a circuit current will flow from a place with higher voltage(+) to a place with lower
voltage(-).
7
2.4.a
Measuring Voltage
When measuring voltage difference between two points in a circuit it is important to connect the multimeter/voltmeter
in parallel. The meter will feel the difference in voltage between the two probes. High impedance through the meter is
important. A meter with low impedance will draw current out of the circuit and change its function. See sections on
resistance and impedance.
2.5
Resistance
A resistance limits the flow of current. We can easily imagine that longer pipes, smaller pipes, small openings or even
bends in a pipe system can limit the flow of water. The same is true for current in wires. The formula which relates
current through a resistance to applied voltage and resistance value is called Ohms law.
2.5.a
Ohms law
The current through a resistance is proportional to the applied voltage and inversely proportional to the resistance
value:
V
⇒V =R∗I
(2.5.1)
I=
R
Impedance is a more general concept that includes resistance. Impedance is denoted with the Z character. Ohms law
still holds:
V =Z ∗I
(2.5.2)
Whenever the book, this article or the lecture slides mention impedance, you can think it’s like resistance. More on
impedance later.
2.5.b
Measuring resistance
Measuring resistance is very similar to voltage, but it is important to disconnect the resistor from the rest of your circuit.
The multimeter will send a small current through the resistor, so having other components connected can reduce the
measured resistance.
2.6
Voltage source
A voltage source, for example a battery, applies a voltage difference to two points. An ideal voltage source will always
deliver the desired voltage, regardless of how much current is drawn. Ideal voltage sources don’t exist, but we can often
assume that the voltage source is ideal (as long as we stay within a specified current range).
2.7
Current source
Similar to a voltage source the current source delivers a specified current regardless of required voltage. Current sources
are not as common as voltage sources.
2.8
Alternating and Direct Current(AC/DC)
Voltage, current or any other signal can be varying in time(alternating) or constant over time(direct). This terminology
is misleading as it usually has nothing to do with current. In electronics the AC or DC terms are most commonly used
about voltages. This terminology has been adopted in many different sciences, such as mathematics, statistics, physics
and signal processing.
3
3.1
Power
Energy
Energy is the ability to perform work. The unit, Joules, is expressed by:
J=
kg × m2
s2
8
(3.1.1)
In mechanics energy is usually thought of as either potential or kinetic, from gravity or motion respectfully. In electronics
we usually talk about energy in terms of chemical potential energy in batteries or in terms of power.
3.2
Power
Power is the rate of energy, work per time:
P =
3.2.a
J
s
(3.2.1)
Watts Law
When pushing current through a resistor or resistive circuit power is converted to heat or light. The amount of power
consumed is directly proportional to both voltage(pressure) and current(amount of water):
P =V ∗I =
4
V2
= I 2R
R
(3.2.2)
Impedance
As mentioned earlier impedance is a more general form of resistance. Impedance consists of resistance and reactance.
This chapter focuses on the mathematical basis of impedance and related terms. Some concepts, like capacitors and
inductors, are mentioned but not explained. These are explained in depth in separate chapters later
4.1
Resistance
Resistance limits the current flow. For a given voltage and resistance Ohm’s law (eq. (2.5.1) on page 8) can determine
the current. Resistance does not vary with frequency. Resistors have resistance. Ideal resistors have only resistance,
and no reactance. In ideal resistors impedance should be constant for different voltages, temperatures and frequencies.
Practical resistors are not perfect in these areas. The symbol for resistance is R.
4.2
Reactance
Reactance can be found in capacitors and inductors, it is expressed in ohms(Ω) just like resistance, but is different as
it changes with frequency. The symbol for reactance is X. For capacitive reactance (capacitors) XC is commonly used,
while XL is commonly used for inductive reactance (inductors).
