Normalization and denormalization with example

Normalization means designing the filter with convenient reference values, usually a cut-off (or half-power) frequency of $\omega_o = 1$ rad/s and a termination of $R = 1\ \Omega$. Denormalization (scaling) converts the normalized element values to the actual frequency and impedance level required.

Importance:

• Filter tables (Butterworth, Chebyshev, Bessel poles and ladder element values) are published only in normalized form; one table serves every frequency and impedance level.
• Calculations use simple numbers like 1, 1.414, 2 instead of values like $10^{-9}$ and $10^{4}$, so errors are fewer.
• Designs can be compared and reused: a 1 rad/s prototype becomes a 1 kHz or 1 MHz filter by simple scaling.
• Scaling lets the designer choose practical element values (e.g. capacitors in nF–μF, resistors in kΩ) without changing the shape of the response.

Scaling relations used for denormalization

With frequency scaling factor $k_f = \omega_{actual}/\omega_{normalized}$ and magnitude scaling factor $k_m = R_{actual}/R_{normalized}$:

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

Example

A normalized second-order Butterworth low-pass filter ($\omega_o$ = 1 rad/s, 1 Ω terminations) has $L = 1.414$ H and $C = 1.414$ F (doubly terminated ladder). Required: $f_o$ = 1 kHz and 600 Ω terminations.

$$\begin{aligned} k_f &= 2\pi\times1000 = 6283.2, \quad k_m = 600 \\ L' &= \frac{600 \times 1.414}{6283.2} = 0.135\ \text{H} \\ C' &= \frac{1.414}{600 \times 6283.2} = 0.375\ \mu\text{F} \end{aligned}$$

 600R    0.135 H
 o-/\/\/--UUUU--+------+
 Vs             |      |
             0.375uF  600R  Vo
                |      |
 o--------------+------+

The design was done once with simple numbers (1.414) and then adapted to the real requirement only by scaling.