Figure 14.29 14.7 Passive Filters For Practice Prob. 14.9

← Back to Fundamentals of Electric Circuits Overview

Figure 14.29 14.7 Passive Filters For Practice Prob. 14.9

The concept of filters has been an integral part of the evolution of electrical engineering from the beginning. Several technological achievements would not have been possible without electrical filters. Because of this prominent role of filters, much effort has been expended on the theory, design, and construction of filters and many articles and books have been written on them. Our discussion in this chapter should be considered introductory.

A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.

As a frequency-selective device, a filter can be used to limit the frequency spectrum of a signal to some specified band of frequencies. Filters are the circuits used in radio and TV receivers to allow us to select one desired signal out of a multitude of broadcast signals in the environment.

A filter is a passive filter if it consists of only passive elements R, L, and C. It is said to be an active filter if it consists of acti ve elements (such as transistors and op amps) in addition to passi ve elements R, L, and C. We consider passive filters in this section and active filters in the next section. LC filters have been used in practical applications for more than eight decades. LC filter technology feeds related areas such as equalizers, impedance-matching networks, transformers, shaping networks, power dividers, attenuators, and directional couplers, and is continuously providing practicing engineers with opportunities to inno vate and experiment. Besides the LC filters we study in these sections, there are other kinds of filters—such as digital filters, electromechanical filters, and microwave filters—which are beyond the level of this text.

0 (b) ω^c ω 1 0 (a) ω^c ω 1 0 (c) ω¹ ω² ω 1 0 Passband Passband Passband Stopband Stopband Stopband Passband Passband Stopband Stopband ω¹ ω² ω 1 │H(ω)│ │H(ω)│ │H(ω)│ │H(ω)│

Figure 14.30

Ideal frequency response of four types of filters: (a) low-pass filter, (b) high-pass filter, (c) band-pass filter, (d) band-stop filter.

(d)

Figure 14.31 A low-pass filter.

Figure 14.32 Ideal and actual frequency response of a low-pass filter.

As shown in Fig. 14.30, there are four types of filters whether passive or active:

• 1. A low-pass filter passes low frequencies and stops high frequencies, as shown ideally in Fig. 14.30(a).
• 2. A high-pass filter passes high frequencies and rejects low frequencies, as shown ideally in Fig. 14.30(b).
• 3. A band-pass filter passes frequencies within a frequenc y band and blocks or attenuates frequencies outside the band, as sho wn ideally in Fig. 14.30(c).
• 4. A band-stop filter passes frequencies outside a frequenc y band and blocks or attenuates frequencies within the band, as sho wn ideally in Fig. 14.30(d).

Table 14.5 presents a summary of the characteristics of these filters. Be aware that the characteristics in Table 14.5 are only valid for first- or second-order filters—but one should not have the impression that only these kinds of filter exist. We now consider typical circuits for realizing the filters shown in Table 14.5.

TABLE 14.5

Summary of the characteristics of ideal filters.

Type of Filter	H(0)	H(∞)	H(ωc) or H(ω0)
Low-pass	1	0	1∕ √ 2
High-pass	0	1	1∕ √ 2
Band-pass	0	0	1
Band-stop	1	1	0

ωc is the cutoff frequency for low-pass and high-pass filters; ω0 is the center frequency for band-pass and band-stop filters.

14.7.1 Low-Pass Filter

A typical low-pass filter is formed when the output of an RC circuit is taken off the capacitor as shown in Fig. 14.31. The transfer function (see also Example 14.1) is

$$\mathbf{H}(\omega) = \frac{\mathbf{V}_o}{\mathbf{V}_i} = \frac{1/j\omega C}{R + 1/j\omega C}$$ $$\mathbf{H}(\omega) = \frac{1}{1 + j\omega RC}$$

(14.50)

Note that H(0) = 1, H(∞) = 0. Figure 14.32 shows the plot of ∣H(ω)∣, along with the ideal characteristic. The half-power frequency, which is equivalent to the corner frequency on the Bode plots but in the context of filters is usually known as the cutoff frequency ωc, is obtained by setting the magnitude of H(ω) equal to 1∕ √ __ 2 , thus,

$$H(\omega_c) = \frac{1}{\sqrt{1 + \omega_c^2 R^2 C^2}} = \frac{1}{\sqrt{2}}$$

or

$$\omega_c = \frac{1}{RC} \tag{14.51}$$

The cutoff frequency is also called the rolloff frequency.

A low-pass filter is designed to pass only frequencies from dc up to the cutoff frequency ωc.

