R1
R2
R3
e1e2
iin
i
Figure 3.32
The presence of a current source in the circuit does not affect the node method greatly; just
include it in
writing KCL equations as a current leaving the node. The circuit has three nodes, requiring us
to define
two node v

R1 + R2 + R3
iin
To find the indicated current, we simply use i =
e2
R3
.
Example 3.6: Node Method Example
In the circuit shown in Figure 3.33, we cannot use the series/parallel combination rules: The
vertical
resistor at node 1 keeps the two horizontal 1

the output if we use impedances. Because impedances depend only on frequency, we find
ourselves in the
frequency domain. A common error in using impedances is keeping the time-dependent
part, the complex
exponential, in the fray. The entire point of using

RC
d
dt
vout + vout = vin
This is the same equation that was derived much more tediously in Section 3.8. Finding the
differential
equation relating output to input is far simpler when we use impedances than with any other
technique.
Exercise 3.11 (Solutio

Figure 3.37: Of the four possible dependent sources, depicted is a voltage-dependent voltage source in
the context of a generic circuit.
Figure 3.38 shows the circuit symbol for the op-amp and its equivalent circuit in terms of a
voltagedependent
voltage

+
v
+
Figure 3.39: The top circuit depicts an op-amp in a feedback amplifier configuration. On the bottom
is the equivalent circuit, and integrates the op-amp circuit model into the circuit.
Note that the op-amp is placed in the circuit upside-down, with

R1
+ Ej2_fC +
E
R2
=0
with the result
E=
R2
R1 + R2 + j2_fR1R2C
Vin
To find the transfer function between input and output voltages, we compute the ratio E
Vin
. The transfer
functions magnitude and angle are
jH (f) j =
q R2
(R1 + R2)2 + (2_fR1R2C)2
\H (f

In some (complicated) cases, we cannot use the simplification techniquessuch as parallel
or series combination
rulesto solve for a circuits input-output relation. In other modules, we wrote v-i relations
and
Kirchhoffs laws haphazardly, solving them more

R
2_L
. We want this cutof
frequency to be much less than 60 Hz. Suppose we place it at, say, 10 Hz. This specification
would
require the component values to be related by R
L
= 20_ = 62:8. The transfer function at 60 Hz
would be _
1
j2_60L + R
_
=
1
R
_

R2
Figure 3.35: Transfer functions of the circuits shown in Figure 3.34. Here, R1 = 1, R2 = 1, and C = 1.
When R2 = R1, as shown on the plot, the passband gain becomes half of the original, and the
cutof
frequency increases by the same factor. Thus, addin

each resistor are the same while the currents are not. Because the voltages are the same,
we can find the
current through each from their v-i relations: i2 = vout
R2 and iL = vout
RL . Considering the node where all three
resistors join, KCL says that the

The unit pulse (Figure 2.5) describes turning a unit-amplitude signal on for a duration of _
seconds, then
turning it off.
p_ (t) =
8><
>:
0; t < 0
1; 0 < t < _
0; t > _
(2.24)
1
t
p(t)
Figure 2.5: The pulse.
We will find that this is the second most impo

Figure 2.15: A time reversal system.
Again, such systems are difficult to build, but the notion of time reversal occurs frequently in
communications
systems.
Exercise 2.5 (Solution on p. 30.)
Mentioned earlier was the issue of whether the ordering of syst

Recall that each elements current and voltage must obey the convention that positive
current is defined to
enter the positive-voltage terminal. With this convention, a positive value of vkik corresponds
to consumed
power, a negative value to created power

v
i
Figure 3.1: The generic circuit element.
Current flows through circuit elements, such as that depicted in Figure 3.1, and through
conductors,
which we indicate by lines in circuit diagrams. For every circuit element we define a voltage
and a current.

t nT
2
_
Solution to Exercise 2.5 (p. 23)
In the first case, order does not matter; in the second it does. Delay means t ! t _ . Timereverse
means t ! t
Case 1 y (t) = Gx(t _ ), and the way we apply the gain and delay the signal gives the same
result.
Ca