Hv hv bjvjvhcvh

1. RLC Circuit and Resonance
Electric Circuit 2
Endah S. Ningrum
Mechatronics-EEPIS-ITS

2. Impedance of Series RLC Circuits


A series RLC circuit contains both inductance and capacitance
Since XL and XC have opposite effects on the circuit phase angle, the
total reactance (Xtot)is less than either individual reactance
C
L
tot
X
X
X 


3. Impedance of Series RLC Circuits




When XL>XC, the circuit is predominantly inductive, causes the total
current to lag the source voltage.
When XC> XL, the circuit is predominantly capacitive, causes the
total current to lag the source voltage.
Total reactance |XL – XC|
Total impedance for a series RLC circuit is:

4. Example


Problem
Determine the total impedance and the phase angle in the next figure, with
f=1kHz, R=560kW, L=100mH, and C=0,56mF
Solution
   
    















628
100
1
2
2
284
56
.
0
1
2
1
2
1
mH
kHz
L
f
X
F
kHz
C
f
X
L
C






5. Example
 Solution







 344
284
628
C
L
tot
X
X
X
   



























6
.
31
560
344
tan
tan
657
344
560
1
1
2
2
2
2
R
X
X
R
Z
tot
tot
tot


6. Analysis of Series RLC Circuits







A series RLC circuit is:
Capacitive when XC>XL
Inductive when XL>XC
Resonant when XC=XL
At resonance Ztot = R
XL is a straight line
y = mx + b
XC is a hyperbola xy = k

7. Example
 Problem
For each of the following frequencies of the source voltage, f i
nd the
impedance and the phase angle for the circuit in the next f i
gure. Note the
change in the impedance and the phase angle with frequency, with,
R=3,3k, L=100mH, and C=0,022F for:
a. f=1khZ b. f=3,5kHz c. f=5kHz

8. Example


Solution
Of a.    
    















628
100
1
2
2
23
.
7
022
.
0
1
2
1
2
1
mH
kHz
L
f
X
k
F
kHz
C
f
X
L
C












 k
k
X
X
X C
L
tot
6
.
6
23
.
7
628
   




























4
.
63
3
.
3
6
.
6
tan
tan
38
.
7
6
.
6
3
.
3
1
1
2
2
2
2
k
k
R
X
k
k
k
X
R
Z
tot
tot
tot


9. Example


Solution
Of b.    
    















k
mH
kHz
L
f
X
k
F
kHz
C
f
X
L
C
20
.
2
100
5
.
3
2
2
07
.
2
022
.
0
5
.
3
2
1
2
1












 130
07
.
2
20
.
2 k
k
X
X
X C
L
tot
   




























26
.
2
3
.
3
130
tan
tan
30
.
3
130
3
.
3
1
1
2
2
2
2
k
R
X
k
k
X
R
Z
tot
tot
tot


10. Example


Solution
Of c.    
    















k
mH
kHz
L
f
X
k
F
kHz
C
f
X
L
C
14
.
3
100
5
2
2
45
.
1
022
.
0
5
2
1
2
1












 k
k
k
X
X
X C
L
tot
69
.
1
45
.
1
14
.
3
   




























1
.
27
3
.
3
69
.
1
tan
tan
71
.
3
69
.
1
3
.
3
1
1
2
2
2
2
k
k
R
X
k
k
k
X
R
Z
tot
tot
tot


11. Voltage Across the Series
Combination of L and C



In a se r ie s RLC circuit , t he
ca p a ci t or v ol t a g e a n d t h e
inductor voltage are always 180°
out of phase with each other
Because they are 180° out of
phase, VC and VL subtract from
each other
The voltage across L and C
combined is always less that the
larger individual voltage across
either element Th e v olt a g e a cr oss t h e ser ies
combination of C and L is always less
than the larger individual voltage
across either C or L

12. Voltage Across the Series
Combination of L and C



In a se r ie s RLC circuit , t he
ca p a ci t or v ol t a g e a n d t h e
inductor voltage are always 180°
out of phase with each other
Because they are 180° out of
phase, VC and VL subtract from
each other
The voltage across L and C
combined is always less that the
larger individual voltage across
either element Inductor voltage and capacitor
voltage effectively subtract
because they are out of phase

13. 

