Bode Magnitude Plots

Bode Magnitude Plots are convenient since each factor in $s$ can be treated individually with logarithms. For example, if we have a complex transfer function:

$$A(s)=\frac{N_1(s)N_2(s)\dots }{D_1(s)D_2(s)\dots }$$

Then, we have:

$$\begin{array}{rl}|A(s){|}_{\text{dB}}& =20{\log }_{10}(|A|)=20{\log }_{10}\left(\left|\frac{N_1{N}_2\dots }{D_1{D}_2\dots }\right|\right)\\ & =20\log N_1+20\log N_2+\cdots -20\log D_1-20\log D_2\dots \\ & =|N_1{|}_{\text{dB}}+|N_2{|}_{\text{dB}}+\cdots -|D_1{|}_{\text{dB}}-|D_2{|}_{\text{dB}}\end{array}$$

For example, if $A(s)=\frac{s/{\omega }_c}{s/{\omega }_c+1}\times (1+R_f/R_i)$ we would have:

$$|A{|}_{\text{dB}}=\left|\frac{s_{{\omega }_c}}{s_{{\omega }_c}+1}\right|+|A_2{|}_{\text{dB}}$$

We can decompose the high-pass filter example as: