Rather than solving the differential equation that arises in
circuits containing capacitors and inductors, let's pretend that
all sources in the circuit are complex exponentials having the
same frequency. Although this pretense can
only be mathematically true, this fiction will greatly ease
solving the circuit no matter what the source really is.
For the above example
RC
RC
circuit (Figure 1), let
v
in
=
V
in
ei2πft
v
in
V
in
2
f
t
.
The complex amplitude
V
in
V
in
determines the size of the source and its phase. The critical
consequence of assuming that sources have this form is that
all voltages and currents in the circuit
are also complex exponentials, having amplitudes governed by
KVL, KCL, and the v-i relations and the
same frequency as the source. To appreciate why this should be
true, let's investigate how each circuit element behaves when
either the voltage or current is a complex exponential. For the
resistor,
v=Ri
v
R
i
.
When
v=Vei2πft
v
V
2
f
t
;
then
i=VRei2πft
i
V
R
2
f
t
.
Thus, if the resistor's voltage is a complex exponential, so is
the current, with an amplitude
I=VR
I
V
R
(determined by the resistor's v-i relation)
and a frequency the same as the voltage. Clearly, if the current
were assumed to be a complex exponential, so would the
voltage. For a capacitor,
i=Cdvd
t
i
C
t
v
.
Letting the voltage be a complex exponential, we have
i=CVi2πfei2πft
i
C
V
2
f
2
f
t
.
The amplitude of this complex exponential is
I=CVi2πf
I
C
V
2
f
.
Finally, for the inductor, where
v=Ldid
t
v
L
t
i
,
assuming the current to be a complex exponential results in the
voltage having the form
v=LIi2πfei2πft
v
L
I
2
f
2
f
t
,
making its complex amplitude
V=LIi2πf
V
L
I
2
f
.
The major consequence of assuming complex exponential
voltage and currents is that the ratio
Z=VI
Z
V
I
for each element does not depend on time, but does depend on source frequency.
This quantity is known as the element's
impedance.
The impedance is, in general, a complex-valued,
frequency-dependent quantity. For example, the magnitude of the
capacitor's impedance is inversely related to frequency, and has
a phase of
−π2
2
.
This observation means that if the current is a complex
exponential and has constant amplitude, the amplitude of the
voltage decreases with frequency.
Let's consider Kirchoff's circuit laws. When voltages around a
loop are all complex exponentials of the same frequency, we have
∑
n
n
v
n
=∑
n
n
V
n
ei2πft=0
n
n
v
n
n
n
V
n
2
f
t
0
(1)
which means
∑
n
n
V
n
=0
n
n
V
n
0
(2). We can easily imagine that the complex
amplitudes of the currents obey KCL.
What we have discovered is that source(s) equaling a complex
exponential of the same frequency forces all circuit variables
to be complex exponentials of the same frequency. Consequently,
the ratio of voltage to current for each element equals the
ratio of their complex amplitudes, which depends only on the
source's frequency and element values.
This situation occurs because the circuit elements are linear
and time-invariant. For example, suppose we had a circuit
element where the voltage equaled the square of the current:
vt=Ki2t
v
t
K
i
t
2
.
If
it=Iei2πft
i
t
I
2
f
t
,
vt=KI2ei2π2ft
v
t
K
I
2
2
2
f
t
,
meaning that voltage and current no longer had the same
frequency and that their ratio was time-dependent.
Because for linear circuit elements the complex amplitude of
voltage is proportional to the complex amplitude of
current—
V=ZI
V
Z
I
— assuming complex exponential sources means circuit
elements behave as if they were resistors, where instead of
resistance, we use impedance.
Because complex amplitudes for
voltage and current also obey Kirchoff's laws, we can solve
circuits using voltage and current divider and the series and
parallel combination rules by considering the elements to be
impedances.
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