Bode Plot of Magnitude

Q: How can I graphically depict ratio of output to input power for a LTI system?

A: Suppose the following:

We could examine $\frac{P_o}{P_i}$

Alternatively, we could examine $log(\frac{P_o}{P_i})$ which has dimensionless units of bels

More commonly, we use $10log(\frac{P_o}{P_i})$ which gives us decibels (dB)

Let's take a closer look at this $10log(\frac{P_o}{P_i})$:

$P_i=\frac{1}{2}Re\left\{V_iI_i^*\right\}$

$P_i=\frac{1}{2}Re\left\{\frac{V_iV_i^*}{R}\right\}$

$P_i=\frac{1}{2}Re\left\{\frac{|V_i|^2}{R}\right\}$

$P_i=\frac{|V_i|^2}{2R}$

Similarly,

$P_o=\frac{|V_o|^2}{2R}$

Hence, $10log\frac{P_o}{P_i}$ =

=$10log\frac{|V_o|^2}{|V_i|^2}$ 

=$20log\frac{|V_o|}{|V_i|}$ 

Recall that $|H(j\omega)|=\frac{|V_o|}{|V_i|}$

So $10log(\frac{P_o}{P_i})=20log|H(j\omega)|$

To visualize the ratio of power with respect to angular frequency, $\omega$ sits on the horizontal axis using a logarithmic scale while $20log|H(j\omega)|$ sits on the vertical axis. From these 2 axes, a Bode plot may be constructed. For a rough and quick idea of the circuits behavior, straight line approximations are commonly used.