# Control Systems - Construction of Bode Plots

In this chapter, let us understand in detail how to construct (draw) Bode plots.

## Rules for Construction of Bode Plots

Follow these rules while constructing a Bode plot.

Represent the open loop transfer function in the standard time constant form.

Substitute, $s=j\omega$ in the above equation.

Find the corner frequencies and arrange them in ascending order.

Consider the starting frequency of the Bode plot as 1/10

^{th}of the minimum corner frequency or 0.1 rad/sec whichever is smaller value and draw the Bode plot upto 10 times maximum corner frequency.Draw the magnitude plots for each term and combine these plots properly.

Draw the phase plots for each term and combine these plots properly.

**Note** − The corner frequency is the frequency at which there is a change in the slope of the magnitude plot.

### Example

Consider the open loop transfer function of a closed loop control system

$$G(s)H(s)=\frac{10s}{(s+2)(s+5)}$$

Let us convert this open loop transfer function into standard time constant form.

$$G(s)H(s)=\frac{10s}{2\left( \frac{s}{2}+1 \right )5 \left( \frac{s}{5}+1 \right )}$$

$$\Rightarrow G(s)H(s)=\frac{s}{\left( 1+\frac{s}{2} \right )\left( 1+\frac{s}{5} \right )}$$

So, we can draw the Bode plot in semi log sheet using the rules mentioned earlier.

## Stability Analysis using Bode Plots

From the Bode plots, we can say whether the control system is stable, marginally stable or unstable based on the values of these parameters.

- Gain cross over frequency and phase cross over frequency
- Gain margin and phase margin

### Phase Cross over Frequency

The frequency at which the phase plot is having the phase of -180^{0} is known as **phase cross over frequency**. It is denoted by $\omega_{pc}$. The unit of phase cross over frequency is **rad/sec**.

### Gain Cross over Frequency

The frequency at which the magnitude plot is having the magnitude of zero dB is known as **gain cross over frequency**. It is denoted by $\omega_{gc}$. The unit of gain cross over frequency is **rad/sec**.

The stability of the control system based on the relation between the phase cross over frequency and the gain cross over frequency is listed below.

If the phase cross over frequency $\omega_{pc}$ is greater than the gain cross over frequency $\omega_{gc}$, then the control system is

**stable**.If the phase cross over frequency $\omega_{pc}$ is equal to the gain cross over frequency $\omega_{gc}$, then the control system is

**marginally stable**.If the phase cross over frequency $\omega_{pc}$ is less than the gain cross over frequency $\omega_{gc}$, then the control system is

**unstable**.

### Gain Margin

Gain margin $GM$ is equal to negative of the magnitude in dB at phase cross over frequency.

$$GM=20\log\left( \frac{1}{M_{pc}}\right )=20logM_{pc}$$

Where, $M_{pc}$ is the magnitude at phase cross over frequency. The unit of gain margin (GM) is **dB**.

### Phase Margin

The formula for phase margin $PM$ is

$$PM=180^0+\phi_{gc}$$

Where, $\phi_{gc}$ is the phase angle at gain cross over frequency. The unit of phase margin is **degrees**.

The stability of the control system based on the relation between gain margin and phase margin is listed below.

If both the gain margin $GM$ and the phase margin $PM$ are positive, then the control system is

**stable**.If both the gain margin $GM$ and the phase margin $PM$ are equal to zero, then the control system is

**marginally stable**.If the gain margin $GM$ and / or the phase margin $PM$ are/is negative, then the control system is

**unstable**.