**Bode Plot Definition**

H.W. Bode introduced a method to present the information of a polar plot of a transfer function GH(s), actually the frequency response GH (jω), as two plots with the angular frequency were at the common axis. The first plot shows the **magnitude** of the transfer function as a function of ω, and the second plot shows the **phase** as a function of ω. This pair of plots is referred to as **Bode Plot or Bode Diagram**.

The magnitude of the transfer function is expressed in **decibels (dB)**, the phase in degrees and the common parameter of **frequency** is plotted on a logarithmic scale in radians. At times, the magnitude of a transfer function is referred to as **gain** and the corresponding plot as a **gain plot**.

**Bode Plot Advantages**

- One apparent advantage of the bode diagram is the relative ease with which it is obtained.
- A second advantage is that this technique is feasible for lower frequencies, where measuring the phase difference between input and output signals is difficult.
- A third advantage results from the introduction of logarithms, thus reducing the process of multiplying two transfer functions to addition.

**Corner frequency or Breakpoint**

In bode-plot, low-frequency asymptote (that is ω<<1/T) and high-frequency asymptote (that is ω>>1/T) cut off at 0 decibels (dB) line where ω=1/T, that is the frequency called corner frequency or break point.

## Bode Plot Matlab

The Bode plot or diagram of a transfer function can be constructed by combining the transfer functions of following elementary factors.

- Gain factor K
- Integrator or Differentiator
- Simple lag or simple lead
- Quadratic lag or quadratic lead

We will discuss above elementary factors one by one:

**Gain factor K**

A constant K may be considered as complex number expressed in polar form with magnitude K and angle 0** ^{.}** if K is positive or -180

**if K is negative. For gain factor K, the bode-plot is obtained as:**

^{.}If the open loop gain is

*$G\left( j\omega \right)H\left( j\omega \right)=K$*

Then it’s **Magnitude (dB)** is

\[{{\left| G\left( j\omega \right)H\left( j\omega \right) \right|}_{dB}}=20log\left| K \right|=\text{Constant}\]

And its **Phase** is

\[\angle G(j\omega )H(j\omega )=\left\{ \begin{matrix} {{0}^{\centerdot }} & K>0 \\-{{180}^{\centerdot }} & K<0 \\\end{matrix} \right.\]

The value in decibels is positive when the magnitude of the constant is greater than 1 and negative when less than 1.A number equal to 1 has a value of 0 in decibels.

The log-magnitude plot for a gain factor K is a straight horizontal line with the magnitude of 20log*K* decibels. By changing the value of gain *K* in the transfer function brings up or brings down the log-magnitude curve by the proportionate amount. The gain factor *K* has no effect on the phase curve of the bode plot.

**Matlab Plot for Constant Gain Factor K**

Here, we implemented the bode plot of gain factor K for the comprehensive understanding of the readers.

% Bode Plot for Constant Gain Factors K=4,10,12 clear all;close all;clc num1 = 4; den = [1]; sys1 = tf(num1,den); grid; bodeplot(sys1) hold on num2 = 10; den = [1]; sys2 = tf(num2,den); grid; bodeplot(sys2) hold on num3 = 20; den = [1]; sys3 = tf(num3,den); grid; bodeplot(sys3) grid on hold off legend('K=4','K=10','K=12','Orientation','horizontal');

**Plot for Gain Factor**

Fig.1: Plot for Gain Factor K

**Integrator and Differentiator**

Pure integrator and differentiator are represented by transfer function 1/s and s respectively. Bode plots are obtained from the corresponding frequency response function 1/jω and jω.

For differentiator, the bode-plot is obtained as:

If the open loop gain is

*$G\left( j\omega \right)H\left( j\omega \right)=j\omega $*

Then it’s **Magnitude (dB)** is

\[{{\left| G\left( j\omega \right)H\left( j\omega \right) \right|}_{dB}}=20log\left| j\omega \right|=20log(\omega )\]

And it’s **Phase** is

\[\angle G(j\omega )H(j\omega )=\angle j\omega ={{90}^{\centerdot }}\]

When we draw bode-plot for differentiator, we can observe that magnitude plot is a straight line with a slope of +20 dB/decade. Whereas, the phase plot is a straight line with angle 90^{o}.

