Analog filter applications and element scaling

Analog filter

An analog filter is a frequency-selective circuit that works on continuous-time signals, built from resistors, inductors, capacitors and active devices such as op-amps. It passes the wanted band of frequencies and attenuates the others.

Applications of filter networks

• Communication receivers and transmitters: channel selection, IF filtering, harmonic suppression.
• Anti-aliasing filters before ADCs and smoothing (reconstruction) filters after DACs.
• Audio: equalizers, tone controls, loudspeaker crossover networks.
• Power systems: harmonic filters, EMI/RFI line filters, DC power supply ripple filters.
• Biomedical instruments: ECG/EEG filters, 50 Hz notch filters.
• Control and instrumentation: removing sensor noise.

Importance of element scaling equations

Filter tables give element values for $\omega_o$ = 1 rad/s and 1 Ω terminations, such as 1 H or 2 F, which are not practical. The scaling equations

$$R' = k_m R, \qquad L' = \frac{k_m}{k_f}L, \qquad C' = \frac{C}{k_m k_f}$$

($k_f = \omega_{new}/\omega_{old}$, $k_m = R_{new}/R_{old}$) convert them to real values without redesigning. They follow from keeping each impedance $k_m$ times larger at a frequency $k_f$ times higher.

Example 1 (frequency and magnitude scaling): normalized third-order Butterworth LPF: 1 Ω, $L_1$ = 1 H, $C_2$ = 2 F, $L_3$ = 1 H, 1 Ω. Required $\omega_o = 10^4$ rad/s and 1 kΩ terminations ($k_f = 10^4$, $k_m = 10^3$):

$$L_1' = L_3' = \frac{10^3 \times 1}{10^4} = 0.1\ \text{H}, \qquad C_2' = \frac{2}{10^3 \times 10^4} = 0.2\ \mu\text{F}$$

Example 2 (choosing $k_m$ for a practical capacitor): for an active RC section with $R$ = 1 Ω and $C$ = 1 F at 1 rad/s, moving to $\omega_o$ = 10⁴ rad/s with $C$ = 10 nF needs $k_m = 1/(10^4 \times 10^{-8}) = 10^4$, so $R$ = 10 kΩ.

Thus scaling makes the normalized design reusable and lets the designer choose standard, available component values.