Bode Plot Basics

Bode form

It is necessary to examine each of these factors above and learn how they contribute to the shape of the system spectrum. We now begin this activity in earnest. The three types of factors are:

• $K_0(j\omega)^n$ where $n$ is integer-valued.

• $(j\omega \tau + 1)^{\pm 1}$

• $\left[\Big(\dfrac{j\omega}{\omega_n}\Big)^2 + 2\zeta \dfrac{j\omega}{\omega_n} + 1\right]^{\pm 1}$

For example consider for some constant $K$ the transfer function,

where $s$ is the complex frequency variable $j\omega$. It can be written as It can then be expanded as:

Type 1: $K_0(j\omega)^n$

Its magnitude is

It is a linear function of $\log \omega$; hence it gives a line of slope $n$ passing through the value $\log |K_0|$ at $\omega = 1$. This line is called a low-frequency asymptote.

In our example above, we had $K_0(j\omega)^{-1}$. Its magnitude plot is shown in the figure below.

For Type 1 factor, its phase is

The phase is therefore a constant independent of $\omega$.

In our example above, we had $K_0(j\omega)^{-1}$ . Its phase plot is shown above. In this example the phase is $-90^\circ$ for all $\omega$.

Type 2: $j\omega\tau + 1$

To study $|j\omega\tau+1|$ and $\angle(j\omega\tau+1)$ as a function of $\omega$, we will look at different cases as below:

• For $\omega \tau \ll 1$, we have $j\omega\tau + 1 \approx 1$.

• For $\omega \tau \gg 1$ we have $j\omega\tau + 1 \approx j\omega\tau$.In the case, Type 2 factor behaves like Type 1 with $K_0 = \tau, n = 1$

• This transition from one extreme to the other can be said to occur at $\omega\tau=1\, \Longleftrightarrow \, \omega = 1/\tau$.This frequency is called the breakpoint frequency.

For the magnitude plot,

• For small $\omega$ below the breakpoint, $M \approx 1$, a horizontal line.

• For large $\omega$ above the breakpoint,

This gives a line of slope $1$ passing through the point $(1/\tau,1)$ on a plot of log-log scale. Note these are just asymptotes; the actual value of $M$ at $\omega = 1/\tau$ is $\sqrt{2}$ The figure below shows the magnitude slope "steps up" by $1$ at the breakpoint.

As for the phase plot,

• For small $\omega$ (below breakpoint), $\phi \approx 0^\circ$.

• For large $\omega$ (above breakpoint),

• At breakpoint ($\omega\tau=1$),

The figure above shows that the phase "steps up" by $90^\circ$ as we go past the breakpoint.

Type 2: $(j\omega\tau+1)^{-1}$

Since this type of factor is the inverse of the one we just discussed, the magnitude and phase plots are reflections of the corresponding plots for the stable real zero with respect to the horizontal axis,

• at breakpoint frequency, its magnitude plot “steps down” by 1 in magnitude slope and

• its phase plot “steps down” by $90^\circ$.

Type 3: $\left(\frac{j\omega}{\omega_n}\right)^2 + 2\zeta \frac{j\omega}{\omega_n} + 1$

Finally we get to the second order term. Here, we start by writing out the complex number in its Cartesian form:

Note that $\omega_n$ in the form of this term is what is called the "natural frequency". As before we will consider different cases: $\omega \ll \omega_n$ and $\omega \gg \omega_n$.

For the magnitude plot we have the following:

• For small frequencies $\omega \ll \omega_n$, we have

  $$\left(1- \left(\frac{\omega}{\omega_n} \right)^2 \right) + 2\zeta\frac{\omega}{\omega_n}j \to 1 \qquad \textrm{as} \qquad \frac{\omega}{\omega_n} \to 0$$
  and so $M \approx 1$. Therefore $\log M = 0$ and we get a horizontal line.

• Similalry, for large frequencies $\omega \gg \omega_n$, we have that the square term dominates and

  $$M \approx \left(\frac{\omega}{\omega_n}\right)^2 \implies \log M \approx 2 \log \omega - 2 \log \omega_n$$
  In this case, the asymptote is a line of slope 2 passing through the point $(\omega, M) = (\omega_n, 1)$.

Therefore, for factor of Type 3 above the magnitude slope “steps up” by 2 as we go through the breakpoint.

For the phase plot we have:

• For $\omega \ll \omega_n$ we are close to positive 1 on the real axis and so $\phi \approx 0$.

• For $\omega = \omega_n$ the real part disappears and a purely imaginary number has $\phi = 90$.

• For $\omega \gg \omega_n$ note that $\operatorname{Re} \to -\omega^2$ and $\operatorname{Im} \to \omega$ and the quantity edges closer to the negative real axis giving us that $\phi \approx 180$.

