**Bode Plot Example of First Order System using Matlab**

In this article, **Bode Plot** of Simple Phase-Lag Network (First Order System) is obtained using Matlab. In order to draw Bode Plot, we need transfer function from which we deduce the equations for **Magnitude** and **Phase**.

\[G(s)=\frac{1}{2s+1}\]

Function in the frequency domain can be written as:

\[G(s)=\frac{1}{2j\omega +1}\]

From above expression, we can deduce the corner frequency or break point as;

\[\omega =\frac{1}{2}\]

**For Magnitude Plot:**

When ω<<1 (very very small value), then

$G(s)\approx 1$

So, for very small value of ω, log magnitude of the transfer function would be;

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

Hence, magnitude response would be constant below break point.

When ω>>1 (very very large value), then

\[G(s)\approx \frac{1}{2j\omega }\]

So, for very large value of ω, log magnitude of the transfer function would be;

\[|G(j\omega ){{|}_{dB}}=20\text{ }log|G(j\omega )|=20\text{ }log\left| \frac{1}{|2j\omega |} \right|=20\text{ }log\left( \frac{1}{2\omega } \right)=20log\left| 1 \right|-20log\left| 2\omega \right|=-20log(2\omega )\]

So, above the break point, the magnitude plot would be a straight line with -20 dB/decade slope

Now, phase of the transfer function G(s) can be calculated as;

\[\angle G(j\omega )=0-{{\tan }^{-1}}(\omega T)={{\tan }^{-1}}(\omega T)\]

**For Phase Plot:**

When ω is very very small (ω≈0), then

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

When ω is very very large (ω→∞), then

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

**Bode Plot Example Matlab Code**

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

% Bode Plot for Phase-Lag Network Example clc % Transfer function K = [1]; T = 2; num = [K]; den = [T 1]; H = tf(num, den) % Bode Plot grid on bode(H) grid %For Asymptotic Plot % num=[1]; % den=[2 1]; % bode_asymptotic(num,den);

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