Problem
Find the voltage across each element in the next f i
gure and draw a
complete voltage phasor diagram. Also f i
nd the voltage across L and C
combined
Solution
Example







 k
k
k
X
X
X C
L
tot
35
60
25
    






 k
k
k
X
R
Z tot
tot
8
.
82
35
75
2
2
2
2

14. Example
 Solution
A
k
V
Z
V
I
tot
S

121
28
.
8
10




   
   
    V
k
A
X
I
V
V
k
A
X
I
V
V
k
A
R
I
V
C
C
L
L
R
26
.
7
60
121
03
.
3
25
121
08
.
9
75
121





















V
V
V
V
V
V L
C
CL
23
.
4
03
.
3
26
.
7 





















 

25
75
35
tan
tan
1
1
k
k
R
X tot


15. Series Resonance



Resonance is a condition in a series RLC circuit in which the
capacitive and inductive reactances are equal in magnitude
The result is a purely resistive impedance
The formula for series resonance is:
At the resonant frequency (f ), the reactances are equal in magnitude
and effectively cancel, leaving Z = R

16. Series Resonance
   
C
L
f
f
C
L
f
C
f
L
f
C
f
L
f
X
X
r
r
r
r
r
r
r
C
L




















2
2
2
2
2
4
1
1
4
2
2
2
1
2






C
L
f r



2
1

17. Series Resonance



At the resonant frequency fr, the voltages across C and L are equal
in magnitude.
Since they are 180º out of phase with each other, they cancel,
leaving 0 V across the C,L combination (point A to point B).
The section of the circuit from A to B effectively looks like a short at
resonance (neglecting winding resistance).

18. V&I Ampitudes in Series RLC Circuit
Below the Resonant Frequency
At the
Resonant
Frequency
Above the Resonant Frequency

19. Series Resonance
Generalized current and voltage magnitudes as a function of frequency in a series RLC
circuit. VC and VL can be much larger than the source voltage. The shapes of the graphs
depend on particular circuit values.

20. Series RLC Impedance as a Function
of Frequency

21. Example


Problem
Find I, VR
, VL
, and VC
at the resonance in the next figure , with VS=50mV,
R=22, XL
=100, and XC
=100
Solution
mA
mV
R
V
I
S
27
.
2
22
50




   
   
    mV
mA
X
I
V
mV
mA
X
I
V
mV
mA
R
I
V
C
C
L
L
R
227
100
27
.
2
227
100
27
.
2
50
22
27
.
2



















22. Example
 Problem
Determine the impedance at the following frequencies, for the R=10,
L=100mH, C=0,01F
a. fr b. 1kHz below fr c. 1kHz above fr

23. Example


Solution for a.
Solution for b.


 10
R
Z
   
kHz
F
mH
C
L
f r
03
.
5
01
.
0
100
2
1
2
1








kHz
kHz
kHz
kHz
f
f r
03
.
4
1
03
.
5
1 




   
    















k
mH
kHz
L
f
X
k
F
kHz
C
f
X
L
C
53
.
2
100
03
.
4
2
2
95
.
3
01
.
0
03
.
4
2
1
2
1












 k
k
k
X
X
X C
L
tot
42
.
1
95
.
3
53
.
2
    






 k
k
X
R
Z tot
tot
42
.
1
42
.
1
10
2
2
2
2

24. Example
 Solution for c.
kHz
kHz
kHz
kHz
f
f r
03
.
6
1
03
.
5
1 




   
    















k
mH
kHz
L
f
X
k
F
kHz
C
f
X
L
C
79
.
3
100
03
.
6
2
2
64
.
2
01
.
0
03
.
6
2
1
2
1












 k
k
k
X
X
X C
L
tot
15
.
1
64
.
2
79
.
3
    






 k
k
X
R
Z tot
tot
15
.
1
15
.
1
10
2
2
2
2

25. Phase Angle of a Series RLC Circuit

26. A Basic Series Resonant Bandpass
Filter



Curre nt is maximum at
resonant frequency
Bandwidth (BW) is the range
b e t w e e n t w o c u t o f f
frequencies (f1 to f2)
W i t h i n t h e b a n d w i d t h
frequencies, the current is
greater than 70.7% of the
highest resonant value
Bandwidth of Series Resonant Circuits

27. Frequency Response of Series Resonant
Bandpass Filter
Example of the frequency
re sponse of a se rie s
resonant band-pass f i
lter
with the input voltage at a
constant 10 V rms. The
winding resistance of the
coil is neglected.

