EE42: Running Checklist of Electronics Terms
14.02.05 – Dick White
Terms are listed roughly in order of their introduction. Most definitions can be found in
your text.
TERM
Charge, current, voltage, resistance , conductance, energy, power
Coulomb, ampere, volt, ohm, siemen (mho), joule, watt
Reference directions
Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL),
Ohm’s Law
Series connection, parallel connection
DC (steady), AC (time-varying)
Independent and dependent ideal voltage and current source
Voltage divider, current divider
Analog (A/D), Digital (D/A)
Multimeter (DMM), Oscilloscope
Prefixes (milli-, etc.)
Linear, nonlinear elements
Superposition (analysis)
Nodal analysis (node, supernode)
Loop analysis (mesh, branch)
Power delivery, dissipation, storage, maximum power transfer
Equivalent circuits (Rs, Cs or Ls in series/parallel; Thevenin,
Norton)
Frequency; angular frequency; period; phase (Hz; radian/s)
Capacitor, inductor, transformer
Phasor, impedance, reactance
Amplifier, filter, transfer function
Steady-state, transient, sinusoidal excitation
Terms2
Week 5a
1
Lecture 5a
Review: Types of Circuit Excitation
Why Sinusoidal Excitation?
Phasors
Week 5a
2
Types of Circuit Excitation
Linear TimeInvariant
Circuit
Linear TimeInvariant
Circuit
Steady-State Excitation
OR
Linear TimeInvariant
Circuit
Digital
Pulse
Source
Linear TimeInvariant
Circuit
Sinusoidal (SingleFrequency) Excitation
Transient Excitation
Week 5a
3
Why is Sinusoidal Single-Frequency Excitation
Important?
1. Some circuits are driven by a single-frequency
sinusoidal source.
Example: The electric power system at frequency of
60+/-0.1 Hz in U. S. Voltage is a sinusoidal function of time
because it is produced by huge rotating generators
powered by mechanical energy source such as steam
(produced by heat from natural gas, fuel oil, coal or
nuclear fission) or by falling water from a dam
(hydroelectric).
Week 5a
4
Bonneville Dam (Columbia River) Where Much
of California’s Electric Power Comes From
Week 5a
5
Turbine-generator sets at Bonneville Dam
Week 5a
6
Where 3-Phase Electricity Comes From
Generator
driven by
falling
water has
3 separate
coils
Rotation of Rotor
Plane of Coil B
N
Plane of
Coil A
S
Coil A
Plane of Coil C
Output voltages
from the 3 coils VoltageA
(they leave the
generating plant
on 3 separate
cables)
B
C
Direct current
in the rotor
(rotating coil)
produces a
magnetic field
that generates
currents in
stationary coils
A, B and C
Time
Time for which rotor position is shown
Week 5a
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Why Sinusoidal Excitation? (continued)
2. Some circuits are driven by sinusoidal sources
whose frequency changes slowly over time.
Example: Music reproduction system (different
notes).
3. And, you can express any periodic electrical signal as a
sum of single-frequency sinusoids – so you can
analyze the response of the (linear, time-invariant)
circuit to each individual frequency component and
then sum the responses to get the total response.
Week 5a
8
Representing a Square Wave as a Sum of Sinusoids
Ti me (ms)
Signal
signal(V)
Relative Amplitude
d
Signal (V)
c
Signal
signal(V)
b
a
Frequenc y (Hz)
(a) Square wave with 1-second period. (b) Fundamental component (dotted) with 1-second period, third-harmonic (solid black)
with1/3-second period, and their sum (blue). (c) Sum of first ten
components. (d) Spectrum with 20 terms.
9
Week 5a
Single-frequency sinusoidal-excitation AC circuit problems
1. The technique we’ll show works on circuits composed of linear
elements (R, C, L) that don’t change with time  “linear
time-invariant circuits”.
2. The circuit is driven with independent voltage and/or current
sources whose voltages or currents vary at a single frequency, f,
measured in Hertz (abbreviated Hz) this is the number of cycles the
voltages or currents execute per second. We can represent the
source voltages or currents as functions of time as
v(t) = V0cos(wt) or i(t) = I0cos(wt),
where w = 2pf is the angular frequency in radians per second.
Example: In the U. S. the AC power frequency, f, is 60 Hz
and the peak voltage V0 is 170 V, so w = 377 radians/s and
v(t) = 170cos(377t) V. More generally, we might have sources
v(t) = V0sin(wt) or i(t) = I0cos(wt = f), where f is a phase angle.
