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FREQUENCY RESPONSE, FILTERS, AND RESONANCE
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[CHAP. 12
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! 0 0:5!x !x 2!x 1
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Fig. 12-6
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The magnitude and angle are plotted in Fig. 12-7. This transfer function approaches unity at high frequency, where the output voltage is the same as the input. Hence the description low-frequency rollo and the name high-pass.
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Fig. 12-7
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A transfer impedance of the RL high-pass circuit under no-load is H1 ! V2 j!L2 I1 or H1 ! ! j R1 !x
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The angle is constant at 908; the graph of magnitude versus ! is a straight line, similar to a reactance plot of !L versus !. See Fig. 12-8.
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Fig. 12-8
Interchanging the positions of R and L results in a low-pass network with high-frequency roll-o (Fig. 12-9). For the open-circuit condition
CHAP. 12]
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
Fig. 12-9
Hv1 ! with !x  R2 =L1 ; that is,
R2 1 R2 j!L1 1 j !=!x
1 jHv j q 1 !=!x 2
H tan 1 !=!x
The magnitude and angle plots are shown in Fig. 12-10. The voltage transfer function Hv1 approaches zero at high frequencies and unity at ! 0. Hence the name low-pass.
Fig. 12-10
The other network functions of this low-pass network are obtained in the Solved Problems.
EXAMPLE 12.2 Obtain the voltage transfer function Hv1 for the open circuit shown in Fig. 12-11. p frequency, in hertz, does jHv j 1= 2 if (a) C2 10 nF, (b) C2 1 nF At what
Fig. 12-11
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
[CHAP. 12
Hv1 ! a
1=j!C2 1 R1 1=j!C2 1 j !=!x
where
!x 
1 2 10 4 R1 C2 C2
rad=s
1 jHv j q 1 !=!x 2 p and so jHv j 1= 2 when ! !x or when f 2 104 =2 3:18 kHz. 2 10 4 2 104 rad=s 10 10 9
10 3:18 31:8 kHz 1
Comparing a and b , it is seen that the greater the value of C2 , the lower is the frequency at which jHv j drops to 0.707 of its peak value, 1; in other words, the more is the graph of jHv j, shown in Fig. 1210, shifted to the left. Consequently, any stray shunting capacitance, in parallel with C2 , serves to reduce the response of the circuit.
HALF-POWER FREQUENCIES The frequency !x calculated in Example 12.2, the frequency at which jHv j 0:707jHv jmax
is called the half-power frequency. In this case, the name is justi ed by Problem 12.5, which shows that the power input into the circuit of Fig. 12-11 will be half-maximum when 1 j!C R1 2 that is, when ! !x . Quite generally, any nonconstant network function H ! will attain its greatest absolute value at some unique frequency !x . We shall call a frequency at which jH ! j 0:707jH !x j a half-power frequency (or half-power point), whether or not this frequency actually corresponds to 50 percent power. In most cases, 0 < !x < 1, so that there are two half-power frequencies, one above and one below the peak frequency. These are called the upper and lower half-power frequencies (points), and their separation, the bandwidth, serves as a measure of the sharpness of the peak.
GENERALIZED TWO-PORT, TWO-ELEMENT NETWORKS
The basic RL or RC network of the type examined in Section 12.2 can be generalized with Z1 and Z2 , as shown in Fig. 12-12; the load impedance ZL is connected at the output port. By voltage division, V2 Z0 V Z1 Z 0 1 or Hv V2 Z0 V1 Z1 Z 0 The other transfer
where Z 0 Z2 ZL = Z2 ZL , the equivalent impedance of Z2 and ZL in parallel. functions are calculated similarly, and are displayed in Table 12-1.
CHAP. 12]
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
Fig. 12-12
Table 12-1
Output Condition Network Function
Hz
V1 I1 Z1
Hv
V2 V1
Hi
I2 I1
Hv Hz
V2 I1 0
Hi I 2 H z V1 1 Z1 0
Short-circuit, ZL 0 Open-circuit ZL 1 Load, ZL
0 Z2 Z1 Z2 Z Z1 Z 0
1 0 Z2 Z2 ZL
Z1 Z2 Z1 Z
Z2 Z
Z 0 ZL Z1 Z 0
THE FREQUENCY RESPONSE AND NETWORK FUNCTIONS
The frequency response of a network may be found by substituting j! for s in its network function. This useful method is illustrated in the following example.
EXAMPLE 12.3 Find (a) the network function H s V2 =V1 in the circuit shown in Fig. 12-13, (b) H j! for LC 2=!2 and L=C R2 , and (c) the magnitude and phase angle of H j! in (b) for !0 1 rad/s. 0 (a) Assume V2 is known. Use generalized impedances Ls and 1=Cs and solve for V1 . From IR V2 =R, VA R Ls IR IC CsVA Then, and Cs R Ls V2 R and R Ls V2 R V2 Cs R Ls 1 Cs R Ls V2 V2 R R R 3
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