14.9 Scaling

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14.9 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. 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.

14.9.1 Magnitude Scaling

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

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

$$\mathbf{Z}_R = R, \qquad \mathbf{Z}_L = j\omega L, \qquad \mathbf{Z}_C = \frac{1}{j\omega C} \tag{14.78}$$

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

$$\mathbf{Z}'_R = K_m \mathbf{Z}_R = K_m R, \qquad \mathbf{Z}'_L = K_m \mathbf{Z}_L = j\omega K_m L$$ $$\mathbf{Z}'_C = K_m \mathbf{Z}_C = \frac{1}{j\omega C/K_m}$$

(14.79)

Comparing Eq. (14.79) with Eq. (14.78), we notice the following changes in the element values: R → KmR, L → Km L, and C → C∕Km. Thus, in magnitude scaling, the new values of the elements and frequency are

$$R' = K_m R, \qquad L' = K_m L$$ $$C' = \frac{C}{K_m}, \qquad \omega' = \omega$$

(14.80)

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

$$\omega_0' = \frac{1}{\sqrt{LC'}} = \frac{1}{\sqrt{K_m LC/K_m}} = \frac{1}{\sqrt{LC}} = \omega_0$$

(14.81)

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 forms of Eqs. (14.2a) and (14.2b), which are dimen sionless quantities.

14.9.2 Frequency Scaling

Frequency scaling is equivalent to relabeling the frequency axis of a frequency response plot. It is needed when translating frequencies such as a resonant frequency, a corner frequency, a bandwidth, etc., to a realistic level. It can be used to bring capacitance and inductance values into a range that is convenient to work with.

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 Eq. (14.78), we see that the impedances of L and C are frequency-dependent. If we apply frequenc y scaling to ZL(ω) and ZC(ω) in Eq. (14.78), we obtain

$$\mathbf{Z}_L = j(\omega K_f)L' = j\omega L \qquad \Rightarrow \qquad L' = \frac{L}{K_f} \tag{14.82a}$$ $$Z_C = \frac{1}{j(\omega K_f)C'} = \frac{1}{j\omega C} \qquad \Rightarrow \qquad C' = \frac{C}{K_f} \tag{14.82b}$$

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

$$R' = R, \qquad L' = \frac{L}{K_f}$$ $$C' = \frac{C}{K_f}, \qquad \omega' = K_f \omega$$

(14.83)

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

$$\omega'_{0} = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{(L/K_{f})(C/K_{f})}} = \frac{K_{f}}{\sqrt{LC}} = K_{f}\omega_{0}$$

(14.84)

and for the bandwidth

$$B' = K_f B \tag{14.85}$$

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

14.9.3 Magnitude and Frequency Scaling

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

$$R' = K_m R, \qquad L' = \frac{K_m}{K_f} L$$ $$C' = \frac{1}{K_m K_f} C, \qquad \omega' = K_f \omega$$

(14.86)

These are more general formulas than those in Eqs. (14.80) and (14.83). We set Km = 1 in Eq. (14.86) when there is no magnitude scaling or Kf = 1 when there is no frequency scaling.

A fourth-order Butterworth low-pass filter is shown in Fig. 14.48(a). The Example 14.14 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.

Figure 14.48

For Example 14.14: (a) Normalized Butterworth low-pass filter, (b) scaled version of the same low-pass filter.

Solution:

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

$$K_f = \frac{\omega_c'}{\omega_c} = \frac{100\pi \times 10^3}{1} = \pi \times 10^5$$

Also, if each 1-Ω resistor is to be replaced by a 10-k Ω resistor, then the magnitude scale factor must be

$$K_m = \frac{R'}{R} = \frac{10 \times 10^3}{1} = 10^4$$

Using Eq. (14.86),

$$L'_1 = \frac{K_m}{K_f} L_1 = \frac{10^4}{\pi \times 10^5} (1.848) = 58.82 \text{ mH}$$

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$$L'_2 = \frac{K_m}{K_f} L_2 = \frac{10^4}{\pi \times 10^5} (0.765) = 24.35 \text{ mH}$$

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$$C'_1 = \frac{C_1}{K_m K_f} = \frac{0.765}{\pi \times 10^9} = 243.5 \text{ pF}$$

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$$C'_2 = \frac{C_2}{K_m K_f} = \frac{1.848}{\pi \times 10^9} = 588.2 \text{ pF}$$

The scaled circuit is shown in Fig. 14.48(b). This circuit uses practical values and will provide the same transfer function as the prototype in Fig. 14.48(a), but shifted in frequency.

A third-order Butterworth filter normalized to ωc = 1 rad/s is shown Practice Problem 14.14 in Fig. 14.49. Scale the circuit to a cutoff frequency of 10 kHz. Use 15-nF capacitors.

Answer: R′ ¹ = R′ ² = 1.061 kΩ, C′ ¹ = C′ 2 = 15 nF, L′ = 33.77 mH.