From Lesson 1, the (open-loop) frequency response of an LTI system with transfer function \( G(s) \) is obtained by the substitution \( s → j\omega \), producing the complex-valued function \( G(j\omega) \).

If \( G(s) = \dfrac{N(s)}{D(s)} \) with real coefficients, then for each real frequency \( \omega \in \mathbb{R} \), the value \( G(j\omega) \) is a complex number. Nyquist and Nichols plots are simply different ways to draw the same information:

• Nyquist plot: plot the locus of \( G(j\omega) \) in the complex plane as \( \omega \) varies (frequency is a parameter).
• Nichols plot: plot \( 20\log_{10}|G(j\omega)| \) versus \( \angle G(j\omega) \) (again frequency is a parameter).

flowchart TD
  A["Start: Transfer function G(s)"] --> B["Choose frequency grid \nw = logspace(...)"]
  B --> C["Compute G(jw) for each w"]
  C --> D["Nyquist: plot \n(Re(G), Im(G))"]
  C --> E["Nichols: plot \n(phase_deg(G), mag_dB(G))"]
  D --> F["Interpret: closeness to (-1,0), \nloop shape, resonance as large radius"]
  E --> G["Interpret: gain/phase trends, \nresonance as magnitude peak, chart overlays"]
        

Throughout, we will use the polar form \( G(j\omega) = r(\omega)e^{j\phi(\omega)} \), where \( r(\omega)=|G(j\omega)| \) and \( \phi(\omega)=\angle G(j\omega) \). The same point in the Nyquist plane has Cartesian coordinates:

\[ \Re\{G(j\omega)\} = r(\omega)\cos(\phi(\omega)),\qquad \Im\{G(j\omega)\} = r(\omega)\sin(\phi(\omega)). \]

Let \( G(s) \) be a proper rational transfer function with real coefficients. The Nyquist locus is the set \( \{G(j\omega): \omega \in \mathbb{R}\} \subset \mathbb{C} \). In practice, we sample \( \omega \) over a wide band and draw the parametric curve.

Writing \( G(j\omega) = \dfrac{N(j\omega)}{D(j\omega)} \), the Nyquist curve is the image of the imaginary axis under the mapping \( s \mapsto G(s) \) restricted to \( s = j\omega \).

2.1 Conjugate Symmetry (Real-Coefficient Systems)

A key property (used in plotting) is that the negative-frequency part is the complex conjugate of the positive-frequency part.

Proposition: If \( G(s) \) has real coefficients, then

\[ G(-j\omega) = \overline{G(j\omega)} \quad \text{for all real } \omega. \]

Proof: For a polynomial with real coefficients, complex conjugation commutes with evaluation:

\[ \overline{N(j\omega)} = N(\overline{j\omega}) = N(-j\omega),\qquad \overline{D(j\omega)} = D(\overline{j\omega}) = D(-j\omega). \]

Therefore, \( \overline{G(j\omega)} = \dfrac{\overline{N(j\omega)}}{\overline{D(j\omega)}} = \dfrac{N(-j\omega)}{D(-j\omega)} = G(-j\omega) \). □

Consequently, it is enough to compute the curve for \( \omega > 0 \) and reflect it across the real axis to obtain \( \omega < 0 \).

2.2 Resonance Interpreted on the Nyquist Curve

For systems with a lightly damped mode (Lesson 3), the magnitude \( |G(j\omega)| \) can become large near the resonant frequency. In the Nyquist plane, “large magnitude” means the curve moves far from the origin.

Example second-order factor (from Lesson 3): \( G_2(s)=\dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \). Its frequency-response magnitude is

\[ |G_2(j\omega)| = \frac{\omega_n^2}{\sqrt{(\omega_n^2-\omega^2)^2+(2\zeta\omega_n\omega)^2}}. \]

For \( \zeta < 1/\sqrt{2} \), the peak occurs at \( \omega_r = \omega_n\sqrt{1-2\zeta^2} \), so around \( \omega \approx \omega_r \) the Nyquist curve typically “bulges” outward.