**Matlab Plot for Differentiator**

Here, we implemented the Bode plot of a differentiator for the comprehensive understanding of the readers.

% Bode Plot for Differentiator clear all;close all;clc % Transfer function K = [1 0]; T = 1; num = [K]; den = [T]; H = tf(num, den) % Bode Plot grid on bode(H);

**Plot for Differentiator**

Fig.2: Plot for Differentiator

Similarly, for **Integrator**, the bode-plot is obtained as:

If the open loop gain is

*$G\left( j\omega \right)H\left( j\omega \right)={}^{1}/{}_{j\omega }$*

Then it’s **Magnitude (dB)** is

${{\left| G\left( j\omega \right)H\left( j\omega \right) \right|}_{dB}}=20log\left| \frac{1}{j\omega } \right|=20log(1)-20log(\omega )=-20log(\omega )$

And its **Phase** is

\[\angle G(j\omega )H(j\omega )=\angle \frac{1}{j\omega }=-{{90}^{\centerdot }}\]

When we draw bode-plot for the integrator, we can observe that magnitude plot is a straight line with a slope of -20 dB/decade. Whereas, the phase plot is a straight line with angle -90^{o}.

**Matlab Plot for Integrator**

Here, we implemented the bode plot of Integrator for the comprehensive understanding of the readers.

% Bode Plot for Integrator clear all;close all;clc % Transfer function K = [1]; T = 1; num = [K]; den = [T 0]; H = tf(num, den) % Bode Plot grid on bode(H)

**Plot for Integrator**

Fig.3: Plot for Integrator

**Simple lag or Simple Lead**

These bode diagrams arise from phase-lag or phase-lead networks whose respective transfer functions are 1/Ts+1 and Ts+1.

For a simple phase-lag network, the bode-plot is obtained as:

If the open loop gain is

\[G\left( j\omega \right)H\left( j\omega \right)=\frac{1}{1+j\omega T}\]

Where T is a real constant.

Then it’s **Magnitude (dB)** is

\[{{\left| G\left( j\omega \right)H\left( j\omega \right) \right|}_{dB}}=20log\left| \frac{1}{1+j\omega T} \right|=20log\left| 1 \right|-20log\left| 1+j\omega T \right|=-20log(\sqrt{{{1}^{2}}+{{\omega }^{2}}{{T}^{2}}})\]

For **Magnitude Plot**

For low frequency asymptote (when s→0)

When ω<<1/T (very very small, like approaches to zero), then magnitude would be:

\[|G(j\omega )H(j\omega ){{|}_{dB}}=20\text{ }log(1)=0dB\]

Which means that magnitude plot would be a straight line at 0 dB at low frequency (ω<<1/T).

Similarly, for high-frequency asymptote (when s→∞)

When ω>>1/T (very very big, like approaches to infinity), then magnitude would be:

\[|G(j\omega )H(j\omega ){{|}_{dB}}=-20\text{ }log(\omega T)\]

Which means that magnitude plot would be a straight line with -20 dB/decade slope at high frequency (ω>>1/T).

And its **Phase** is

\[\angle G(j\omega )H(j\omega )=\angle \frac{1}{1+j\omega T}=-{{\tan }^{-1}}(\omega T)\]

For **Phase** plot

When ω<<0.1/T, then

\[\angle G(j\omega )H(j\omega )={{0}^{\centerdot }}\]

Which means that phase plot would be a straight line at 0** ^{. }**until

**ω=0.1/T.**

^{ }When ω=0.1/T, then

\[\angle G(j\omega )H(j\omega )=-{{45}^{\centerdot }}\]

Which means that phase plot would be a straight line with -45** ^{.}** /decade slope until

**ω=10/T.**

^{ }When ω>>10/T, then

\[\angle G(j\omega )H(j\omega )=-{{90}^{\centerdot }}\]

Which means that phase plot would be a straight line at -90^{o}.

**Matlab Plot for Simple Phase Lag Network**

Here, we implemented the Bode plot of a Simple Phase Lag network for the comprehensive understanding of the readers.

