L6

10/08/26 Mon 15:37pm

Filters

there’s 5

1. low pass
2. high pass
3. band pass
4. band reject / notch filter
5. All pass reject (for the purpose of phase shifting)

Ignore this ⬆ ( if you want to see the pdf go to my notes’s web page)

Deriving the Low Pass Filter

$$V_A=\frac{V_i\times \frac{1}{SC}}{R+\frac{1}{SC}}$$ $$V_A=\frac{V_i}{SRC+1}$$ $$V_0=V_A\times \left(1+\frac{R_2}{R_1}\right)$$ $$V_0=V_i\left(1+\frac{R_2}{R_1}\right)\left(\frac{1}{SCR+1}\right)$$ $$V_0=V_i\left(1+\frac{R_2}{R_1}\right)\left(\frac{1}{1+j\omega CR}\right)$$ $$\frac{V_0}{V_i}=\left(1+\frac{R_2}{R_1}\right)\left(\frac{1}{1+j\omega RC}\right)$$ $$|G|=\left(1+\frac{R_2}{R_1}\right)\frac{1}{\sqrt{1+{\omega }^2{R}^2{C}^2}}$$ $$|G{|}_{max}^{\omega =0}=1+\frac{R_2}{R_1}$$ $$|G{|}_{min}^{\omega =\infty }=0$$ $$|G{|}^{\omega ={\omega }_c}=\frac{1}{\sqrt{2}}\left(1+\frac{R_2}{R_1}\right)$$

Steps in Designing a Filter

1. Assume C value = 0.01 $\text{micro}F$
2. Find the value of R using $f_c=\frac{1}{2\pi RC}$
3. From the Maximum gain formula i.e. $1+\frac{R_2}{R_1}$ by assuming any one of them.

$$V_a=\frac{V_{i}R}{\frac{1}{SC}+R}$$ $$V_a=\frac{V_{i}R}{1+RCS}$$ $$V_0=V_i\frac{R}{R+\frac{1}{SC}}=V_{i}R\frac{CS}{1+RCS}$$ $$\frac{V_0}{V_i}=|G|=\left(1+\frac{R_2}{R_1}\right)\left(\frac{\omega RC}{\sqrt{1+R^2{C}^2{\omega }^2}}\right)$$