Nyquist Plot

A Nyquist plot shows the frequency response of a system as a curve in the complex plane. For each frequency $\omega$, you evaluate $G(j\omega)$ or, in feedback problems, the loop transfer function $L(j\omega)$. The real part becomes the horizontal coordinate, the imaginary part becomes the vertical coordinate, and one point now carries both magnitude and phase.

The fastest way to read it is this: each point is one frequency, the distance from the origin is the magnitude, and the angle from the positive real axis is the phase. That makes a Nyquist plot useful even if you are only trying to understand the shape of the frequency response.

What A Nyquist Plot Tells You

Start with a transfer function in the variable $s$. To get the frequency response, substitute

$$s = j\omega$$

and evaluate the resulting complex expression for different values of $\omega$.

If

$$G(j\omega) = x(\omega) + jy(\omega),$$

then the Nyquist plot is the curve traced by the point

$$(x(\omega), y(\omega))$$

in the complex plane.

This matters because the plot keeps two pieces of information together:

• $|G(j\omega)|$ tells you the magnitude.
• $\arg(G(j\omega))$ tells you the phase.

On one graph, you can see where the response starts, how it turns, and whether it approaches the origin or another important point.

The Intuition Behind The Curve

Think of frequency as moving a pointer through the complex plane. At each frequency, the system produces one complex response. As $\omega$ changes, that response moves, and the full path is the Nyquist plot.

If the system has real coefficients, the negative-frequency branch is the mirror image of the positive-frequency branch across the real axis. That condition matters. You should only use mirror symmetry when the transfer function coefficients are real.

Worked Example: $G(s) = \frac{1}{1+s}$

Take the transfer function

$$G(s) = \frac{1}{1+s}.$$

Substitute $s = j\omega$:

$$G(j\omega) = \frac{1}{1+j\omega}.$$

Now rewrite it in rectangular form:

$$G(j\omega) = \frac{1-j\omega}{1+\omega^2} = \frac{1}{1+\omega^2} - j\frac{\omega}{1+\omega^2}.$$

So the real and imaginary parts are

$$x(\omega) = \frac{1}{1+\omega^2}, \qquad y(\omega) = -\frac{\omega}{1+\omega^2}.$$

Now the shape is easy to read.

When $\omega = 0$,

$$G(j0) = 1,$$

so the plot starts at the point $1$ on the real axis.

As $\omega \to \infty$,

$$G(j\omega) \to 0,$$

so the curve moves toward the origin.

For positive $\omega$, the imaginary part is negative, so the positive-frequency branch lies in the lower half-plane.

You can go one step further and identify the exact curve. These points satisfy

$$x^2 + y^2 = x,$$

which is equivalent to

$$\left(x - \frac{1}{2}\right)^2 + y^2 = \left(\frac{1}{2}\right)^2.$$

So the positive-frequency branch traces the lower half of a circle centered at $\left(\frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. Because this system has real coefficients, the negative-frequency branch mirrors it across the real axis and completes the circle.

This example is the main idea in a clean form: a Nyquist plot is just the geometric path traced by a complex-valued function of frequency.

How To Read A Nyquist Plot Quickly

When you first see a Nyquist plot, ask four questions:

1. Where does the curve start when $\omega = 0$?
2. Where does it go as $\omega$ becomes large?
3. Which half-plane does the positive-frequency branch occupy?
4. Does the curve pass near or around any critical point that matters for the task?

For basic interpretation, the first three questions are usually enough. For closed-loop stability in unity feedback, the critical point is $-1 + 0j$, and the meaning of encirclements depends on the open-loop poles as well as the plotted function. That condition should be stated before using the Nyquist stability criterion.

Common Nyquist Plot Mistakes

Treating It Like An Ordinary $x$-$y$ Plot

The horizontal and vertical coordinates are not two unrelated measured quantities. They are the real and imaginary parts of one complex response.

Ignoring Which Way Frequency Increases

The same curve shape can mean different things if you do not know which way frequency increases along the path.

Assuming Mirror Symmetry Without Checking

For systems with real coefficients, symmetry lets you reconstruct the negative-frequency branch. If that condition does not hold, you should not assume a simple mirror image.

Using Stability Rules Without Stating The Setup

The Nyquist stability criterion is powerful, but it depends on what function is being plotted and on properties of the open-loop system. The encirclement count is only meaningful after that setup is made explicit.

When A Nyquist Plot Is Used

Nyquist plots are most common in control systems, where you want magnitude and phase on one picture instead of splitting them into separate graphs. They are useful for comparing frequency response, judging how feedback may behave, and checking how close a system may be to an important stability boundary.

They also appear in signal and circuit analysis whenever the complex frequency response itself is the main object of interest. Even outside formal stability tests, the plot is a fast way to see how a system moves through the complex plane as frequency changes.

Try A Similar Problem

Try your own version with

$$G(s) = \frac{1}{(1+s)^2}.$$

Compute $G(j\omega)$, separate the real and imaginary parts, and sketch where the plot starts, which half-plane the positive-frequency branch enters, and where it ends. If you want to go one step further, check whether the curve still has a simple geometric shape or not.