Nonlinear Circuits
R. Spencer
1
Nonlinear Circuits
1
– A Primer for E17
Richard Spencer, November 18, 2010 (modified May, 2011)
Consider a network with input
x
(
t
)
and output
y
(
t
)
. If an input
x
1
(
t
)
produces an output
y
1
(
t
)
and an input
x
2
(
t
)
produces an output
y
2
(
t
)
, then the network is
linear
if and only
if an input
x
(
t
)
=
ax
1
(
t
)
+
bx
2
(
t
)
produces an output
y
(
t
)
=
ay
1
(
t
)
+
by
2
(
t
)
. If, in addition
to being linear, the response of the network is independent of when an input is applied,
then the network is called a
linear time-invariant
(LTI) network. We have many powerful
tools for analyzing LTI systems. For example, as shown briefly in the powerpoint
presentation on frequency-domain analysis in class (you can download it from the class
website)
2
, we can represent a complicated periodic input by a Fourier series and then find
the output if we know the frequency-domain transfer function for the circuit,
H
(
j
ω
).
The vast majority of analog circuits perform functions that we desire to be linear; in other
words, we want the output to be a linear function of the input. But, all active electronic
devices (e.g., transistors) and some passive devices (e.g., diodes) are nonlinear.
Designing truly linear systems using devices with nonlinear characteristics is impossible.
Nevertheless, we are able to design circuits that behave approximately linearly by
restricting the signals to a small enough range that the nonlinear devices appear linear to
a sufficient degree of accuracy (this is called
small-signal
analysis).
Nonlinear systems pose a problem for the designer because the mathematical tools
available for dealing with them are not as well developed as those for linear systems, and
are frequently too complicated to use effectively in design. In fact, nonlinear analysis
often requires that the problem be solved numerically rather than analytically. The way
around these mathematical difficulties is to approximate the devices as being linear for
some set of circumstances.
The purpose of this primer is to introduce you to the basic concepts of analyzing and
designing circuits that contain nonlinear elements. The small-signal approximation is
shown in Section 5.7 of your text, so we won’t cover that here (although we will make
some comments to tie that material together with what is presented here). Instead, we will
focus on how to use a piecewise linear model to analyze some circuits when the small-
signal approximation is inappropriate.
Consider, for example, a semiconductor diode. The schematic symbol, equation and
iv
characteristic of a diode are shown in Figure 1. There are two constants in this equation.
The
saturation current
,
I
S
, is a device- and temperature-dependent constant. Although it
varies over several orders of magnitude from one device to another, a typical value is
10
-14
A. The
thermal voltage
,
V
T
, is a fundamental constant that is proportional to
absolute temperature and is about 25 mV at room temperature (a cold room, but a
convenient value for
V
T
). The diode is obviously a nonlinear device. In fact, the
usefulness of the device derives from its nonlinear operation. A diode is essentially a one-