0195136047.pdf

3.4 FREQUENCY RESPONSE 161

H(ω), dB

θ(ω), deg

0

90

0

180

80

40

0.1 1 10 100

ω /α^2 + β^2

Slope = 90 °/decade

Slope = 40 dB/decade

ω /α^2 + β^2

Figure 3.4.8Asymptotic Bode plot for

the quadraticH(s)= 1 + 2

α

α^2 +β^2

s+

s^2

α^2 +β^2

.

Y(jω) ̄ =^1

R+jωL+ 1 /j ω C

=

1

R+j(ωL− 1 /ωC)

(3.4.7)

At theseries resonant frequencyω 0 = 1 /

√

LC,

Y(jω ̄ 0 )=^1

R

=Y 0

corresponds to maximum admittance (or minimum impedance). The ratioY/Y 0 can be written as

Y ̄

Y 0

(j ω)=

R

R+j

(

ωL−

1

ωC

)=

1

1 +j

(

L

R

)(

ω−

1

ωLC

)

=

1

1 +j

ω 0 L

R

(

ω

ω 0

−

ω 0

ω

) (3.4.8)

Introducing aquality factorQS=ω 0 L/Randper-unit source frequency deviationδ=(ω−
ω 0 )/ω 0 ,

Y ̄

Y 0

(jω)=

1

1 +jQS

(

ω

ω 0

−

ω 0

ω

)=

1

1 +jδQS

(

2 +δ

1 +δ

) (3.4.9)

Forδ<<1, i.e., for small frequency deviations around the resonant frequency, Equation (3.4.9)
becomes, near resonance,

Y ̄

Y 0

(jω)=

1

1 +j 2 δQS

(3.4.10)

which is the equation of theuniversal resonance curveplotted in Figure 3.4.9. The curve
applies equally well to the parallelGLCcircuit withZ/Z ̄ 0 as the ordinate, when the value of
Qp=ω 0 C/G. The bandwidth and the half-power points in a resonant circuit (either series or
parallel) correspond toω 0 /Qand 2δQ=±1 when the magnitudeY/Y 0 orZ/Z 0 is 0.707.