CHAPTER 14. Introduction to Frequency Selective Circuits

CHAPTER 14 Introduction to Frequency Selective Circuits

Frequency-selective circuits Varying source frequency on circuit voltages and currents. The result of this analysis is the frequency response of a circuit. Frequency-selective circuits are also called filters, such as telephones, radios, televisions, and satellites, employ frequency-selective circuits. Figure 14.1 The action of a filter on an input signal results in an output signal.

14.1 Some Preliminaries The signals passed from the input to the output fall within a band of frequencies called the passband. Input voltages outside this band have their magnitudes attenuated by the circuit and are thus effectively prevented from reaching the output terminals of the circuit. Frequencies not in a circuit s passband are in its stopband. Frequency-selective circuits are categorized by the location of the passband. Figure 14.2 A circuit with voltage input and output.

One way of identifying the type of frequencyselective circuit is to examine a frequency response plot. A frequency response plot shows how a circuit s transfer function (both amplitude and phase) changes as the source frequency changes. A frequency response plot has two parts. One is a graph of H(jω) versus frequency ω. This part of the plot is called the magnitude plot. The other part is a graph of θ(jω) versus frequency ω. This part is called the phase angle plot.

One passband and one stopband, which are defined by the cutoff frequency. Low-pass filter, which passes signals at frequencies lower than the cutoff frequency from the input to the output. High-pass filter, which passes signals at frequencies higher than the cutoff frequency. Bandpass filter, which passes a source voltage to the output only when the source frequency is within the band defined by the two cutoff frequencies. Bandreject filter, which passes a source voltage to the output only when the source frequency is outside the band defined by the two cutoff frequencies.

Figure 14.3 Ideal frequency response plots of the four types of filter circuits. (a) An ideal low-pass filter. (b) An ideal high-pass filter. (c) An ideal bandpass filter. (d) An ideal bandreject filter.

Passive filters Passive filters, their filtering capabilities depend only on the elements: resistors, capacitors, and inductors.

14.2 Low-Pass Filters The Series RL Circuit Qualitative Analysis Figure 14.4 (a) A series RL lowpass filter. (b) The equivalent circuit at ω = 0 and (c) The equivalent circuit at ω =.

Figure 14.5 The frequency response plot for the series RL circuit in Fig. 14.4(a).

Defining the Cutoff Frequency The definition for cutoff frequency widely used by electrical engineers is the frequency for which the transfer function magnitude is decreased by the factor 1 2 from its maximum value: where H max is the maximum magnitude of the transfer function.

The Power Delivered

At the cutoff frequency the average power delivered by the circuit is one half the maximum average power. Thus ω c is also called the half-power frequency.

The Series RL Circuit Quantitative Analysis Figure 14.6 The s-domain equivalent for the circuit in Fig. 14.4(a).

Cutoff frequency for RL filters

Example 14.1 Electrocardiology is the study of the electric signals produced by the heart. These signals maintain the heart s rhythmic beat, and they are measured by an instrument called an electrocardiograph. This instrument must be capable of detecting periodic signals whose frequency is about 1 Hz (the normal heart rate is 72 beats per minute). The instrument must operate in the presence of sinusoidal noise consisting of signals from the surrounding electrical environment, whose fundamental frequency is 60 Hz the frequency at which electric power is supplied.

Example 14.1 Choose values for R and L in the circuit of Fig. 14.4(a) such that the resulting circuit could be used in an electrocardiograph to filter out any noise above 10 Hz and pass the electric signals from the heart V o at or near 1 Hz. Then compute the magnitude of at 1 Hz, 10 Hz, and 60 Hz to see how well the filter performs.

Example 14.1

A Series RC Circuit as a low-pass filter 1. Zero frequency (ω = 0): The impedance of the capacitor is infinite, and the capacitor acts as an open circuit. The input and output voltages are thus the same. 2. Frequencies increasing from zero: The impedance of the capacitor decreases relative to the impedance of the resistor, and the source voltage divides between the resistive impedance and the capacitive impedance. The output voltage is thus smaller than the source voltage. 3. Infinite frequency (ω = ): The impedance of the capacitor is zero, and the capacitor acts as a short circuit. The output voltage is thus zero. Figure 14.7 A series RC low-pass filter.

Example 14.2 For the series RC circuit in Fig. 14.7: a) Find the transfer function between the source voltage and the output voltage. b) Determine an equation for the cutoff frequency in the series RC circuit. c) Choose values for R and C that will yield a lowpass filter with a cutoff frequency of 3 khz. Figure 14.7 A series RC low-pass filter.

Example 14.2 Figure 14.8 The s- domain equivalent for the circuit in Fig. 14.7.

Example 14.2

Transfer function for a low-pass filter Figure 14.9 Two lowpass filters, the series RL and the series RC, together with their transfer functions and cutoff frequencies.

Relating the Frequency Domain to the Time Domain This result is a direct consequence of the relationship between the time response of a circuit and its frequency response.

14.3 High-Pass Filters The Series RC Circuit Qualitative Analysis Figure 14.10 (a) A series RC high-pass filter; (b) the equivalent circuit at ω = 0 and (c) the equivalent circuit at ω =.

Figure 14.11 The frequency response plot for the series RC circuit in Fig. 14.10(a).

The Series RC Circuit Quantitative Analysis Figure 14.12 The s-domain equivalent of the circuit in Fig. 14.10(a).

Example 14.3 Show that the series RL circuit in Fig. 14.13 also acts like a high-pass filter: a) Derive an expression for the circuit s transfer function. b) Use the result from (a) to determine an equation for the cutoff frequency in the series RL circuit. c) Choose values for R and L that will yield a highpass filter with a cutoff frequency of 15 khz.

Example 14.3 Figure 14.13 The circuit for Example 14.3.