Consider unity negative feedback with open-loop transfer \( G(s) \). The closed-loop transfer function is \( T(s) = \dfrac{G(s)}{1+G(s)} \), and the closed-loop characteristic equation is:

\[ 1 + G(s) = 0. \]

On the imaginary axis \( s = j\omega \), a marginal condition occurs when \( 1 + G(j\omega) = 0 \), i.e.,

\[ G(j\omega) = -1 + j0. \]

Thus, if the Nyquist curve passes exactly through the point \( -1 \), the closed loop is at the boundary of stability (this connects to the stability concepts covered in Chapter 11).

3.1 Gain and Phase Conditions for Crossing -1

Using polar form \( G(j\omega)=r(\omega)e^{j\phi(\omega)} \), the equality \( G(j\omega)=-1 \) is equivalent to the simultaneous conditions:

\[ r(\omega)=1,\qquad \phi(\omega) = (2k+1)\pi,\; k\in\mathbb{Z}. \]

In Bode terminology, these correspond to “0 dB magnitude” and “-180° phase” at the same frequency. Nyquist and Nichols plots visualize these conditions geometrically.

3.2 (Brief) Encirclement Logic from the Argument Principle

Chapter 11 developed the idea that closed-loop stability is tied to the roots of \( 1+G(s) \). A core theoretical bridge is the argument principle applied to \( 1+G(s) \) along the Nyquist contour (details are expanded in stability-focused lessons).

If \( P \) is the number of right-half-plane poles of \( 1+G(s) \) (equivalently the RHP poles of \( G(s) \)) and \( Z \) is the number of right-half-plane zeros of \( 1+G(s) \) (i.e., RHP closed-loop poles), then the net number \( N \) of clockwise encirclements of \( -1 \) by the Nyquist plot satisfies:

\[ N = Z - P. \]

In the common introductory case where \( P=0 \) (open-loop stable), closed-loop stability requires \( Z=0 \), hence the Nyquist plot must have \( N=0 \) net encirclements of \( -1 \).

The Nichols plot uses the same polar data \( r(\omega), \phi(\omega) \), but places them on axes that are closer to Bode interpretation:

• Horizontal axis: \( \phi(\omega) \) in degrees.
• Vertical axis: \( 20\log_{10}(r(\omega)) \) in dB.

Therefore, the Nichols curve is simply the parametric curve: \( (\phi(\omega), 20\log_{10} r(\omega)) \). Mapping back to Nyquist is immediate via:

\[ r(\omega) = 10^{\frac{M_{dB}(\omega)}{20}},\qquad \Re\{G(j\omega)\} = r(\omega)\cos(\phi(\omega)),\qquad \Im\{G(j\omega)\} = r(\omega)\sin(\phi(\omega)). \]

4.1 Closed-Loop Magnitude as a Function of Open-Loop Polar Data

For unity feedback, \( T(s)=\dfrac{G(s)}{1+G(s)} \). At frequency \( \omega \):

\[ T(j\omega)=\frac{G(j\omega)}{1+G(j\omega)}. \]

Let \( G(j\omega)=re^{j\phi} \). Then \( 1+G(j\omega)=1+re^{j\phi} \) and:

\[ |1+re^{j\phi}|^2 = (1+re^{j\phi})(1+re^{-j\phi}) = 1 + r^2 + 2r\cos\phi. \]

Hence the closed-loop magnitude ratio \( M(\omega)=|T(j\omega)| \) satisfies:

\[ M^2 = \frac{|G|^2}{|1+G|^2} = \frac{r^2}{1+r^2+2r\cos\phi}. \]

This single formula is the mathematical reason Nichols charts are useful: lines of constant \( M \) (constant closed-loop magnitude) can be overlaid on the Nichols plane and compared against the open-loop Nichols curve.

4.2 Deriving the Constant-\( M \) Contour Equation (Introductory Derivation)

Fix a constant \( M \). From \( M^2=\dfrac{r^2}{1+r^2+2r\cos\phi} \), rearrange:

\[ M^2(1+r^2+2r\cos\phi)=r^2 \;\;\Longleftrightarrow\;\; (M^2-1)r^2 + 2M^2(\cos\phi)\,r + M^2 = 0. \]

This is a quadratic equation in \( r \) for each phase \( \phi \). Solving gives:

\[ r = \frac{-M^2\cos\phi \pm M\sqrt{1 - M^2\sin^2\phi}}{M^2 - 1}, \quad \text{provided } 1 - M^2\sin^2\phi \ge 0. \]

