Scaling

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

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

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:

Comparing the above equations, 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

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. This technique is similar to how we applied magnitude scaling accept we apply Kf to the frequency. In that manner, we obtain the new impedances below.

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 & Frequncy Scaling

If a circuit is scaled in magnitude and frequency at the same time, then we can use the equations below. These equations are the general form. Notice that if Km = 1 in the equations, there is no Magnitude Scaling and if Kf = 1, there is no frequency scaling. So you could just use this set of equations all the time.