4.3
Complex impedance
Impedance is then a complex number, a sum of resistance and reactance:
Z = R + jX
(4.3.1)
Z = R + jXL
(4.3.2)
Z = R − jXC
(4.3.3)
For inductive reactance:
And for capacitive reactance:
The absolute value or magnitude of impedance is the length of the vector:
p
|Z| = R2 + X 2
9
(4.3.4)
4.4
Conductance
Conductance is the opposite of resistance:
1
(4.4.1)
R
It quantifies how well a wire, component or material can conduct electricity. Conductance is constant with respect to
frequency. Conductance is more complicated for components which also have reactance:
G=
G=
R
|Z|2
(4.4.2)
See section 4.6 to understand why.
4.5
Susceptance
Susceptance is the opposite of reactance:
−1
(4.5.1)
X
Just like conductance susceptance is more complicated for components which have both resistance and reactance:
B=
B=
−X
|Z|2
(4.5.2)
Read section 4.6 to better understand susceptance.
4.6
Admittance
Admittance is the opposite of impedance:
1
(4.6.1)
Z
It is important to note that because of the imaginary unit it is not always trivial to convert between impedance and
admittance:
1
1
R − jX
R − jX
Y =
=
= 2
=
(4.6.2)
Z
R + jX
R + X2
|Z|2
Y =
And for completeness, the difference between capacitors and inductors is shown:
1
1
R + jXC
R + jX
=
=
= 2
2
Z
R − jXC
R + XC
|Z|2
1
1
R − jXL
R − jX
Y =
=
= 2
=
Z
R + jXL
R + XL2
|Z|2
Y =
The minus sign in impedance for capacitive reactance can also be confusing. In cases where an impedance is purely
resistive or non-resistive this conversion is simpler:
1
1
=
=G
ZR
R
1
1
1
YC =
=
=j
= jBC
ZC
−jXC
XC
1
1
1
YL =
=
=j
= −jBL
ZC
jXL
−XL
YR =
10
4.7
Updated Ohm’s law
With conductance and admittance we can express ohm’s law in new, useful ways.
V = R ∗ I,
G=
and
1
R
This is useful for expressing current through parallel resistances:
I=
V
⇒I =V ∗G
R
(4.7.1)
Ohm’s law also applies to impedance and admittance:
V =R∗I ⇒V =Z ∗I
(4.7.2)
I =V ∗G⇒I =V ∗Y
(4.7.3)
These formulae are easy to derive/remember and incredibly useful in some types of circuits.
4.8
Impedance and Admittance table
Below is a table that summarizes this section:
Ohm, Ω
Siemens, S
Real(DC)
Resistance, R
Conductance, G
Imaginary(AC)
Capacitive and Inductive Reactance XC , XL
Susceptance, B
Total(Combined)
Impedance, Z
Admittance, Y
4.9
5
5.1
Impedance vector - magnitude and phase
Circuits
Fundamentals
A circuit consists of nodes, paths, loops, branches and components. A component can be a resistor, capacitor, battery,
LED etc. Components are connected by wires, a node consists of all the wires that are connected together without any
(significant) impedance. There is one node on each pin of a component. Pins or wires are in the same node if they are
directly connected, and thus have the same voltage. A path is a road (through components) where a current can flow.
A loop is a closed path that connects to itself. A closed circuit has one or more loops and paths.
5.2
Series circuits
When connecting resistors in series we sum their resistance values to get the total:
RT OT = R1 + R2 + ... + Rn
(5.2.1)
The same is true for non-resistive impedances.
5.3
Parallel circuits
When connecting resistors in parallel we sum their conductances to find total conductance:
GT OT = G1 + G2 + ... + Gn
11
(5.3.1)
This makes sense; connecting two wires or equal resistors will double the conductance. We can easily find the resistance
of two parallell resistors:
1
1
1
RT OT =
=
= 1
(5.3.2)
1
GT OT
G1 + G2
R1 + R2
Or the common shortcut:
RT OT =
5.4
R1 R2
R1 + R2
(5.3.3)
Kirchhoffs Voltage Law
The sum of all voltages going around a loop in a circuit is always 0.
V1 + V2 + ... + Vn = 0
(5.4.1)
This is because when we end up at the same node, we must also end up at the original voltage level.