28. Formula for Bandwidth


Bandwidth for either series or parallel resonant circuits
is the range of frequencies between the upper and lower
cutoff frequencies for which the response curve (I or Z)
is 0.707 of the maximum value
Ideally the center frequency is:

29. Half Power Point
 The 70.7% cutoff point is also referred to as:
• The Half Power Point
• -3dB Point

30. Half Power Point
 
   
      max
2
max
2
max
2
2
/
1
2
max
2
2
/
1
2
/
1
2
max
max
5
,
0
5
,
0
707
,
0
707
,
0
P
R
I
R
I
P
R
I
R
I
P
R
I
P
f
f
f
f
f
f
















31. Half Power Point


















in
out
in
out
P
P
dB
V
V
dB
log
10
log
20
  dB
V
V
3
707
,
0
log
20
707
,
0
log
20
max
max










 

32. Selectivity



Selectivity defines how well a
resonant circuit responds to a
certain frequency and
discriminates against all other
frequencies
The narrower the bandwidth
steeper the slope, the greater
the selectivity
This is related to the Quality
(Q) Factor (performance) of
the inductor at resonance. A
higher Q Factor produces a
tighter bandwidth

33. Selectivity
W
L
L
L
R
R
X
R
X
R
I
X
I
Q
Q








2
2
power
true
power
reactive
Dissipated
Energy
Strored
Energy
How Q Affects Bandwitdh
Q
f
BW
r

Note: XL
at the resonant frequency

34. A Basic Series Resonant Bandstop
Filter
A basic series resonant band-stop
Filter Circuit
Generalized response curve for a
band-stop filter

35. Frequency Response of Series
Resonant Bandstop Filter
E x a m p l e o f t h e
frequency response of
a series resonant band
-stop filter with Vin at a
constant 10 V rms.
T h e w i n d i n g
resistance is neglected.

36. Parallel RLC Circuits
 A parallel resonant circuit
stores energy in the magnetic
f ie l d of t he coi l a nd t he
electric f i
eld of the capacitor.
The energy is transferred back
and forth between the coil
and capacitor

37. Parallel RLC Circuits













G
B
Y
Z
B
G
Y
B
B
B
tot
tot
C
L
tot
1
2
2
tan
1

   












R
CL
CL
R
tot
I
I
I
I
I
1
2
2
tan

1.
2.
3.
Where :
B : Susceptance
Y : Admitance
G : Conductance

38. Current Across the Parallel
Combination of L and C












R
CL
CL
R
tot
I
I
I
I
I
1
2
2
tan

L
C
CL
I
I
I 

Where :

39. Current Across the Parallel Combination of
L and C

40. Conversion of Series-Parallel to
Parallel
 
 
1
1
2
2
2









 

Q
R
R
Q
Q
L
L
W
eq
p
eq
  mH
mH
mH
L
L
L
eq
eq
1
.
10
01
.
1
10
10
1
10
10
2
2









 


For Q>=10

41. Parallel Resonant Circuits



For parallel resonant circuits, the impedance is maximum (in theory,
infinite) at the resonant frequency
Total current is minimum at the resonant frequency
Bandwidth is the same as for the series resonant circuit; the critical
frequency impedances are at 0.707Zmax
A basic parallel resonant band-pass filter

42. Parallel Resonant Circuits



For parallel resonant circuits, the impedance is maximum (in theory,
infinite) at the resonant frequency
Total current is minimum at the resonant frequency
Bandwidth is the same as for the series resonant circuit; the critical
frequency impedances are at 0.707Zmax
Generalized frequency response curves for a parallel resonant band-pass filter

43. Parallel Resonant Circuits
Example of the
response of a
parallel resonant
band-pass filter
with the input
voltage at a
constant 10 V rms

44. Parallel Resonant Circuits
A basic parallel resonant band-stop filter

45. Applications
 A simplified portion of a TV receiver showing filter usage

46. Applications
 A simplified diagram of a superheterodyne AM radio broadcast
receiver showing the application of tuned resonant circuits