Simple Magnitude and Frequency Scaling in Filter Circuits

magnitude and frequency scaling

In designing and analyzing filters and resonant circuits or in circuit analysis in general, it is sometimes convenient to work with element values of 1 Ω, 1 H, or 1 F, and then transform the values to realistic values by scaling. Now we will make use of Magnitude and Frequency Scaling.

We have taken advantage of this idea by not using realistic element values in most of our examples and problems; mastering circuit analysis is made easy by using convenient component values. We have thus eased calculations, knowing that we could use scaling to then make the values realistic.

There are two ways of scaling a circuit: magnitude or impedance scaling, and frequency scaling. Both are useful in scaling responses and circuit elements to values within the practical ranges. While magnitude scaling leaves the frequency response of a circuit unaltered, frequency scaling shifts the frequency response up or down the frequency spectrum.

Magnitude Scaling

Magnitude scaling is the process of increasing all impedance in a network by a factor, the frequency response remaining unchanged.

Recall that impedances of individual elements R, L, and C are given by

magnitude and frequency scaling
(1)

In magnitude scaling, we multiply the impedance of each circuit element by a factor Km and let the frequency remain constant. This gives the new impedances as

magnitude and frequency scaling
(2)

Comparing Equation.(2) with Equation.(1), we notice the following changes in the element values: R KmR, L KmL, and C C/Km. Thus, in magnitude scaling, the new values of the elements and frequency are

magnitude and frequency scaling
(3)

The primed variables are the new values and the unprimed variables are the old values. Consider the series or parallel RLC circuit. We now have

magnitude and frequency scaling
(4)

showing that the resonant frequency, as expected, has not changed. Similarly, the quality factor and the bandwidth are not affected by magnitude scaling.

Also, magnitude scaling does not affect transfer functions in the previous post in ‘Learn What is Transfer Function Quickly’, which are dimensionless quantities.

Frequency Scaling

Frequency scaling is the process of shifting the frequency response of a network up or down the frequency axis while leaving the impedance the same.

We achieve frequency scaling by multiplying the frequency by a factor
Kf while keeping the impedance the same.

From Equation.(1), we see that the impedances of L and C are frequency-dependent. If we apply frequency scaling to ZL(ω) and ZC(ω)
in Equation.(1), we obtain

magnitude and frequency scaling
(5a)
magnitude and frequency scaling
(5b)

since the impedance of the inductor and capacitor must remain the same
after frequency scaling. We notice the following changes in the element
values:
L L/Kf and C C/Kf . The value of R is not affected, since its impedance does not depend on frequency.

Thus, in frequency scaling, the new values of the elements and frequency are

magnitude and frequency scaling
(6)

Again, if we consider the series or parallel RLC circuit, for the resonant
frequency

magnitude and frequency scaling
(7)

and for the bandwidth

magnitude and frequency scaling
(8)

but the quality factor remains the same (Q’ = Q).

Magnitude and Frequency Scaling

If a circuit is scaled in magnitude and frequency at the same time, then

magnitude and frequency scaling
(9)

These are more general formulas than those in Equations.(3) and (6). We set Km = 1 in Equation.(9) when there is no magnitude scaling or Kf = 1 when there is no frequency scaling.

Magnitude and Frequency Scaling Example

A fourth-order Butterworth lowpass filter is shown in Figure.(1a). The filter is designed such that the cutoff frequency ωc = 1 rad/s. Scale the circuit for a cutoff frequency of 50 kHz using 10-kΩ resistors.

magnitude and frequency scaling
Figure 1. (a) Normalized Butterworth lowpass filter, (b) scaled version of the same lowpass filter.

Solution:
If the cutoff frequency is to shift from ωc = 1 rad/s to ωc = 2π(50) krad/s, then the frequency scale factor is

magnitude and frequency scalingAlso, if each 1-Ω resistor is to be replaced by a 10-kΩ resistor, then the
magnitude scale factor must be

magnitude and frequency scaling

Using Equation.(9)

magnitude and frequency scalingThe scaled circuit is as shown in Figure.(1b). This circuit uses practical
values and will provide the same transfer function as the prototype in Figure.(1a), but shifted in frequency