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EXAMPLE 12.8 Then,
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Repeat Example 12.7 for H s 10s= s2 30s 106 . Again, from !2 106 , !0 1000 rad/s. 0 q 302 =4 106 30=2 985:1 rad=s q !h 302 =4 106 30=2 1015:1 rad=s !l a 30 rad=s and Q 1000=30 33:3
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Note that !l and !h can also be approximated with good accuracy by !l !0 =2 1000 30=2 985 rad=s and !h !0 =2 1000 30=2 1015 rad=s
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NATURAL FREQUENCY AND DAMPING RATIO
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The denominator of the bandpass function given in (7) may be written as s2 a s b s2 2!0 s !2 0 p p where !0 b is called the natural frequency and  a= 2 b is called the damping ratio. For  > 1, the circuit has two distinct poles on the negative real axis and is called overdamped. For  1, the circuit has a real pole of order two at !0 and is critically damped. For  < 1, the circuit has a pair of p p conjugate poles at !0 j!0 1 2 and !0 j!0 1 2 . The poles are positioned on a semicircle in the left half plane with radius !0 . The placement angle of the poles is  sin 1  (see Fig. 12-17). The circuit is underdamped and can contain damped oscillations. Note that the damping ratio is equal to half of the inverse of the quality factor.
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Fig. 12-17
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RLC SERIES CIRCUIT; SERIES RESONANCE
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The RLC circuit shown in Fig. 12-18 has, under open-circuit condition, an input or driving-point impedance   1 Zin ! R j !L !C
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FREQUENCY RESPONSE, FILTERS, AND RESONANCE
Fig. 12-18
The circuit is said to be in series resonance (or low-impedance resonance) when Zin ! is real (and so jZin ! j is a minimum); that is, when !L 1 0 !C or 1 ! !0  p LC
Figure 12-19 shows the frequency response. The capacitive reactance, inversely proportional to !, is higher at low frequencies, while the inductive reactance, directly proportional to !, is greater at the higher frequencies. Consequently, the net reactance at frequencies below !0 is capacitive, and the angle on Zin is negative. At frequencies above !0 , the circuit appears inductive, and the angle on Zin is positive.
Fig. 12-19
By voltage division, the voltage transfer function for Fig. 12-18 is Hv1 ! R RYin ! Zin !
The frequency response (magnitude only) is plotted in Fig. 12-20; the curve is just the reciprocal of that in Fig. 12-19(a). Note that roll-o occurs both below and above the series resonant frequency !0 . The points where the response is 0.707, the half-power points (Section 12.3), are at frequencies !l and !h . The bandwidth is the width between these two frequencies: !h !l . A quality factor, Q0 !0 L=R, may be de ned for the series RLC circuit at resonance. (See Section 12.12 for the general development of Q.) The half-power frequencies can be expressed in terms of the circuit elements, or in terms of !0 and Q0 , as follows: s s !  2 R R 1 1 1 !0 !h 1 2L 2L LC 4Q2 2Q0 0 s s !  2 R R 1 1 1 !0 1 !l 2L 2L LC 4Q2 2Q0 0
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
[CHAP. 12
Fig. 12-20
See Problem 12.5.
Subtraction of the expressions gives R !0 L Q0
which suggests that the higher the quality, the narrower the bandwidth.
QUALITY FACTOR It is
A quality factor or gure of merit can be assigned to a component or to a complete circuit. de ned as   maximum energy stored Q  2 energy dissipated per cycle
a dimensionless number. This de nition is in agreement with de nitions given in Sections 12.9 and 12.11. A practical inductor, in which both resistance and inductance are present, is modeled in Fig. 12-21. 2 The maximum stored energy is 1 LImax , while the energy dissipated per cycle is 2   2 I 2 R 2 Ieff R max ! !
Fig. 12-21
Hence,
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