% Bode Plot for Phase-Lag Network clear all;close all;clc % Transfer function K = [1]; T = 1; num = [K]; den = [T 1]; H = tf(num, den) % Bode Plot grid on bode(H) grid

**Plot for Simple Phase Lag Network**

Fig.4: Plot for Simple Phase-Lag Network

Similarly, for a simple phase-lead network, the bode-plot is obtained as:

If the open loop gain is

$G\left( j\omega \right)H\left( j\omega \right)=1+j\omega T$

Where T is a real constant.* *

Then it’s **Magnitude (dB)** is

\[{{\left| G\left( j\omega \right)H\left( j\omega \right) \right|}_{dB}}=20log\left| 1+j\omega T \right|=20log(\sqrt{{{1}^{2}}+{{\omega }^{2}}{{T}^{2}}})\]

For **magnitude plot**

For low frequency asymptote (when s→0)

When ω<<1/T (very very small, like approaches to zero), then magnitude would be:

\[|G(j\omega )H(j\omega ){{|}_{dB}}=20\text{ }log(1)=0dB\]

Which means that magnitude plot would be a straight line at 0 dB at low frequency (ω<<1/T).

Similarly, for high-frequency asymptote (when s→∞)

When ω>>1/T (very very big, like approaches to infinity), then magnitude would be:

\[|G(j\omega )H(j\omega ){{|}_{dB}}=20\text{ }log(\omega T)\]

Which means that magnitude plot would be a straight line at with +20 dB/decade slope at high frequency (ω>>1/T).

And its **Phase** is

\[\angle G(j\omega )H(j\omega )=\angle 1+j\omega T={{\tan }^{-1}}(\omega T)\]

For **Phase plot**

When ω<<0.1/T, then

\[\angle G(j\omega )H(j\omega )={{0}^{\centerdot }}\]

Which means that phase plot would be a straight line at 0** ^{. }**until

**ω=0.1/T.**

^{ }When ω=0.1/T, then

\[\angle G(j\omega )H(j\omega )=-{{45}^{\centerdot }}\]

Which means that phase plot would be a straight line with +45** ^{.}** /decade slope until

**ω=10/T.**

^{ }When ω>>10/T, then

\[\angle G(j\omega )H(j\omega )={{90}^{\centerdot }}\]

Which means that phase plot would be a straight line at 90^{o}.

**Matlab Plot for simple Phase Lead Network**

Here, we implemented the bode plot of a Simple Phase Lead network for the comprehensive understanding of the readers.

% Bode Plot for Phase-Lead Network clc % Transfer function K = [1]; T = 1; num = [K 1]; den = [T]; H = tf(num, den) % Bode Plot grid on bode(H) grid

**Plot for Simple Phase Lead Network**

Fig.5: Plot for Simple Phase-Lead Network

**Quadratic Factor (Second Order)**

The transfer function for typical quadratic function can be written as

\[G\left( j\omega \right)H\left( j\omega \right)=\frac{\omega _{n}^{2}}{{{s}^{2}}+2\zeta {{\omega }_{n}}s+\omega _{n}^{2}}\]

The **Magnitude Plot** is obtained from

\[|G(j\omega )H(j\omega ){{|}_{dB}}=20\log \left| \frac{\omega _{n}^{2}}{{{\left( j\omega \right)}^{2}}+2\zeta {{\omega }_{n}}\left( j\omega \right)+\omega _{n}^{2}} \right|\]

\[|G(j\omega )H(j\omega ){{|}_{dB}}=20\log \left| \frac{1}{{{\left( {}^{j\omega }/{}_{{{\omega }_{n}}} \right)}^{2}}+2\zeta {{\omega }_{n}}\left( {}^{j\omega }/{}_{{{\omega }_{n}}} \right)+1} \right|\]

The above equation can be simplified as

\[|G(j\omega )H(j\omega ){{|}_{dB}}=-20~log{{\left[ {{\left( 1-{{\left( {}^{\omega }/{}_{{{\omega }_{n}}} \right)}^{2}} \right)}^{2}}+{{\left( {}^{2\zeta \omega }/{}_{{{\omega }_{n}}} \right)}^{2}} \right]}^{{}^{1}/{}_{2}}}\]

And the **Phase Plot** is obtained from

\[\angle G(j\omega )H(j\omega )=-ta{{n}^{-1}}\left( \frac{{}^{2\zeta \omega }/{}_{{{\omega }_{n}}}}{1-{{\left( {}^{\omega }/{}_{{{\omega }_{n}}} \right)}^{2}}} \right)\]

In above expression, the damping ratio ζ is a parameter and the frequency is normalized as ω/ ω_{n }.

