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Problem set #9, EE 223, 4/01/2003 – 4/08/
Chapter 16, Problem 1.
A parallel RLC circuit has R =1 kΩ, C =47 μF, and L =11 mH. (a) Compute Q 0. (b) Determine the resonant frequency (in Hz). (c) Sketch the voltage response as a function of frequency if the circuit is excited by a steady-state 1-mA sinusoidal current source.
We have a parallel RLC with R = 1 kΩ, C = 47 μF and L = 11 mH.
(a) Qo = R(C/L) ½^ = 65.
(b) f o = ωo / 2 π = (LC) -½^ / 2 π = 221.3 Hz
(c) The circuit is excited by a steady-state 1-mA sinusoidal source:
The admittance Y ( s ) facing the source is Y ( s ) = 1/R + 1/ s L + s C
= C( s^2 + s /RC + 1/LC)/ s so Z ( s ) = ( s /C) / ( s^2 + s /RC + 1/LC) and
Z ( j ω) = (1/C) ( j ω) / (1/LC – ω^2 + j ω/RC).
Since V = 10 -3^ Z , we note that | V | > 0 as ω → 0 and also as ω → ∞.
Bandwidth B = ω 0 / Q 0 = 21.27 s -1^ (≅3.4 Hz) ⇒ f 1 = 219.6 Hz and f 2 = 223.0 Hz
A parallel RLC circuit is measured to have a Q 0 of 200. Determine the remaining component value if ( a )R=1Ω and C=1 μF; ( b ) L =12 fH and C=2.4nF; ( c )R =121.7 kΩ and L =100 pH.
(a) R = 1 Ω and C = 1 μF. Qo = R(C/L) ½^ = 200 so L = C(R/ Qo ) 2 = 25 pH
(b) L = 12 fH and C = 2.4 nF R = Qo (L/ C) ½^ = 447.2 mΩ
(c) R = 121.7 kΩ and L = 100 pH C = (Qo / R) 2 L = 270 aF
Let R =1 MΩ, L =1 H, C =1 μF, and I =10 μA in the circuit of Fig. 16.1. (Parallel Resonant Circuit) ( a )Find ω 0 and Q 0. (b) Plot | V | as a function of ω , 995 <ω<1005 rad/s.
Parallel: R 10 , L^6 1, C 10 6 , I 10 0 A
s^ μ
(a) 3 6 6
1000 rad/s; Q RC 10 1000 LC
ω o o ω o
= = = = +^ − =
(b) V = I / Y = 10-5^ / (10-6^ + j (10-6ω - 1/ω)) = 10 -5^ / (10-3^ (10-3^ + j (10-3ω - 1000/ω))
| V| = 10-2^ / √(10-6^ + (10 -3ω - 1000/ω) 2 )
B = ω 0 / Q 0 = 1 s -1^ ⇒ ω 1 = 999.5 s -1^ and ω 2 = 1000.5 s -
ω
V
