Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f,
any output of the circuit (the current through any component, or the voltage between any two
points) is also sinusoidal with frequency f. Note that the output need not be in phase with the
input.[1]
An equivalent definition of a linear circuit is that it obeys the superposition principle. This means
that the output of the circuit F(x) when a linear combination of signals ax1(t) + bx2(t) is applied
to it is equal to the linear combination of the outputs due to the signals x1(t) and x2(t) applied
separately:
Informally, a linear circuit is one in which the values of the electronic components, the
resistance, capacitance, inductance, gain, etc. don't change with the level of voltage or
current in the circuit.
Examples
A linear circuit is one that has no nonlinear electronic components in it. Examples of linear
circuits are ideal amplifiers, differentiators, and integrators, or any circuit composed exclusively
of ideal resistors, capacitors, inductors, op-amps (in the "non-saturated" regime), and other
"linear" circuit elements.
Some examples of circuits and components that operate in a nonlinear way: diodes, transistors
when they are operated in saturation, iron core inductors and transformers when the core is
saturated, mixers, modulators, and digital logic circuits.
Significance
Because they obey the superposition principle, linear circuits can be analyzed with powerful
mathematical frequency domain techniques, including Fourier analysis and the Laplace
transform. These also give an intuitive understanding of the qualitative behavior of the circuit,
characterizing it using terms such as gain, phase shift, resonant frequency, bandwidth, Q_factor,
poles, and zeros. The analysis of a linear circuit can often be done by hand using a scientific
calculator.
In contrast, nonlinear circuits usually don't have exact solutions. They must be analyzed using
approximate numerical methods by electronic circuit simulation computer programs such as
Spice, if accurate results are desired. These can give solutions for any specific circuit, but not
much insight into the operation of the circuit in general with different component values or
inputs. The behavior of such linear circuit elements as resistors, capacitors, and inductors can be
specified by a single number (resistance, capacitance, inductance, respectively). In contrast, a
nonlinear element's behavior is specified by its detailed transfer function, which may be given as
a graph. So specifying the characteristics of a nonlinear circuit requires more information than is
needed for a linear circuit.
Linear circuits and systems form a separate category within electronic manufacturing.
Manufacturers of transistors and integrated circuits divide their product lines into 'linear' and
'digital' lines, for example. In their linear components, manufacturers work to reduce nonlinear
behavior to a minimum, to make the real component conform as closely as possible to the 'ideal'
model used in circuit theory.
Small signal approximation
Nonlinear elements such as transistors tend to behave linearly when small AC signals are applied
to them. So in analysing many circuits where the signal levels are small, for example those in TV
and radio receivers, nonlinear elements can be replaced with a linear small-signal model,
allowing linear analysis techniques to be used.
Conversely, many linear circuit elements show nonlinearity as the signal level is increased. If
nothing else, the power supply voltage to the circuit usually puts a limit on the magnitude of
voltage output from a circuit. Above that limit, the output ceases to scale in magnitude with the
input, failing the definition of linearity.
References
1. ^ Zumbahlen, Hank (2008). Linear circuit design handbook. Newnes.
ISBN 0750687037.
2. Linear and Non-Linear Circuit Elements
3. Author: J.B. Hoag
4. Imagine, if you will, a sealed box containing electrical circuits of some sort or other, and
having four binding posts, two for the "input" and two for the " output." Let the output
current I be observed for a succession of voltages E applied to the input terminals, and
let a plot be made of these two quantities, with I vertically and E horizontally. If the I-
E curve which is obtained is a straight line, the electrical devices in the box are to be
called linear elements; if a curved line results, the box contains nonlinear elements.
5. If the "circuit elements" in the box consist merely of a series resistance, the symmetrical
straight line of Fig. 23 A will be obtained.
Fig. 23 A. Symmetrical ohmic element
6. Let a fluctuating voltage (e-t) be applied to the input terminals. Then the fluctuating
output current (i-t), as shown in this figure, will have the same shape as that of the input
voltage.
7. If the circuit elements consist of crystals, such as galena or iron pyrites, or if they consist
of copper-oxide rectifiers, the unsymmetrical, non-linear (= non-ohmic) curve of Fig. 23
B will be obtained.
Figure 1non ohomic unsymmetrical element
8. A pure sinusoidal voltage at the input will not yield an output current of the same
(sinusoidal) wave-form. Instead, the current will be partially rectified, as shown in the
figure, and will be distorted or "full of harmonics".
9. If the box contains a diode, the detector or rectifier property shown in Fig. 23 C will
result.
Fig. 23 C. Half-wave non-ohmic
element
10. These conditions were discussed in some detail in earlier chapters. The rectifying action
of a triode operating on the upper knee of its characteristic curve is shown in Fig. 23 D.
Fig. 23 D. Rectifier action on the upper
knee of a triode
11. Figure 23 E shows the output currents versus input grid voltages of triodes operated from
a point in the middle of the straight portion of the characteristic curve. Notice that when
too great an input voltage is used, the upper and lower knees of the characteristic curve
are in use and the tops and bottoms of the output current are "squared off".
Fig. 23 E. Distortion in Class A
amplifiers. Also, the principle of limiters
12. This principle is undesirable in some applications, such as amplifiers, where it is called
distortion, but it is useful in other applications, such as current-limiting devices and
square-wave generators.
13. When the grid of a triode is made quite positive, it diverts considerable numbers of
electrons to itself. Then the plate current falls off, as indicated by the drop in the upper
end of the curve in Fig. 23 F.
Fig. 23 F. Grid current distortion
14. An extremely large grid voltage fluctuation gives rise to the peculiar dip in the top of the
square wave.
Fig. 23 G. A "characteristic" curve which yields
an unusual double-pulse output
15. Figure 23 G shows the output current versus input voltage for an electrical device called a
"thyrite bridge circuit". Note that the output has a double hump, or frequency, for each
half-cycle of the applied voltage.
Fig. 23 H. Class B amplification
16. Figure 23 H shows the characteristic curve of an ordinary triode whose C-bias is so
negative that the operating point is located at the cutoff point, i.e., for Class B
amplification. The output current in this case is rectified. Class C amplifiers are operated
with the C-bias well to the left of the cutoff point.