Week 5a
10
We could solve our circuit equations using such functions of
time, but we’d have to do a lot of tedious trigonometric
transformations. Instead we use a mathematical trick to
eliminate time dependence from our equations!
The trick is based on a fundamental fact about linear, timeinvariant circuits excited with sinusoidal sources: the
frequencies of all the voltages and currents in the circuit are
identical.
Week 5a
11
RULE: “Sinusoid in”-- “Same-frequency sinusoid out” is true
for linear time-invariant circuits. (The term “sinusoid” is intended
to include both sine and cosine functions of time.)
Circuit of linear elements
(R, L, C)
Output:
Excitation:
vS(t) = VScos(wt + f)
Iout(t) = I0cost(wt + a)
Given
Given
Given
SAME
?
SAME
w
?
Intuition: Think of sinusoidal excitation (vibration) of a linear
mechanical system – every part vibrates at the same frequency, even
though perhaps at different phases.
Week 5a
12
You can solve AC circuit analysis problems that involve
Circuits with linear elements (R, C, L) plus independent and
dependent voltage and/or current sources operating at a
single angular frequency w = 2pf (radians/s) such as
v(t) = V0cos(wt) or i(t) = I0cos(wt).
By using any of Ohm’s Law, KVL and KCL equations,
doing superposition analysis, nodal analysis or mesh
analysis,
AND
Using instead of the terms below on the left (general
excitation), the terms below on the right (sinusoidal
excitation):
Week 5a
13
Resistor I-V relationship
General excitation
vR = iRR
VR = IRR where R is the resistance in ohms,
VR = phasor voltage across the resistor,
IR = phasor current through the resistor,
and boldface indicates complex quantity.
Capacitor I-V relationship
General excitation
iC = CdvC/dt
Sinusoidal excitation
Sinusoidal excitation
IC = VC / ZC where IC = phasor current
through the capacitor, VC = phasor voltage
across the capacitor, the capacitive
impedance ZC in ohms is ZC = 1/jwC ,
j = (-1)1/2, and boldface capital letters are
complex quantities.
(Note: EE’s use j for (-1)1/2 instead of i,
since i might suggest current)
Week 5a
14
Inductor I-V relationship
General excitation
vL = LdiL/dt
Sinusoidal excitation
VL = IL ZL where VL is the phasor voltage
across the inductor, IL is the phasor
current through the inductor, the
inductive impedance in ohms ZL is
ZL = jwL , j = (-1)1/2 and boldface capital
letters are complex quantities.
Week 5a
15
Example 1
We’ll explain what phasor currents and voltages are shortly, but first let’s
look at an example of using them:
Here’s a circuit containing an AC voltage source with angular frequency
w, and a capacitor C. We represent the voltage source and the current
that flows (in boldface print) as phasors VS and I -- whatever they are!
+
VS
-
I
C
We can obtain a formal solution for the unknown current in this circuit
by writing KVL:
-VS + IZC = 0
We can solve symbolically for I:
I = VS/ZC = jwCVS
Week 5a
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Note that so far we haven’t had to include the variable of time
in our equations -- no sin(wt), no cos(wt), etc. -- so our
algebraic task has been almost trivial. This is the reason
for introducing phasor voltages and currents, and impedances!
In order to “reconstitute” our phasor currents and voltages to
see what functions of time they represent, we use the rules
below. Note that often (for example, when dealing with the
gain of amplifiers or the frequency characteristics of filters),
we may not even need to go back from the phasor domain to
the time domain – just finding how the magnitudes of voltages
and currents vary with frequency w may be the only
information we want.
Week 5a
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Rules for “reconstituting” phasors (returning to the
time domain)
Rule 1: Use the Euler relation for complex numbers:
imaginary
ejf = cos(f) + jsin(f), where j = (-1)1/2
j
1
f
real
Rule 2: To obtain the actual current or voltage i(t) or v(t)
as a function of time
1. Multiply the phasor I or V by ejwt, and
2. Take the real part of the product
For example, if I = 3 amps, a real quantity, then
i(t) = Re[Iejwt] = Re[3ejwt] = 3cos(wt) amps where Re
means “take the real part of”
Week 5a
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Rule 3: If a phasor current or voltage I or V is not purely
real but is complex, then multiply it by ejwt and take the
real part of the product.
For example, if V = V0ejf, then v(t) = Re[Vejwt] =
Re[V0ejfejwt] = Re[V0ej(wt + f)] =
V0cos(wt + f)
Week 5a
19
Finishing Example 1
+
vS(t) = 4 cos(wt)
-
i(t)
C
Apply this approach to the capacitor circuit above, where the
voltage source has the value
vS(t) = 4 cos(wt) volts.