After computing \( r \), convert to dB via \( 20\log_{10}r \). While full Nichols charts include many standard overlays, the key point here is that closed-loop magnitude can be derived from open-loop \( (r,\phi) \) data in a purely algebraic way.

flowchart TD
  A["Frequency w"] --> B["Open-loop point G(jw) in complex plane"]
  B --> C["Polar: r=|G|, phi=arg(G)"]
  C --> D["Nichols coordinates: \n(phi_deg, 20log10(r))"]
  B --> E["Unity feedback: \nT = G/(1+G)"]
  C --> F["Closed-loop magnitude: \nM^2 = r^2/(1+r^2+2r cos(phi))"]
        

We implement Nyquist/Nichols plotting around an example plant that includes a lightly damped resonant mode and an additional real pole: \( G(s) = K \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\dfrac{1}{\tau s + 1} \).

5.1 Python (python-control)

File: Chapter12_Lesson4.py

# Chapter12_Lesson4.py
# Nyquist and Nichols Plots (Introductory Level) — Frequency Response and Resonance
#
# Dependencies:
#   pip install control numpy matplotlib
#
# Output files:
#   - Chapter12_Lesson4_nyquist.png
#   - Chapter12_Lesson4_nichols.png
#   - Chapter12_Lesson4_freqresp.csv

import numpy as np
import matplotlib.pyplot as plt

try:
    import control  # python-control
except ImportError as e:
    raise SystemExit("Missing dependency 'control'. Install with: pip install control") from e


def poly_eval_complex(coeffs, s):
    """Evaluate polynomial with real coefficients at complex s (highest power first)."""
    y = 0.0 + 0.0j
    for c in coeffs:
        y = y * s + c
    return y


def tf_freqresp_manual(num, den, w):
    """Manual complex frequency response G(jw) for transfer function num(s)/den(s)."""
    Gjw = np.zeros_like(w, dtype=np.complex128)
    for k, wk in enumerate(w):
        s = 1j * wk
        Gjw[k] = poly_eval_complex(num, s) / poly_eval_complex(den, s)
    return Gjw


# Example open-loop plant:
# G(s) = K * wn^2 / (s^2 + 2*zeta*wn*s + wn^2) * 1/(tau*s + 1)
K = 5.0
wn = 10.0
zeta = 0.20
tau = 0.05

num1 = [wn**2]
den1 = [1.0, 2*zeta*wn, wn**2]
num2 = [1.0]
den2 = [tau, 1.0]

sys = K * control.tf(num1, den1) * control.tf(num2, den2)

w = np.logspace(-1, 2.5, 1500)  # rad/s
Gjw = np.array([control.evalfr(sys, 1j*wk) for wk in w], dtype=np.complex128)

# Cross-check via manual polynomial evaluation
num = np.polymul(np.array(num1), np.array(num2)) * K
den = np.polymul(np.array(den1), np.array(den2))
Gjw_manual = tf_freqresp_manual(num, den, w)
print("Max |Gjw - Gjw_manual| =", np.max(np.abs(Gjw - Gjw_manual)))

mag = np.abs(Gjw)
phase = np.angle(Gjw, deg=True)
mag_db = 20*np.log10(mag)

np.savetxt(
    "Chapter12_Lesson4_freqresp.csv",
    np.column_stack([w, Gjw.real, Gjw.imag, mag_db, phase]),
    delimiter=",",
    header="omega_rad_s,ReG,ImG,Mag_dB,Phase_deg",
    comments=""
)

# Nyquist
plt.figure()
plt.plot(Gjw.real, Gjw.imag, linewidth=1.0, label="G(jw), w>0")
plt.plot(Gjw.real, -Gjw.imag, linewidth=1.0, label="G(jw), w<0 (symmetry)")
plt.scatter([-1.0], [0.0], s=35, marker="x", label="-1 point")
plt.axhline(0.0, linewidth=0.8)
plt.axvline(0.0, linewidth=0.8)
plt.gca().set_aspect("equal", adjustable="box")
plt.xlabel("Re{G(jw)}")
plt.ylabel("Im{G(jw)}")
plt.title("Nyquist Curve")
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.savefig("Chapter12_Lesson4_nyquist.png", dpi=200)