The magnitude and phase plots are shown below:

Note that the extremal value (peak or trough) does not happen at $\omega = \omega_n$ but rather at a different frequency called the resonant frequency.

Type 3: $\left[\left(\frac{j\omega}{\omega_n}\right)^2 + 2\zeta \frac{j\omega}{\omega_n} + 1 \right]^{-1}$

Similar to the type 2 case, now the phase and magnitude plots are reflected about the $x$ axis as shown in the plot below.

This brings us to the end of our discussion on creating phase/magnitude plots for the Bode primitives. Next lecture we will do a full fledged example but before that we wrap up the current one with a discussion of resonant frequency.

Second order systems

Systems that are of Type 3, i.e. having the form

$$\left[ \left(\dfrac{j\omega}{\omega_n}\right)^2 + 2\zeta \left(\dfrac{j\omega}{\omega_n}\right) + 1 \right]^{-1} \qquad \Leftrightarrow \qquad \dfrac{\omega_n^2}{s^2+2\zeta\omega_ns + \omega_n^2}$$

are called second-order systems since they are associated with second order ODE's of the form

where $x(t)$ is the input to the system. In the above:

• $\omega_n>0$ is called the natural frequency

• $\zeta \ge 0$ is called the damping coefficient

For a system with a $G(j\omega)$ as above, we can derive the magnitude response as:

if we set $z:= \frac{\omega}{\omega_n}$ as a new variable we can examine the new function $M(z)$. As a function of $z$, the function $M(z)$ attains a maximum when $z = \pm \sqrt{1-2 \zeta ^2}$.

This means

since $\omega_n$ cannot be negative. Furthermore, we see that there is an extremum for real valued $\omega$ only for $\zeta \in \left[0, 2^{-1/2}\right]$. This is called the resonant frequency defined above - a frequency at which bad things can happen.

Finally, second order systems are bi-parametric - i.e. while the general shape of the magnitude/phase plots are as discussed above, the exact nature depends on both $\omega_n$ as well as $\zeta$.

This link allows one to change these parameters to visualize how the plots change.

Bode plot examples

Let us consider a few examples of plotting Bode diagrams (i.e. the magnitude/phase plots) for a few system transfer functions.

Example 1

Let us start with the example we considered in the last lecture. Namely,

$$G(s) = \frac{s+3}{s(s^2 + 2s + 4)}$$

Recall that in Bode form it had was written as:

Type 1 factor

The Type 1 factor here is $3/4 \left(j \omega \right) ^{-1}$. It has a negative unity exponent and $K=3/4$. We already showed in the previous lecture what the Type 1 factor for this transfer function will contribute to the Bode plot. These figures are reproduced here for consistency.

Type 2 factor

For the Type 2 factor in this transfer function, that is,

we see that it has a positive unity as exponent and $\tau = 1/3$. Thus the magnitude and phase plot for this term are as follows:

The magnitude plot is horizontal until near the breakpoint and then proceeds upwards with slope 1 while the phase plot climbs from $0$ to $90^\circ$ by hitting $45^\circ$ at the breakpoint.

Type 3 factor

The type 3 factor in this case is

which has negative unity as exponent, $\omega_n = 2$ and $\zeta = \dfrac{1}{2}$. Therefore for the magnitude plot, the graph is horizontal until it reaches near the natural frequency whereupon it descends downwards with slope 2. Similarly the phase plot goes from 0 to $-180 ^\circ$ passing through $-90 ^\circ$ at the natural frequency.

Combining plots

The final plot is a pointwise addition of all the graphs above. For example we know the phase plot will start at $-90 = -90 + 0 + 0$ where the terms on the right of the equality are contributions from the Type 1, Type 2 and Type 3 factors. Similarly, we know it will end at $-180 = -90 + 90 -180$. If the breakpoint frequency $1/\tau$ of the Type 2 term and the natural frequency $\omega_n$ of the Type 3 term where the same, we could have expected the phase plot to cross $-135^\circ = -90^\circ +45^\circ -90^\circ$ at this joint frequency. However, $\omega_n < 1/\tau$ and so the phase plot crosses $-135^\circ$ much earlier.

In the same way we can reason about the magnitude plot. We know it will start out with a slope of negative unity. But once the frequency passes beyond the breakpoint/natural frequency, the Type 2 term will add a slope of positive unity; at the same time the Type 3 term will contribute $-2$ to the slope. So the slope of the graph beyond the breakpoint will be $-1 + 1 - 2 = -2$. Therefore the overall plot is two segments of slope -1 and -2 joined around the the breakpoint/natural frequency with a small kink.

Example 2

Consider the following transfer function:

$$G(s) = \dfrac{2000\left(s + 0.5\right)}{s \left(s+10\right)\left(s+50\right)}$$

This function has one factor of Type 1 and three factors of Type 2.