The phasor voltage VS is then purely real: VS = 4. The
phasor current is I = VS/ZC = jwCVS = (wC)VSejp/2, where
we use the fact that j = (-1)1/2 = ejp/2; thus, the current in a
capacitor leads the capacitor voltage by p/2 radians (90o).
Note: Often (especially in this class) we may not care about
the phase angle, and will focus just on the amplitude of the
voltage or current that we obtain. This will be particularly
true of filters and amplifiers.
Week 5a
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The actual current that flows as a function of time, i(t),
is obtained by substituting VS = 4 into the equation for I above,
multiplying by ejwt, and taking the real part of the product.
i(t) = Re[j (wC) x 4ejwt] = Re[4(wC)ej(wt + p/2)]
i(t) = 4(wC)cos(wt + p/2) amperes
Note: We obtained the current as a function of time (the
current waveform) without ever having to work with
trigonometric identities!
Week 5a
21
Analysis of an RC Filter
Consider the circuit shown below. We want to use
phasors and complex impedances to find how the
ratio |Vout/Vin| varies as the frequency of the input
sinusoidal source changes. This circuit is a filter;
how does it treat the low frequencies and the high
frequencies?
+
+
R
Vout
C
Vin
I
-
Assume the input voltage is vin(t) = Vincos(wt) and represent
it by the phasor Vin. A phasor current I flows clockwise in the
circuit.
Week 5a
22
Write KVL:
-Vin + IR +IZC = 0 = -Vin + I(R + ZC)
The phasor current is thus
I = Vin/(R + ZC)
The phasor output voltage is Vout = I ZC.
Thus
Vout = Vin[ZC /(R + ZC)]
If we are only interested in the dependence upon frequency
of the magnitude of (Vout / Vin) we can write
| Vout / Vin | = |ZC/(R + ZC)| = 1/|1 + R/ ZC |
Substituting for ZC, we have 1 + R/ ZC = 1 + jwRC,
2
whose magnitude is (RC )  1 Thus,
·
V out
1
--------- = --------------------------------2
Vin
 wRC  + 1
Week 5a
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Explore the Result
·
V out
1
--------- = --------------------------------2
Vin
 wRC  + 1
If wRC << 1 (low frequency) then | Vout / Vin | = 1
If wRC >> 1 (high frequency) then | Vout / Vin | ~ 1/wRC
If we plot | Vout / Vin | vs. wRC we obtain roughly the plot
below, which was plotted on a log-log plot:
|Vo ut /Vi n |

1
w =
wRC
The plot shows that this is a low-pass filter. Its cutoff frequency
is at the frequency w for which wRC = 1.
Week 5a
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Notice that we’ve obtained a lot of information about how this
particular RC circuit performs just by looking at the magnitude
of the ratio of phasor output voltage to phasor input voltage
(i.e., we haven’t had to study the phase angles associted with
those phasor voltages. (In more detailed studies the phase angles
can be important – but not in this course.)
Does this behavior make sense from what we know about
capacitors? YES!
At low frequency a capacitor is like an open circuit –
so the output voltage would equal the input voltage
At high frequency a capacitor is like a short circuit –
so the output voltage would be very small.
Week 5a
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Here is a useful web site to explore:
http://www.phys.unsw.edu.au/~jw/AC.html
You’ll find some demonstrations dealing with phasors
and impedances there.
And Appendix A in Hambley is a review of complex numbers.
Week 5a
26
Why Does the Phasor Approach Work?
1. Phasors are discussed at length in your text (Hambley 3rd
Ed., pp. 195-201) with an interpretation that sinusoids can
be visualized as the real axis projection of vectors rotating
in the complex plane, as in Fig. 5.4. This is the most basic
connection between sinusoids and phasors.
2. We present phasors as a convenient tool for analysis of
linear time-invariant circuits with a sinusoidal excitation.
The basic reason for using them is that they eliminate the
time dependence in such circuits, greatly simplifying the
analysis.
3. Your text discusses complex impedances in Sec. 5.3, and
circuit analysis with phasors and complex impedances in
Sec. 5.4. Skim over this LIGHTLY.
Week 5a
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Motivations for Including Phasors in EECS 40
1. It enables us to include a lab where you measure the
behavior of RC filters as a function of frequency, and
use LabVIEW to automate that measurement.
2. It enables us to (probably) include a nice operational
amplifier lab project near the end of the course to
make an “active” filter (the RC filter is passive).
3. It enables you to find out what impedances are and
use them as real EEs do.
4. The subject was also supposedly included (in a way)
in EECS 20 which some of you may have taken.
Week 5a
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