# Nichols
phase_unwrapped = np.unwrap(np.deg2rad(phase))
phase_unwrapped_deg = np.rad2deg(phase_unwrapped)

plt.figure()
plt.plot(phase_unwrapped_deg, mag_db, linewidth=1.0)
plt.xlabel("Phase (deg)")
plt.ylabel("Magnitude (dB)")
plt.title("Nichols Curve")
plt.grid(True)
plt.tight_layout()
plt.savefig("Chapter12_Lesson4_nichols.png", dpi=200)

# Margins for unity feedback
gm, pm, wgc, wpc = control.margin(sys)
print("Gain margin (abs) =", gm)
print("Phase margin (deg) =", pm)
print("Gain crossover w_gc (rad/s) =", wgc)
print("Phase crossover w_pc (rad/s) =", wpc)
      

5.2 C++ (from-scratch frequency response sampling)

File: Chapter12_Lesson4.cpp

/* Chapter12_Lesson4.cpp
   Nyquist and Nichols Plots (Introductory Level) — Frequency Response and Resonance

   Build (example):
     g++ -O2 -std=c++17 Chapter12_Lesson4.cpp -o Chapter12_Lesson4

   Output:
     - Chapter12_Lesson4_freqresp_cpp.csv
*/

#include <complex>
#include <vector>
#include <iostream>
#include <fstream>
#include <cmath>

static std::complex<double> poly_eval(const std::vector<double>& coeffs,
                                     const std::complex<double>& s) {
    std::complex<double> y(0.0, 0.0);
    for (double c : coeffs) {
        y = y * s + c;
    }
    return y;
}

static std::vector<double> logspace(double a, double b, int n) {
    std::vector<double> w;
    w.reserve(n);
    for (int k = 0; k < n; ++k) {
        double t = (n == 1) ? 0.0 : (double)k / (double)(n - 1);
        double p = a + (b - a) * t;
        w.push_back(std::pow(10.0, p));
    }
    return w;
}

int main() {
    const double K = 5.0;
    const double wn = 10.0;
    const double zeta = 0.20;
    const double tau = 0.05;

    // num(s) = K*wn^2
    std::vector<double> num = { K * wn * wn };

    // den(s) = (s^2 + a1*s + a0)(b1*s + b0) expanded
    const double a1 = 2.0 * zeta * wn;
    const double a0 = wn * wn;
    const double b1 = tau;
    const double b0 = 1.0;

    std::vector<double> den = {
        b1,
        (b0 + a1 * b1),
        (a1 * b0 + a0 * b1),
        (a0 * b0)
    };

    const int N = 1500;
    std::vector<double> w = logspace(-1.0, 2.5, N);

    std::ofstream ofs("Chapter12_Lesson4_freqresp_cpp.csv");
    ofs << "omega_rad_s,ReG,ImG,Mag_dB,Phase_deg\n";

    for (double wk : w) {
        std::complex<double> s(0.0, wk); // j*w
        std::complex<double> Gjw = poly_eval(num, s) / poly_eval(den, s);

        double mag = std::abs(Gjw);
        double mag_db = 20.0 * std::log10(mag);
        double ph = std::arg(Gjw) * 180.0 / M_PI;

        ofs << wk << "," << Gjw.real() << "," << Gjw.imag()
            << "," << mag_db << "," << ph << "\n";
    }
    return 0;
}
      

5.3 Java (Apache Commons Math Complex)

File: Chapter12_Lesson4.java

/* Chapter12_Lesson4.java
   Nyquist and Nichols Plots (Introductory Level) — Frequency Response and Resonance

   Dependency (Maven):
     <dependency>
       <groupId>org.apache.commons</groupId>
       <artifactId>commons-math3</artifactId>
       <version>3.6.1</version>
     </dependency>

