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431/531 Class Notes 3
3 Circuit Analysis in Frequency Domain
We now need to turn to the analysis of passive circuits (involving EMFs, resistors, capaci-
tors, and inductors) in frequency domain. Using the technique of the complex imp edance,
we will b e able to analyze time-dep endent circuits algebraically, rather than by solving dif-
ferential equations. We will start by reviewing complex algebra and setting some notational
conventions. It will probably not b e particularly useful to use the text for this discussion,
and it could lead to more confusion. Skimming the text and noting results might b e useful.
3.1 Complex Algebra and Notation
Let
V b e the complex representation of V. Then we can write
V = <(
V ) + {=(
V ) = V e
{
= V [cos + { sin ]
where { =
p
1. V is the (real) amplitude:
V =
q
V
V
h
2
(
V ) + =
2
(
V )
i
1 = 2
where denotes complex conjugation. The op eration of determining the amplitude of a
complex quantity is called taking the modulus. The phase is
= tan
1
h
V )=<(
V )
i
So for a numerical example, let a voltage have a real part of 5 volts and an imaginary part
of 3 volts. Then
V = 5 + 3 { =
p
34 e
{ tan
1 (3=5) .
Note that we write the amplitude of
V , formed by taking its mo dulus, simply as V. It is
often written j
V j. We will also use this notation if there might b e confusion in some context.
Since the amplitude will in general b e frequency dep endent, it will also b e written as V (! ).
We will most often b e interested in results expressed as amplitudes, although we will also
lo ok at the phase.
3.2 Ohm's Law Generalized
Our technique is essentially that of the Fourier transform, although we will not need to
actually invoke that formalism. Therefore, we will analyze our circuits using a single Fourier
frequency comp onent,! = 2 f. This is p erfectly general, of course, as we can add (or
integrate) over frequencies if need b e to recover a result in time domain. Let our complex
Fourier comp onents of voltage and current b e written as
V = V e
{(! t+ 1 ) and
I = I e
{(! t+ 2 ) .
Now, we wish to generalize Ohm's Law by replacing V = I R by
V =
I
Z , where
Z is the
(complex) imp edance of a circuit element. Let's see if this can work. We already know that
a resistor R takes this form. What ab out capacitors and inductors?
Our expression for the current through a capacitor, I = C (dV =dt) b ecomes
I = C
d
dt
V e
{(! t+ 1 )
= {! C
V
Thus, we have an expression of the form
V =
I
Z
C
for the imp edance of a capacitor,
Z
C
, if
we make the identi cation
Z
C
= 1 =({! C ).
For an inductor of self-inductance L, the voltage drop across the inductor is given by
Lenz's Law: V = L(dI =dt). (Note that the voltage drop has the opp osite sign of the induced
EMF, which is usually how Lenz's Law is expressed.) Our complex generalization leads to
V = L
d
dt
I = L
d
dt
I e
{(! t+ 2
)
= {! L
I
So again the form of Ohm's Law is satis ed if we make the identi cation
ZL = {! L.
To summarize our results, Ohm's Law in the complex form
V =
I
Z can b e used to
analyze circuits which include resistors, capacitors, and inductors if we use the following:
resistor of resistance R:
Z
R
= R
capacitor of capacitance C :
Z
C
= 1 =({! C ) = {=(! C )
inductor of self-inductance L:
ZL = {! L
3.2.1 Combining Imp edances
It is signi cant to p oint out that b ecause the algebraic form of Ohm's Law is preserved,
imp edances follow the same rules for combination in series and parallel as we obtained for
resistors previously. So, for example, two capacitors in parallel would have an equivalent
imp edance given by 1 =
Z
p
Z
1
Z
2
. Using our de nition
Z
C
= {=! C , we then recover
the familiar expression Cp = C 1 + C 2. So we have for any two imp edances in series (clearly
generalizing to more than two):
Zs =
Z 1 +
Z 2
And for two imp edances in parallel:
Z
p
h
Z
1
Z
2
i
1
Z 1
Z 2
Z
1
Z
2
And, accordingly, our result for a voltage divider generalizes (see Fig. 9) to
V
out
V
in
Z
2
Z
1
Z
2
Now we are ready to apply this technique to some examples.
3.3 A High-Pass RC Filter
The con guration we wish to analyze is shown in Fig. 10. Note that it is the same as Fig. 7
of the notes. However, this time we apply a voltage which is sinusoidal:
V
in
(t) = V in
e
{(! t+ ) .
As an example of another common variation in notation, the gure indicates that the input
is sinusoidal (\AC") by using the symb ol shown for the input. Note also that the input and
output voltages are represented in the gure only by their amplitudes Vin and Vout , which
also is common. This is ne, since the metho d we are using to analyze the circuit (complex
imp edances) shouldn't necessarily enter into how we describ e the physical circuit.
The decib el scale works as follows: db= 20 log 10
(A
1
=A
2
), where A 1
and A 2
represent any
real quantity, but usually are amplitudes. So a ratio of 10 corresp onds to 20 db, a ratio of 2
corresp onds to 6 db,
p
2 is approximately 3 db, etc.
3.4 A Low-Pass RC Filter
An analogy with the analysis ab ove, we can analyze a low-pass lter, as shown in Fig. 11.
Vin
R
C
Vout
Figure 11: A low-pass lter.
You should nd the following result for the transfer function:
T (! )
j
V
out
j
j
Vin j
V
out
Vin
[ 1 + (! RC )
2 ]
1 = 2
You should verify that this indeed exhibits \low pass" b ehavior. And that the 3 db
frequency is the same as we found for the high-pass lter:
2 f3db = !3db = 1 =(RC ) (9)
We note that the two circuits ab ove are equivalent to the circuits we called \di erentiator"
and \integrator" in Section 2. However, the concept of high-pass and low-pass lters is much
more general, as it do es not rely on an approximation.
An aside. One can compare our results for the RC circuit using the complex imp edance
technique with what one would obtain by starting with the di erential equation (in time) for
an RC circuit we obtained in Section 2, taking the Fourier transform of that equation, then
solving (algebraically) for the transform of V out
. It should b e the same as our result for the
amplitude V out
using imp edances. After all, that is what the imp edance technique is doing:
transforming our time-domain formuation to one in frequency domain, which, b ecause of
the p ossibility of analysis using a single Fourier frequency comp onent, is particularly simple.
This is discussed in more detail in the next notes.