Type 1 term

This term is $\dfrac{2}{j \omega}$. Here $K=2$ and $n=-1$. This factor contributes a line of slope $-1$ passing through $(\omega, M) = (1, 2)$. Since $n=-1$ its phase plot starts at $-90^\circ$.

Type 2 terms

These terms are:

with $\tau_a = 2, \tau_b = 1/10$ and $\tau_c = 1/50$. We can consider their contributions to the magnitude plots as follows:

• The magnitude plot will start with a slope of -1.

• When $\omega = 0.5$, the first Type 2 factor contributes a slope increase of 1.

• When $\omega = 10$, the second Type 2 factor contributes a slope decrease of 1.

• When $\omega = 50$, the third Type 2 factor again contributes a slope decrease of 1.

We can similarly reason about the phase plots:

• From the Type 1 factor we know the graph will start at $-90^\circ$.

• At $\omega =0.5$, the first Type 2 term adds a phase contribution of $90^\circ$.

• At $\omega=10$, the second Type 2 factor adds a phase of $-90^\circ$.

• At $\omega=50$, the third Type 2 factor adds yet another contribution of $-90^\circ$.

• Together, this means the phase plot ends at $-180^\circ$.

Example 3

Consider the following transfer function which is already in Bode form:

$$G(s) = \dfrac{0.01 \left(s^2 + 0.01s + 1\right)}{s^2 \left(\dfrac{s^2}{64} + 0.02 \dfrac{s}{8} + 1 \right)}.$$

This transfer function has a single Type 1 factor and two Type 3 factors.

For its magnitude plot we have that,

• The Type 1 factor with $K=0.01$ and $n=-2$ contributes a line of slope -2 passing through the point $(\omega, M) = (1, 0.01)$.

• The Type 3 factor with positive exponent has natural frequency $\omega_n=1$ and damping coefficient $\zeta=0.005$. It contributes a slope increase of 2 beyond $\omega =\omega_n$ along with a resonance dip.

• The Type 3 factor with negative exponent has natural frequency $\omega_n=8$ and $\zeta=0.01$. It contributes a slope decrease of 2 beyond $\omega = \omega_n$ along with a resonance peak.

For its phase plot we have that,

• The Type 1 factor has exponent negative 2, therefore the phase plot starts at $-2 \times 90 ^\circ = 180^\circ$.

• At $\omega_n = 1$ the first Type 3 factor contributes a phase increase of $180^\circ$.

• At $\omega_n = 8$ the second Type 3 factor contributes a phase decrease of $180 ^\circ$.

• Since $\zeta$ is very small for, both Type 3 factors have sharp transitions.

Bode plots using software

Note: Some of the following commands/code require the use of the Control System Toolbox in MATLAB; so you might have to use the Engineering Workstations made available by the Grainger College of Engineering.

One can solve for the above examples in MATLAB using the following code.

Example 1 - MATLAB

In this example we make use of the Control System Toolbox in MATLAB while in the next example we will make plots without relying on the toolbox. Recall that the transfer function (2) is:

The below code generates the final Bode plot.

num=[1, 3];         % Numerator coefficients
den=[1, 2, 4, 0];   % Denominator coefficients
sys = tf(num, den); % Transfer function model
bodeplot(sys);      % Generate bodeplot
title("Final Bode Plot - Example 1")
grid on;

The below code also shows the term by term contributions.

typ1 = tf(3,[4, 0]);		%TF from just Type 1 term
typ2 = tf([1/3, 1], 1);		%TF from just Type 2 term
typ3 = tf(1, [1/4, 1/2, 1]); 	%TF from just Type 3 term

bodeplot(typ1);
hold on
bodeplot(typ2);
bodeplot(typ3);
legend('Type 1', 'Type 2', 'Type 3')

Example 2 - MATLAB

In this example (i.e. transfer function of (3)),

we generate the magnitude/phase plots without using any toolboxes.

den = @(w) (w.*j).*(w.*j+10).*(w.*j+50);
num = @(w) 2000.*(w.*j + 0.5);
tf = @(w) num(w)./den(w);


w = logspace(-2,3, 1000);
subplot(1, 2, 1)
loglog(w, abs(tf(w)));
xlabel("Frequency [rad/s]")
ylabel("Magnitude")
title("Magnitude plot")
grid on

subplot(1, 2, 2)
semilogx(w, angle(tf(w))*180/pi);
grid on
xlabel("Frequency [rad/s]")
ylabel("Phase angle")
title("Phase plot")
sgtitle("Magnitude/phase plot for Example 2 above")

Example 3 - MATLAB

Left as an exercise. Plot in MATLAB the transfer function of (4).