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SEE 1023
Circuit Theory
Chapter 3
Other Resonant Circuits
Other Resonant Circuits
Any circuit, not limited to series and parallel circuit,
which consists of L and C tends to resonate at its resonant
frequency.
Under resonant conditions:
• Imaginary part of the Z or Y is zero
• Input voltage and input current are in phase
Based on the above conditions, we can find the resonant
frequency, fo. However, it is difficult to find the other
resonant parameters.
Other Resonant Circuits
Circuit 1: Practical parallel resonant circuit
Rl
Is
C
L
Find resonant frequency, fo
Other Resonant Circuits
Solution
Rl
Is
C
Z1
Z1  Rl  jX l
L
where
X l  L
R  jX l
Xl
1
1
R
Y1 

 2
 2
j 2
2
2
Z1 Rl  jX l Rl  X l
Rl  X l
Rl  X l2
Solution
YT 
R
Rl2

X l2
j
Xl
Rl2

X l2
 jC


L
 2
 j C  2
2
2
Rl  X l
Rl  (L) 

R
At resonance, the imaginary part of YT is zero:


L
0
C  2
2
Rl  (L) 

o 
1  Rl 
 
LC  L 
2
1
fo 
2
1  Rl 
 
LC  L 
2
Practical Parallel Resonant Circuit
What about other resonant parameters?
Q, 1, 2 and BW
It is hard to find those quantities.
The easy way is to use an approximated method.
This method is best suited when quality factor
of the coil, Ql > 10.
Practical Parallel Resonant Circuit
Approximated method:
We transform Rl and L in series to Rp and Xp in parallel.
Rl
Rp
L
Xl
1
R
Y  2
j 2
2
Z Rl  X l
Rl  X l2

1
Rl2

Rl
X l2

1
Rl2  X l2
j
Xl
Lp
Xp
1
1
1
Y 

Z R p jX p
Practical Parallel Resonant Circuit
By matching both admittances:

Rl2  X l2
X l2
Rp 
 Rl  Rl
 Rl 1  Ql2
Rl
Rl
where
X l o L
Ql 

Rl
Rl

Quality factor of the coil
 Rl
Xp 

 X l  X l 
 Xl


 Ql2  1 
1

 X l 1  2   X l 
 Q 
 Q2 
l 

 l 
Rl2

Xl
X l2

Xl
X l2
 Ql2  1 

L p  L
 Q2 
 l 
Rl2




2
Practical Parallel Resonant Circuit
By matching both admittances:

Rl2  X l2
X l2
Rp 
 Rl  Rl
 Rl 1  Ql2
Rl
Rl
where
X l o L
Ql 

Rl
Rl

Quality factor of the coil
 Rl
Xp 

 X l  X l 
 Xl


 Ql2  1 
1

 X l 1  2   X l 
 Q 
 Q2 
l 

 l 
Rl2

Xl
X l2

Xl
X l2
 Ql2  1 

L p  L
 Q2 
 l 
Rl2




2
Other Resonant Circuits
Example 1
10 W
3 nF
180 mH
Find resonant parameters
Practical Parallel Resonant Circuit
Solution
2
o 
Ql 
1  Rl 
    1.36  10 6 rad / s
LC  L 
o L
Rl
 24 .474
 10
For Ql > 10, the approximated method can be used without
significantly loss of accuracy.


R p  Rl 1  Ql2  6  10 3
 1  Ql2 
  L  180 mH
L p  L
2
 Q 
 l 
Practical Parallel Resonant Circuit
Solution
Rp
Lp
C
R p  6 10 3 L p  180 mH C  3 nF
Now we can analyze the circuit as in an ideal
parallel RLC resonant circuit using the above
values.
Other Resonant Circuits
Circuit 2
1H
vs
10 W
Find resonant frequency, fo
0.2 F