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103 104 105 Radian frequency (logarithmic scale)
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Figure 622 Frequency response of narrow-band band-pass lter of Example 66 Comments: It should be apparent that, while at the higher and lower frequencies most of
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the amplitude of the input signal is ltered from the output, at the mid-band frequency (4,500 rad/s) most of the input signal amplitude passes through the lter The rst band-pass lter analyzed in this example would pass the mid-band range of the audio spectrum, while the second would pass only a very narrow band of frequencies around the center frequency of 4,500 rad/s Such narrow-band lters nd application in tuning circuits, such as those employed in conventional AM radios (although at frequencies much higher than that of the present example) In a tuning circuit, a narrow-band lter is used to tune in a frequency associated with the carrier of a radio station (for example, for a station found at a setting of AM 820, the carrier wave transmitted by the radio station is at a frequency of 820 kHz) By using a variable capacitor, it is possible to tune in a range of carrier frequencies and therefore select the preferred station Other circuits are then used to decode the actual speech or music signal modulated on the carrier wave; some of these will be discussed in 8
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One application of narrow-band lters is in rejecting interference due to AC line power Any undesired 60-Hz signal originating in the AC line power can cause serious interference in sensitive instruments In medical instruments such as the electrocardiograph, 60-Hz notch lters are often
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Circuits
provided to reduce the effect of this interference2 on cardiac measurements Figure 623 depicts a circuit in which the effect of 60-Hz noise is represented by way of a 60-Hz sinusoidal generator connected in series with a signal source (VS ), representing the desired signal In this example we design a 60-Hz narrow-band (or notch) lter to remove the unwanted 60-Hz noise
60 Hz notch filter RS
+ V 60 Hz _ + VS _
C RL
+ VL _
Figure 623 60-Hz notch lter
Solution: Known Quantities RS = 50 Find Appropriate values of L and C for the notch lter Assumptions None Analysis To determine the appropriate capacitor and inductor values, we
write the expression for the notch lter impedance: Z = ZL Z C = j L j C 1 j L + j C = j L 1 2 LC
Note that when 2 LC = 1, the impedance of the circuit is in nite! The frequency 0 = 1 LC
is the resonant frequency of the LC circuit If this resonant frequency were selected to be equal to 60 Hz, then the series circuit would show an in nite impedance to 60-Hz currents, and would therefore block the interference signal, while passing most of the other frequency components We thus select values of L and C that result in 0 = 2 60 Let L = 100 mH Then C= 1 = 7036 F 2 0 L Vo (j ) RL = = Vi (j ) RS + R L + Z RL RS + R L + j L 1 2 LC
The frequency response of the complete circuit is given below: HV (j ) =
and is plotted in Figure 624
2 See
Example 133 and Section 152 for further information on electrocardiograms and line noise, respectively
6
Frequency Response and System Concepts
Notch filter amplitude response 08 Amplitude ratio 06 04 02 0 100
103 Frequency, Hz
Notch filter phase response 100 50 Degrees 0 _50
103 Frequency, Hz
Figure 624 Frequency response of 60-Hz notch lter
Comments It would be instructive for you to calculate the response of the
notch lter at frequencies in the immediate neighborhood of 60 Hz, to verify the attenuation effect of the notch lter
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Seismic Transducer
This example illustrates the application of the frequency response idea to a practical displacement transducer The frequency response of a seismic displacement transducer is analyzed, and it is shown that there is an analogy between the equations describing the mechanical transducer and those that describe a second-order electrical circuit The con guration of the transducer is shown in Figure 625 The transducer is housed in a case rigidly af xed to the surface of a body whose motion is to be measured Thus, the case will experience the same displacement as the body, xi Inside the case, a small mass, M, rests on a spring characterized by stiffness K, placed in parallel with a damper, B The wiper arm of a potentiometer is connected to the oating mass, M; the potentiometer is attached to the transducer case, so that the voltage Vo is proportional to the relative displacement of the mass with respect to the case, xo
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