Example 14.3

Example 14.3 Figure 14.14 The s- domain equivalent of the circuit in Fig. 14.13.

Example 14.4 Examine the effect of placing a load resistor in parallel with the inductor in the RL high-pass filter shown in Fig. 14.15: a) Determine the transfer function for the circuit in Fig. 14.15. b) Sketch the magnitude plot for the loaded RL highpass filter, using the values for R and L from the circuit in Example 14.3(c) and letting. On the same graph, sketch the magnitude plot for the unloaded RL high-pass filter of Example 14.3(c).

Example 14.4 Figure 14.15 The circuit for Example 14.4. Figure 14.16 The s- domain equivalent of the circuit in Fig. 14.15.

Example 14.4

Example 14.4 Figure 14.17 The magnitude plots for the unloaded RL high-pass filter of Fig 14.13 and the loaded RL highpass filter of Fig. 14.15.

Transfer function for a high-pass filter Figure 14.18 Two high-pass filters, the series RC and the series RL, together with their transfer functions and cutoff frequencies.

14.4 Bandpass Filters Ideal bandpass filters have two cutoff frequencies which identify the passband. For realistic bandpass filters, cutoff frequencies are defined as the frequencies for which the magnitude of the transfer function equals (1/ 2)H max.

Center Frequency, Bandwidth, and Quality Factor Center frequency ω o, defined as the frequency for which a circuit s transfer function is purely real. Resonant frequency, is the same name given to the frequency that characterizes the natural response of the second-order circuits. When a circuit is driven at the resonant frequency, we say that the circuit is in resonance. O c1 c2 Bandwidth, b, is the width of the passband. Quality factor, is the ratio of the center frequency to the bandwidth.

The Series RLC Circuit Qualitative Analysis Figure 14.19 (a) A series RLC bandpass filter; (b) the equivalent circuit for ω = 0; and (c) the equivalent circuit for ω =.

The frequency response plot for the series RLC bandpass filter circuit Figure 14.20 The frequency response plot for the series RLC bandpass filter circuit in Fig. 14.19.

The Series RLC Circuit Quantitative Analysis Figure 14.21 The s-domain equivalent for the circuit in Fig. 14.19(a).

Center frequency

Cutoff frequencies, series RLC filters

Relationship between center frequency and cutoff frequencies

Relationship between bandwidth and cutoff frequencies

Quality factor The ratio of center frequency to bandwidth.

Cutoff frequencies

Example 14.5 A graphic equalizer is an audio amplifier that allows you to select different levels of amplification within different frequency regions. Using the series RLC circuit in Fig. 14.19(a), choose values for R, L, and C that yield a bandpass circuit able to select inputs within the 1 10 khz frequency band. Such a circuit might be used in a graphic equalizer to select this frequency band from the larger audio band (generally 0 20 khz) prior to amplification.

Example 14.5

Example 14.6 a) Show that the RLC circuit in Fig. 14.22 is also a bandpass filter by deriving an expression for the transfer function b) Compute the center frequency, c) Calculate the cutoff frequencies, and the bandwidth, and the quality factor, Q. d) Compute values for R and L to yield a bandpass filter with a center frequency of 5 khz and a bandwidth of 200 Hz, using a capacitor.

Example 14.6 Figure 14.22 The circuit for Example 14.6.

Example 14.6 Figure 14.23 The s-domain equivalent of the circuit in Fig. 14.22.

Example 14.6

Example 14.7 For each of the bandpass filters we have constructed, we have always assumed an ideal voltage source, that is, a voltage source with no series resistance. Even though this assumption is often valid, sometimes it is not, as in the case where the filter design can be achieved only with values of R, L, and C whose equivalent impedance has a magnitude close to the actual impedance of the voltage source. Examine the effect of assuming a nonzero source resistance, R i, on the characteristics of a series RLC bandpass filter.

Example 14.7 a) Determine the transfer function for the circuit in Fig. 14.24. b) Sketch the magnitude plot for the circuit in Fig. 14.24, using the values for R, L, and C from Example 14.5 and setting R i = R. On the same graph, sketch the magnitude plot for the circuit in Example 14.5, where R i = 0. Figure 14.24 The circuit for Example 14.7.

Example 14.7 Figure 14.25 The s- domain equivalent of the circuit in Fig. 14.24.

Example 14.7

Example 14.7 Figure 14.26 The magnitude plots for a series RLC bandpass filter with a zero source resistance and a nonzero source resistance.

Figure 14.27 Two RLC bandpass filters, together with equations for the transfer function, center frequency, and bandwidth of each.

Transfer function for RLC bandpass filter

Relating the Frequency Domain to the Time Domain

14.5 Bandreject Filters Bandreject filters are characterized by the same parameters as bandpass filters: the two cutoff frequencies, the center frequency, the bandwidth, and the quality factor.

The Series RLC Circuit Qualitative Analysis Figure 14.28 (a) A series RLC bandreject filter. (b) The equivalent circuit for ω = 0 (c) The equivalent circuit for ω =.

Figure 14.29 The frequency response plot for the series RLC bandreject filter circuit in Fig. 14.28(a).

The Series RLC Circuit Quantitative Analysis

Figure 14.30 The s- domain equivalent of the circuit in Fig. 14.28(a).

Example 14.8 Using the series RLC circuit in Fig. 14.28(a), compute the component values that yield a bandreject filter with a bandwidth of 250 Hz and a center frequency of 750 Hz. Use a 100 nf capacitor. Compute values for R, L, c1, c2 and Q. Figure 14.28 (a) A series RLC bandreject filter.

Example 14.8

Transfer function for RLC bandreject filter Figure 14.31 Two RLC bandreject filters, together with equations for the transfer function, center frequency, and bandwidth of each.