A low-pass filter can also be formed when the output of an RL circuit is taken off the resistor. Of course, there are many other circuits for low-pass filters.

14.7.2 High-Pass Filter

A high-pass filter is formed when the output of an RC circuit is taken off the resistor as shown in Fig. 14.33. The transfer function is

$$\mathbf{H}(\omega) = \frac{\mathbf{V}_o}{\mathbf{V}_i} = \frac{R}{R + 1/j\omega C}$$ $$\mathbf{H}(\omega) = \frac{j\omega RC}{1 + j\omega RC}$$

(14.52)

Note that H(0) = 0, H(∞) = 1. Figure 14.34 shows the plot of ∣H(ω)∣. Again, the corner or cutoff frequency is

$$\omega_c = \frac{1}{RC} \tag{14.53}$$

A high-pass filter is designed to pass all frequencies above its cutoff frequency ωc.

A high-pass filter can also be formed when the output of an RL circuit is taken off the inductor.

14.7.3 Band-Pass Filter

The RLC series resonant circuit provides a band-pass filter when the output is taken off the resistor as shown in Fig. 14.35. The transfer function is

$$\mathbf{H}(\omega) = \frac{\mathbf{V}_o}{\mathbf{V}_i} = \frac{R}{R + j(\omega L - 1/\omega C)}$$

(14.54)

We observe that H(0) = 0, H(∞) = 0. Figure 14.36 shows the plot of ∣H(ω)∣. The band-pass filter passes a band of frequencies (ω¹ < ω< ω2) centered on ω0, the center frequency, which is given by

$$\omega_0 = \frac{1}{\sqrt{LC}}\tag{14.55}$$

A band-pass filter is designed to pass all frequencies within a band of frequencies, ω¹ < ω< ω2.

Because the band-pass filter in Fig. 14.35 is a series resonant circuit, the half-power frequencies, the bandwidth, and the quality factor are determined as in Section 14.5. A band-pass filter can also be formed by cascading the low-pass filter (where ω2 = ωc) in Fig. 14.31 with the The cutoff frequency is the frequency at which the transfer function H drops in magnitude to 70.71% of its maximum value. It is also regarded as the frequency at which the power dissipated in a circuit is half of its maximum value.

Figure 14.34

Ideal and actual frequency response of a high-pass filter.

Figure 14.35

Figure 14.36 Ideal and actual frequency response of a band-pass filter.

high-pass filter (where ω1 = ωc) in Fig. 14.33. However, the result would not be the same as just adding the output of the low-pass filter to the input of the high-pass filter, because one circuit loads the other and alters the desired transfer function.

14.7.4 Band-Stop Filter

A filter that prevents a band of frequencies between two designated values (ω1 and ω2) from passing is variably known as a band-stop, bandreject, or notch filter. A band-stop filter is formed when the output RLC series resonant circuit is taken off the LC series combination as shown in Fig. 14.37. The transfer function is

nster function is
\n

$$\mathbf{H}(\omega) = \frac{\mathbf{V}_o}{\mathbf{V}_i} = \frac{j(\omega L - 1/\omega C)}{R + j(\omega L - 1/\omega C)}$$

\n(14.56)

Notice that H(0) = 1, H(∞) = 1. Figure 14.38 shows the plot of ∣H(ω)∣. Again, the center frequency is given by

$$\omega_0 = \frac{1}{\sqrt{LC}}\tag{14.57}$$

while the half-power frequencies, the bandwidth, and the quality factor are calculated using the formulas in Section 14.5 for a series reso nant circuit. Here, ω0 is called the frequency of rejection, while the corresponding bandwidth (B = ω² − ω1) is known as the bandwidth of rejection. Thus,

A band-stop filter is designed to stop or eliminate all frequencies within a band of frequencies, ω¹ < ω< ω2.

Notice that adding the transfer functions of the band-pass and the band-stop gives unity at any frequency for the same values of R, L, and C. Of course, this is not true in general b ut true for the circuits treated here. This is due to the fact that the characteristic of one is the inverse of the other.

In concluding this section, we should note that:

• 1. From Eqs. (14.50), (14.52), (14.54), and (14.56), the maximum gain of a passive filter is unity. To generate a gain greater than unity, one should use an active filter as the next section shows.
• 2. There are other ways to get the types of filters treated in this section.
• 3. The filters treated here are the simple types. Many other filters have sharper and complex frequency responses.

Example 14.10 Determine what type of filter is shown in Fig. 14.39. Calculate the corner or cutoff frequency. Take R = 2 kΩ, L = 2 H, and C = 2 μF.