   Output:
     - Chapter12_Lesson4_freqresp_java.csv
*/

import org.apache.commons.math3.complex.Complex;

import java.io.FileWriter;
import java.io.IOException;
import java.io.PrintWriter;

public class Chapter12_Lesson4 {

    static Complex polyEval(double[] coeffs, Complex s) {
        Complex y = Complex.ZERO;
        for (double c : coeffs) {
            y = y.multiply(s).add(new Complex(c, 0.0));
        }
        return y;
    }

    static double[] logspace(double a, double b, int n) {
        double[] w = new double[n];
        for (int k = 0; k < n; k++) {
            double t = (n == 1) ? 0.0 : (double) k / (double) (n - 1);
            double p = a + (b - a) * t;
            w[k] = Math.pow(10.0, p);
        }
        return w;
    }

    public static void main(String[] args) throws IOException {
        double K = 5.0, wn = 10.0, zeta = 0.20, tau = 0.05;

        double[] num = new double[]{K * wn * wn};

        double a1 = 2.0 * zeta * wn, a0 = wn * wn, b1 = tau, b0 = 1.0;
        double[] den = new double[]{ b1, (b0 + a1*b1), (a1*b0 + a0*b1), (a0*b0) };

        double[] w = logspace(-1.0, 2.5, 1500);

        try (PrintWriter pw = new PrintWriter(new FileWriter("Chapter12_Lesson4_freqresp_java.csv"))) {
            pw.println("omega_rad_s,ReG,ImG,Mag_dB,Phase_deg");
            for (double wk : w) {
                Complex s = new Complex(0.0, wk);
                Complex Gjw = polyEval(num, s).divide(polyEval(den, s));

                double magDb = 20.0 * Math.log10(Gjw.abs());
                double phaseDeg = Math.atan2(Gjw.getImaginary(), Gjw.getReal()) * 180.0 / Math.PI;

                pw.printf(java.util.Locale.US, "%.10f,%.10f,%.10f,%.10f,%.10f%n",
                        wk, Gjw.getReal(), Gjw.getImaginary(), magDb, phaseDeg);
            }
        }
    }
}
      

5.4 MATLAB / Simulink

File: Chapter12_Lesson4.m

% Chapter12_Lesson4.m
% Nyquist and Nichols Plots (Introductory Level) — Frequency Response and Resonance
%
% Requirements:
%   - Control System Toolbox (tf, nyquist, nichols, margin)
% Optional:
%   - Simulink (programmatic model creation section at the bottom)

clear; clc; close all;

K = 5.0;
wn = 10.0;
zeta = 0.20;
tau = 0.05;

s = tf('s');
G = K * (wn^2)/(s^2 + 2*zeta*wn*s + wn^2) * 1/(tau*s + 1);

w = logspace(-1, 2.5, 1500);

figure; nyquist(G, w); grid on; title('Nyquist Curve');
saveas(gcf, 'Chapter12_Lesson4_nyquist_matlab.png');

figure; nichols(G, w); grid on; title('Nichols Curve');
saveas(gcf, 'Chapter12_Lesson4_nichols_matlab.png');

[Gm, Pm, Wcg, Wcp] = margin(G);
disp(['Gain margin (abs) = ', num2str(Gm)]);
disp(['Phase margin (deg) = ', num2str(Pm)]);
disp(['Gain crossover Wcg (rad/s) = ', num2str(Wcg)]);
disp(['Phase crossover Wcp (rad/s) = ', num2str(Wcp)]);

[mag, ph] = bode(G, w);
mag = squeeze(mag); ph = squeeze(ph);
mag_db = 20*log10(mag);

T = table(w(:), mag_db(:), ph(:), 'VariableNames', {'omega_rad_s','Mag_dB','Phase_deg'});
writetable(T, 'Chapter12_Lesson4_freqresp_matlab.csv');

% Optional: programmatic Simulink model section is included (commented) in the downloadable .m file.
      

5.5 Wolfram Mathematica

File: Chapter12_Lesson4.nb

(* Chapter12_Lesson4.nb
   Nyquist and Nichols Plots (Introductory Level) — Frequency Response and Resonance
*)

Needs["ControlSystems`"]

K = 5.0; wn = 10.0; zeta = 0.20; tau = 0.05;
s = LaplaceTransform`Private`s;

G = K * (wn^2)/(s^2 + 2*zeta*wn*s + wn^2) * 1/(tau*s + 1);
sys = TransferFunctionModel[G, s];

wmin = 10^-1; wmax = 10^2.5;

NyquistPlot[sys, {wmin, wmax}, PlotLabel -> "Nyquist Curve", GridLines -> Automatic]
NicholsPlot[sys, {wmin, wmax}, PlotLabel -> "Nichols Curve", GridLines -> Automatic]

Margin[sys]
      

Problem 1 (Nyquist symmetry): Let \( G(s)=\dfrac{N(s)}{D(s)} \) with real-coefficient polynomials \( N, D \). Prove that \( G(-j\omega)=\overline{G(j\omega)} \).

