Bode Plots
(1) Consider the RC circuit drawn to the right as a specific illustration; the transfer function is shown
below the circuit.
The polar form of the complex number is of particular interest
here. First we discuss the magnitude, and subsequently the
phase.
A direct description of the magnitude, besides the algebraic
expression, is a plot as a function of frequency. But suppose
that accurate detail is not the prime concern, but rather a means
of quickly ascertaining the general characteristics of the plot.
Even more; it is a means that can be extended to more involved
situations.
A Bode plot provides just this sort of presentation. Sketching
a Bode plot is not difficult; understanding the underlying
concepts is perhaps a bit more involved. First, instead of
dealing directly with the magnitude we deal with the log10 of the magnitude, or more conventionally with
20log10, i.e., in dB measure. Thus
There are two advantages of dealing with the logarithm. One is the compression action of the logarithm; a
very large frequency range on a linear scale is reduced to a much smaller range on a log scale. Plotting a
response over a frequency range from say 100 Hz to 100 kHz involves three equally sized decades intervals
on a log scale, rather more on a linear scale.
Another advantage, to be considered in some detail later, also is a consequence of the fact that the log of a
product of terms is the sum of the log of the individual terms.
To return to the illustrative log expression. Observe that for (ω/ωo)2 << 1 the magnitude (in dB) is
approximately log(1) = 0. Conversely, for (ω/ωo)2 >> 1 the magnitude -> 20log(ω/ωo). The argument of
this latter expression doubles every time the frequency doubles, or equivalently, since doubling means
adding 20log 2 = 6 dB there is an increase of 6 dB/octave. (Equivalently, for a change in frequency by a
factor of 10 the increase is 20log 10 = 20db/decade.
In the figure to the right the two asymptotes are sketched. One asymptote is the axis. The other is a line
rising at 6 dB/octave (45° line) from the point
ω/ωo=1. There is one other representative value
easily recognized. At ω/ωo=1 the exact value of the
magnitude is 3 dB. Given the two asymptotes and
the value at the break point it is easy to draw a fairly
representative curve for the magnitude (in dB) of
the response.
An illustrative PSpice computation is presented
later.
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(2) Now consider a slightly more involved case, as illustrated to
the right. Our objective here is to sketch the variation of the
magnitude of the impedance Z with frequency. It is not difficult
to derive an algebraic expression for the input impedance, and
plotting the functional dependence of the magnitude (in particular)
on frequency can be done, for example, using PSpice. However
the point here is to quickly sketch a fair representation of the
magnitude plot. Expressed in dB |Z| is a sum of three terms:
A Bode plot for the impedance magnitude is drawn below. First the contribution of each term is sketched.
One term is a constant (log is negative since the ratio is less than 1). The asymptote for a second term
starts at ω1 and rises at 6 dB/octave. Similarly the asymptote for the last term starts at ω2, and falls at 6
dB/octave. The complete expression is described by the sum, as shown by the solid line. For ω ≤ ω1 the
sum of the asymptotes is simply equal to the constant term. At ω1 the sum rises at 6db/octave, and
continues rising to ω2 where the rise is compensated by an equal fall. The value at this point can be
calculated after determining either the number of decades or octaves between ω1 and ω2; this is
Then multiply the number of octaves by 6, or the number of decades by 20.
It is not difficult, although usually not particularly necessary, to fit a curve to the asymptotic sketch,
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M H Miller
(3) A Bode plot for the phase shift is done similarly. Note that when a product of several factors is
involved the phases add algebraically;
convert the terms to polar form. Hence
we can deal with phase directly; no log
is involved.
Consider a term such as 1+ (s/ωo), for
which the phase is given by tan1(ω/ωo). The asymptotic approximation
to this expression is drawn to the right;
note the 'key' frequencies ωo/10, ωo, and
10ωo.
The actual curve differs from the
asymptote by just a few degrees (less
than 5.7° at the break points).
(4) As a specific example consider the circuit drawn below. A PSpice plot on which is superimposed a
Bode plot of the voltage transfer function also is drawn below/
Note: 22.7 radians/second -> 3.6 kHz
7.58 radians/second ->1.2 kHz
There are 1.58 octaves between the breakpoints, corresponding to
a 9.5 dB change.
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M H Miller
5) A second PSpice illustrative computation uses the circuit drawn
to the right.
The input impedance magnitude and phase plots follow. Note the
Bode approximations on the input magnitude plot. The pole is (for
C1 = 0.01µ F) at 1/(R2C1) -> 3.39 kHz and the zero is at 1/(R1||R2
C1 -> 19.3 kHz. The dashed line (on the right) is the Bode plot for
C1 = 0.01µ F. The magnitude drops at 6db/octave from the pole
until the frequency corresponds to the zero. (Check the slope by
noting that the magnitude is down 20 dB at 33.9 kHz (one decade higher).
The corresponding phase plot is shown below. Some interpretation is required. Despite the arrow
direction in the diagram the netlist given to PSpice specifies a current flow in the opposite direction. You
can adjust this if you run your own analysis.
You may find it useful to sketch the phase contributions of the zero and pole separately to verify (Bode
approximation) that the pole starts with zero phase shift at 3.39KHz/10, drops 45° /decade to 45° at 3.39
kHz, and continues at the same slope to 33.9 kHz. The 'zero' plot is similar except for the zero frequency
and that the slope is positive. Adding the phase produces a combined plot (shown for C1 = 0.01µ F)
which follows the pole plot until it intercepts the zero plot at 1.93KHz, at which point the rise and fall
cancel each other. The cancellation is maintained until 33.9KHz, after which the zero plot is followed until
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0 phase at 193KHz. The dashed line is the overall phase approximation. Incidentally the number of
decades from 339Hz to 1.93KHz is log(1.93/.339) = 0.76 decade, and the phase change at 45° /decade is
34° . (This is a drop from the artificial phase change due to the netlist description, so the break on the plot
is at 180 - 34 = 146° .
The plots following are for the resistor stepping. The analysis is very much similar to that presented above.
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M H Miller