5.5
Kirchhoffs Current Law
The sum of currents into/out of a node is always 0.
I1 + I2 + ... + In = 0
(5.5.1)
This is because there has to come as much charge out of as into a node. Beware that sign(direction) of current is
important!
5.6
Voltage divider
A voltage divider takes an input voltage and produces a lower output voltage, that is a fraction of the input.
Vx = VS
Rx
RT OT
(5.6.1)
Two or more resistors are used to split the voltage into different levels. The relative values of the resistors decide the
voltages. For example: if one resistor is twice as big as the other, the voltage drop over it will be twice as big as the
voltage drop over the other. A potentiometer is often used to create an adjustable output voltage.
5.6.a
Load
It is important to note that when connecting the voltage divider to a load the output voltage changes(!). Adding a load
to the output will decrease the resistance down to ground and lower the voltage. A voltage divider with lower resistor
values is better at supplying a lot of current to a load without changing output voltage (much).
5.7
Current divider
Current dividers are very similar but not as common as voltage dividers. When connecting resistors in parallel the
current is divided among them. Recall the update ohm’s law in eq. (4.7.1) on page 11:
I =V ∗G
Thus the current through one of the parallel resistors is:
IX = VIN ∗ GX
Total conductance is simply:
GT OT = G1 + G2 + ... + Gn
And total current through all the resistors is:
IT OT = VIN ∗ GT OT
12
The more standard current divider formula is obtained by specifying input current instead of voltage:
IX = (VIN ) ∗ GX = (RT OT ∗ IIN ) ∗ GX =
Or in terms of resistance:
IX =
RT OT
∗ IIN
RX
It is crucial to remember that this is a parallel circuit, so:
RX ≥ RT OT
6
Periodic signals
6.1
Sinusoid
6.1.a
Amplitude
6.1.b
Frequency & Period
6.1.c
Phase shift
6.2
Square wave
6.2.a
Pulse width & Duty Cycle
6.2.b
Rise & fall time
7
Capacitors
7.1
Physical properties
7.2
Stored capacitor charge
7.3
Time constant
7.4
Capacitive reactance
7.5
Voltage & Current
7.6
Parallell capacitors
7.7
Series capacitors
7.8
Practical capacitors
7.9
RC circuits
7.10
Capacitive voltage divider
7.11
Power consumption in capacitors
13
GX
∗ IIN
GT OT
(5.7.1)
(5.7.2)
8
Inductors
8.1
Physical properties
8.2
Induced voltage
8.3
Inductive reactance
8.4
Practical Inductors
8.5
RL circuits
9
Filters and frequency Response
9.1
DC characteristics
9.2
Time domain
9.3
Frequency domain
9.4
Harmonics
9.5
Fourier series and Fourier transform
9.6
Low-Pass filter
9.7
High-Pass filter
9.8
Band-pass filter
9.9
Band-stop filter
10
Resources / Further Reading
This Introduction to Analog Electronics was inspired by previous works by different authors. The list below should
serve as a recommended reading list for anyone who wants to learn more.
• "Electronics Fundamentals: Circuits, Devices and Applications" [1]
• "INF1411 Formelsamling" [2]
• "Formelsamling i Elektronikk" [3]
• "Analog Integrated Cicruit Design" [4]
References
[1]
[2]
Thomas L. Floyd. Electronics Fundamentals: Circuits, Devices and Applications. Reading, Massachusetts: AddisonWesley, 1993.
Ole Christian Schmidt Hagenes. INF1411 Formelsamling. 2015. url: https://github.com/ochagenes/INOR/
blob/master/2_semester/INF1411/formelsamling.pdf.
[3]
Ole Herman Schumacher Elgesem. Formelsamling i Elektronikk. 2015. url: http://olehelg.at.ifi.uio.no/
files/INF1411_formulae.pdf.
[4]
Tony Chan Carusone. Analog Integrated Circuit Design. PublisherX, 19XX.
14