The bode-plot shown in the following figure was obtained from Matlab Software. Asymptotic approximations were drawn by two lines: one for a value of ω/ ω_{n }<<1 and one for ω/ ω_{n }>>1._{ }

At low frequencies, that is, for values for ω such that ω<< ω_{n}, the magnitude in decibels can be approximated by

\[|G(j\omega )H(j\omega ){{|}_{dB}}=-20\log {{1}^{{}^{1}/{}_{2}}}=0\]

Similarly, for higher frequencies that value of magnitude is approximated by

\[|G(j\omega )H(j\omega ){{|}_{dB}}=-20\log {{\left[ {{\left( -{{\left( {}^{\omega }/{}_{{{\omega }_{n}}} \right)}^{2}} \right)}^{2}} \right]}^{{}^{1}/{}_{2}}}=-40\log {}^{\omega }/{}_{{{\omega }_{n}}}\]

Which is a line with slope of -40 dB/decade on semi-log paper. These asymptotes meet when ω= ω_{n }=1; this point is called a break frequency or break point.

Similarly, for phase plot

When ω<< ω_{n }, then

$\angle G(j\omega )H(j\omega )={{0}^{\centerdot }}$

Which means that phase plot would be a straight line at 0^{o}.

When ω>> ω_{n }, then

$\angle G(j\omega )H(j\omega )=-{{180}^{\centerdot }}$

Which means that phase plot would be a straight line at 180^{o}.

When ω= ω_{n }, then

$\angle G(j\omega )H(j\omega )=-{{90}^{\centerdot }}$

Which means that phase plot would be a straight line with -90** ^{.}** /decade slope.

**Matlab Plot for Second Order System**

Here, we implemented the bode plot of a second-order network for the comprehensive understanding of the readers.

%Bode-Plot for Second-Order System clear all;close all;clc fn = 1; % Define natural frequency (Hz) wn = 2*pi*fn; % Natural frequency conversion in rad/s Zeta = 0.05; % Damping Factor Num = [0 0 wn^2]; % Numerator Den = [1 2*Zeta*wn wn^2]; % Denominator Gs= tf(Num,Den) % Transfer Function bode(Gs); %Bode-Plot grid on

**Plot for Second Order System**

Fig.6: Plot for Second Order System

**Special Terms**

There are certain terms, which we need to familiar with to fully understand the bode plot.

**Phase Cross Over Frequency**

It is the frequency, where phase shift becomes -180^{o}.

**Gain Cross Over Frequency **

It is the frequency at which amplitude ratio becomes 1 or log modulus of transfer function becomes 0.

**Bode Plot and Stability **

The characteristic equation of a typical system can be written as,

$1+GH\left( s \right)=0$

Where

$GH\left( s \right)=-1$

And in frequency domain,

$GH\left( j\omega \right)=-1$

How far -1 is from open loop transfer function GH (jω) measures the stability of a system. This distance can be measured in terms called phase margin and gain margin.

Phase and gain margin are usually measured from open loop response and cannot be obtained from the frequency response of a closed loop system directly.

**Gain Margin**

It can be described as an increase in the open-loop system gain |GH (jω)| when system phase is at 180** ^{.}** Which will cause marginal stability of a system.

In order to measure the gain margin,

- Find the point, where system phase response crosses -180
^{.} - At some point obtain amplitude response
- The distance below 0dB at this point shows the system gain margin.

**Phase margin**

In order to measure the phase margin,

- Find the point, where system’s open loop amplitude crosses 0 dB
- At same point, obtain the phase response
- The distance above -180
at this point shows the system phase margin^{. }

For more comprehensive understanding, see the solved Example for Bode Plot using Matlab.

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