Solution: Use conjugation commutation with polynomial evaluation: \( \overline{N(j\omega)}=N(-j\omega) \) and \( \overline{D(j\omega)}=D(-j\omega) \). Then:

\[ \overline{G(j\omega)}=\frac{\overline{N(j\omega)}}{\overline{D(j\omega)}}= \frac{N(-j\omega)}{D(-j\omega)}=G(-j\omega). \]

Problem 2 (Cartesian from Nichols point): Suppose a Nichols coordinate at some frequency is magnitude \( M_{dB}=-6 \) dB and phase \( \phi=-135^\circ \). Compute the corresponding Nyquist point \( \Re\{G\},\Im\{G\} \).

Solution: Convert dB to magnitude:

\[ r = 10^{\frac{M_{dB}}{20}} = 10^{\frac{-6}{20}}. \]

Then use \( \Re\{G\}=r\cos\phi \), \( \Im\{G\}=r\sin\phi \):

\[ \Re\{G\}=10^{\frac{-6}{20}}\cos(-135^\circ),\qquad \Im\{G\}=10^{\frac{-6}{20}}\sin(-135^\circ). \]

Problem 3 (Condition to hit -1): Show that the Nyquist curve passes through \( -1 \) at some frequency if and only if there exists \( \omega_0 \) such that \( |G(j\omega_0)|=1 \) and \( \angle G(j\omega_0) = (2k+1)\pi \).

Solution: If \( G(j\omega_0)=-1 \), then in polar form \( -1 = 1\cdot e^{j(2k+1)\pi} \), hence \( r(\omega_0)=1 \) and \( \phi(\omega_0)=(2k+1)\pi \). Conversely, if those hold then \( G(j\omega_0)=1\cdot e^{j(2k+1)\pi}=-1 \). □

Problem 4 (Derive constant-\( M \) relation): For unity feedback \( T=\dfrac{G}{1+G} \), let \( G=re^{j\phi} \). Derive \( M^2=\dfrac{r^2}{1+r^2+2r\cos\phi} \).

Solution: Start from \( T=\dfrac{re^{j\phi}}{1+re^{j\phi}} \):

\[ M^2 = |T|^2 = \frac{|re^{j\phi}|^2}{|1+re^{j\phi}|^2}=\frac{r^2}{|1+re^{j\phi}|^2}. \]

Compute the denominator magnitude squared by multiplying by the conjugate:

\[ |1+re^{j\phi}|^2 = (1+re^{j\phi})(1+re^{-j\phi}) = 1+r^2+2r\cos\phi. \]

Substitute into the expression for \( M^2 \) to finish. □

Problem 5 (Resonance and Nyquist radius): Consider the second-order factor \( G_2(s)=\dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \). Show that if \( \zeta \) is small, then near \( \omega\approx\omega_n \) the Nyquist curve is far from the origin.

Solution: Evaluate the magnitude (Section 2.2):

\[ |G_2(j\omega)| = \frac{\omega_n^2}{\sqrt{(\omega_n^2-\omega^2)^2+(2\zeta\omega_n\omega)^2}}. \]

At \( \omega=\omega_n \), this reduces to \( |G_2(j\omega_n)|=\dfrac{\omega_n^2}{2\zeta\omega_n^2}=\dfrac{1}{2\zeta} \), which is large when \( \zeta \) is small. Since Nyquist radius equals \( r(\omega)=|G(j\omega)| \), the curve is far from the origin near resonance. □

Nyquist and Nichols plots are alternative parameterizations of the same frequency response \( G(j\omega) \). Nyquist visualizes complex-plane geometry (including the special role of the point \( -1 \) for unity feedback), while Nichols visualizes magnitude (dB) versus phase and provides an algebraic bridge to closed-loop magnitude through \( M^2=\dfrac{r^2}{1+r^2+2r\cos\phi} \). Resonance manifests as large \( |G(j\omega)| \), appearing as outward bulging in Nyquist and as peaks in Nichols magnitude.