Solution:

The transfer function is

ction is
\n

$$\mathbf{H}(s) = \frac{\mathbf{V}_o}{\mathbf{V}_i} = \frac{R||1/sC}{sL + R||1/sC}, \qquad s = j\omega$$

\n(14.10.1)

Figure 14.37 A band-stop filter.

band-stop filter.

Ideal and actual frequency response of a

But

R\left\|\frac{1}{sC} = \frac{R/sC}{R+1/sC} = \frac{R}{1+sRC}

Substituting this into Eq. (14.10.1) gives

uting this into Eq. (14.10.1) gives
\n

$$\mathbf{H}(s) = \frac{R/(1+sRC)}{sL + R/(1+sRC)} = \frac{R}{s^2RLC + sL + R}, \qquad s = j\omega$$

or

$$R/(1 + sRC) \t s2RLC + sL + R'$$ $$H(\omega) = \frac{R}{-\omega^2 RLC + j\omega L + R}$$

(14.10.2)

Because H(0) = 1 and H(∞) = 0, we conclude from Table 14.5 that the circuit in Fig. 14.39 is a second-order low-pass filter. The magnitude of H is ^H =  ___________________ ^{R √}

a second-order low-pass filter. The magnitude of

$$H = \frac{R}{\sqrt{(R - \omega^2 R LC)^2 + \omega^2 L^2}}$$

(14.10.3)

The corner frequency is the same as the half-power frequency, that is, where H is reduced by a factor of 1 ∕ √ __ 2 . Because the dc value of H(ω) is 1, at the corner frequency, Eq. (14.10.3) becomes after squaring

where H is reduced by a factor of 1 /

$$\sqrt{2}$$

. Because the d
is 1, at the corner frequency, Eq. (14.10.3) becomes after

$$H^2 = \frac{1}{2} = \frac{R^2}{(R - \omega_c^2 R LC)^2 + \omega_c^2 L^2}$$

or

$$2 = (1 - \omega_c^2 LC)^2 + \left(\frac{\omega_c L}{R}\right)^2$$

Substituting the values of R, L, and C, we obtain

$$2 = (1 - \omega_c^2 4 \times 10^{-6})^2 + (\omega_c 10^{-3})^2$$

Assuming that ωc is in krad/s,

$$2 = (1 - 4\omega_c^2)^2 + \omega_c^2 \qquad \text{or} \qquad 16\omega_c^4 - 7\omega_c^2 - 1 = 0$$

Solving the quadratic equation in ω^c 2 , we get ω^c 2  = 0.5509 and −0.1134. Because ωc is real,

$$\omega_c = 0.742 \text{ krad/s} = 742 \text{ rad/s}$$

For the circuit in Fig. 14.40, obtain the transfer function Vo(ω)∕Vi(ω). Practice Problem 14.10 Identify the type of filter the circuit represents and determine the corner frequency. Take R1 = 100 Ω = R2, L = 2 mH.

Answer:

$$\frac{R_2}{R_1 + R_2} \left( \frac{j\omega}{j\omega + \omega_c} \right)$$

, high-pass filter

$$\omega_c = \frac{R_1 R_2}{(R_1 + R_2)L} = 25 \text{ krad/s}.$$

For Practice Prob. 14.10.

Example 14.11 If the band-stop filter in Fig. 14.37 is to reject a 200-Hz sinusoid while passing other frequencies, calculate the v alues of L and C. Take R = 150 Ω and the bandwidth as 100 Hz.

Solution:

We use the formulas for a series resonant circuit in Section 14.5.

$$B = 2\pi(100) = 200\pi \text{ rad/s}$$

But

B = __ R L ⇒ L = __ R B = _____ ¹⁵⁰ 200π = 0.2387 H

Rejection of the 200-Hz sinusoid means that f0 is 200 Hz, so that ω0 in Fig. 14.38 is

$$\omega_0 = 2\pi f_0 = 2\pi (200) = 400\pi$$

Given that ω0 = 1∕ √ ___ LC ,

$$= 1/\sqrt{LC},$$

\n

$$C = \frac{1}{\omega_0^2 L} = \frac{1}{(400\pi)^2 (0.2387)} = 2.653 \pi F$$

Practice Problem 14.11 Design a band-pass filter of the form in Fig. 14.35 with a lower cutoff frequency of 20.1 kHz and an upper cutoff frequency of 20.3 kHz. Take R = 30 kΩ. Calculate L, C, and Q.

Answer: 23.87 H, 2.6 pF, 101.