14.13 Summary

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14.13 Summary

• 1. The transfer function H(ω) is the ratio of the output response Y(ω) to the input excitation X(ω); that is, H(ω) = Y(ω)∕X(ω).
• 2. The frequency response is the variation of the transfer function with frequency.
• 3. Zeros of a transfer function H(s) are the values of s = jω that make H(s) = 0, while poles are the values of s that make H(s) → ∞.
• 4. The decibel is the unit of log arithmic gain. For a voltage or current gain G, its decibel equivalent is GdB = 20 log10 G.
• 5. Bode plots are semilog plots of the magnitude and phase of the transfer function as it v aries with frequenc y. The straight-line approximations of H (in dB) and ϕ (in degrees) are constructed using the corner frequencies defined by the poles and zeros of H(ω).
• 6. The resonant frequency is that frequency at which the imaginary part of a transfer function vanishes. For series and parallel RLC circuits.

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

7. The half-power frequencies (ω1, ω2) are those frequencies at which the power dissipated is one-half of that dissipated at the resonant frequency. The geometric mean between the half-power frequencies is the resonant frequency, or

$$\omega_0 = \sqrt{\omega_1 \omega_2}$$

8. The bandwidth is the frequenc y band between half-po wer frequencies:

$$B=\omega_2-\omega_1$$

9. The quality f actor is a measure of the sharpness of the resonance peak. It is the ratio of the resonant (angular) frequency to the bandwidth,

$$Q = \frac{\omega_0}{B}$$

• 10. A filter is a circuit designed to pass a band of frequencies and reject others. Passive filters are constructed with resistors, capacitors, and inductors. Active filters are constructed with resistors, capacitors, and an active device, usually an op amp.
• 11. Four common types of filters are low-pass, high-pass, band-pass, and band-stop. A low-pass filter passes only signals whose frequencies are below the cutoff frequency ωc. A high-pass filter passes only signals whose frequencies are abo ve the cutof f frequency ωc. A band-pass filter passes only signals whose frequencies are within a prescribed

range (ω¹ < ω< ω2). A band-stop filter passes only signals whose frequencies are outside a prescribed range (ω1 > ω> ω2).

12. Scaling is the process whereby unrealistic element v alues are magnitude-scaled by a factor Km and/or frequency-scaled by a factor Kf to produce realistic values.

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

• 13. PSpice can be used to obtain the frequency response of a circuit if a frequency range for the response and the desired number of points within the range are specified in the AC Sweep.
• 14. The radio receiver—one practical application of resonant circuits employs a band-pass resonant circuit to tune in one frequenc y among all the broadcast signals picked up by the antenna.
• 15. The touch-tone telephone and the crosso ver network are tw o typical applications of filters. The touch-tone telephone system employs filters to separate tones of different frequencies to activate electronic switches. The crossover network separates signals in dif ferent frequency ranges so that they can be delivered to different devices such as tweeters and woofers in a loudspeaker system.

Review Questions

14.1 A zero of the transfer function

$$H(s) = \frac{10(s + 1)}{(s + 2)(s + 3)}$$

is at

(a) 10 (b) −1 (c) −2 (d) −3

• 14.2 On the Bode magnitude plot, the slope of 1∕(5 + jω) 2 for large values of ω is
• (a) 20 dB/decade (b) 40 dB/decade

(c) −40 dB/decade (d) −20 dB/decade

14.3 On the Bode phase plot for 0.5 < ω< 50, the slope of [1 + j10ω− ω² ∕25]2 is

(a) 45°/decade	(b) 90°/decade

• (c) 135°/decade (d) 180°/decade
  • 14.4 How much inductance is needed to resonate at 5 kHz with a capacitance of 12 nF?

(a) 2,652 H	(b) 11.844 H
(c) 3.333 H	(d) 84.43 mH

14.5 The difference between the half-power frequencies is called the:

(a) quality factor	(b) resonant frequency
--------------------	------------------------

• (c) bandwidth (d) cutoff frequency
  • 14.6 In a series RLC circuit, which of these quality factors has the steepest magnitude response curve near resonance?

(a) Q = 20	(b) Q = 12
(c) Q = 8	(d) Q = 4

14.7 In a parallel RLC circuit, the bandwidth B is directly proportional to R.

(a) True (b) False

14.8 When the elements of an RLC circuit are both magnitude-scaled and frequency-scaled, which quality is unaffected?

(a) resistor	(b) resonant frequency
(c) bandwidth	(d) quality factor

14.9 What kind of filter can be used to select a signal of one particular radio station?

(a) low-pass (b) high-pass

• (c) band-pass (d) band-stop
• 14.10 A voltage source supplies a signal of constant amplitude, from 0 to 40 kHz, to an RC low-pass filter. A load resistor, connected in parallel across the capacitor, experiences the maximum voltage at:

(a) dc	(b) 10 kHz
(c) 20 kHz	(d) 40 kHz

Answers: 14.1b, 14.2c, 14.3d, 14.4d, 14.5c, 14.6a, 14.7b, 14.8d, 14.9c, 14.10a.