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FOWLER CHAPTER 13
LECTURE 13 RCL CIRCUITS
CHAPTER 13 COMBINED RESISTANCE, INDUCTANCE AND CAPACITANCE(RCL) CIRCUITS
IMPEDANCE (Z): COMBINED OPPOSITION TO RESISTANCE AND REACTANCE.
MEASURED IN OHMS.
ALL 3 RCL COMPOENTS ARE IN SERIES
ELI THE ICEMAN
FOR CAPACITORS: CURRENT LEADS VOLTAGE
FOR INDUCTORS: VOLTAGE LEADS CURRENT
FOR SERIES RCL CIRCUIT CURRENT IS THE SAME IN EACH COMPOENT
VOLTAGE IS ALWAYS OUT OF PHASE IN EACH COMPOENT.
ANOTHER WAY TO REPRESENT COMPLEX WAVEFORMS IS BY THE USE OF VECTORS
SERIES RCL CIRCUITS
1. CURRENT IN RCL CIRCUITS.
CURRENT FLOW IN ALL PARTS OF THIS CIRCUIT ARE THE SAME AND IN PHASE.
IT  I R  I L  I C
2. VOLTAGE IN RCL CIRCUITS
R
VR
0
I
VL 90
VS
FOR INDUCTORS: VOLTAGE LEADS CURRENT
L
0
C
VR  I  R
0
VC  90 
VL  I  X L
I
X L  6.28 fL
I
FOR CAPACITORS: CURRENT LEADS VOLTAGE
VC  I  X C
XC 
1
6.28 fC
THESE V/I PHASE DIAGRAMS ARE DIFFICULT TO FOLLOW, LETS LOOK AT THIS IN ANOTHER LIGHT.
33
R
FIND TOTAL Z
FIRST FIND X L AND X C
VS
X L  6.28 fL  6.28  60 Hz  20mH  7.5
115V
60 Hz
XC 
THINK IN TERMS OF
VECTORS OR PHASORS.
L
20mH
C
10 F
1
1

 265.3
6.28 fC 6.28  60 Hz 10 F
7.5
XL
33 R
33 R
33 R
X  X L  XC
X  7.5  265.3
X  257.8
X 257.8
X C 265.3
X 257.8
Z
Z
USE PYTHAGOREAN THEOREM TO FIND THE INPEDANCE Z OF THIS CIRCUIT
R 33
Z 2  R2  X 2
OR
Z  R  X  R  X L  X C 
2
X 257.8
2
2
2
FOR OUR EXAMPLE
Z
Z  332   257.8
2
X  X L  XC
X  7.5  265.3
X  257.8
Z  260
SINCE I T  I R  I C  I L
USING OHM ' S LAW
FIND VT, SINCE
R
33
115V
60 Hz
L
20mH
C
10 F
VS 115V

 0.44 A  440mA
Z 260
VT  VR  VL  VC
NONE OF THE VOLTAGES ARE IN PHASE.
MUST BE ADDED AS VECTORS.
VR  IT R
VS
IT 
VC  I T X C
VL  IT X L
VR  IT R  0.44 A  33  14.52V
VL  IT X L  .44 A 7.5  3.3V
VC  IT X C  .44 A  265.3  116.73V
MUST USE PHASORS TO FINE VT FOR THIS SERIES CIRCUIT
VL  3.3V
VR  14.52
VR  14.52
VR  14.52

VC  116.73
V  VL  VC
V  3.3V  116.73V
V  113.43V
V

VT
VT  V 2 R  VT
VS  113.43
VT
2
VT  V 2 R  VL  VC 
2
VT  14.522   113.43
2
VT  225  12,996
VT  115V
PARALLEL RCL CIRCUITS
VOLTAGE ACROSS ANY PARALLEL CIRCUIT ELEMENT WILL BE THE SAME AND IN PHASE. SO;
VS  VR  VL  VC
R
VS
33
L
115V
60 Hz
20mH
I L LAGS VL BY 90
TO FIND IT , SOLVE FOR I R , I L, I C
FIND X L AND X C FIRST
X L  6.28 fL  6.28  60 Hz  20mH  7.5
XC 
C
1
1

 265.3
6.28 fC 6.28  60 Hz 10F
10 F
I C LEADS VC BY 90
FIND I R , I L , I C USING OHM ' S LAW
V 115V
IR  
 3.5 A
R 33
V
115V
IL 

 15.3 A
X L 7.5
V
115V
IC 

 0.43 A
X C 265.3
FIND IT, AGAIN SINCE IR,IL, IC ARE ALL OUT OF PHASE
MUST USE VECTORS TO FIND A SOLUTION.
I C  0.43 A
I R  3.5 A
I R  3.5 A


IT
I L  15.3 A
I R  3.5 A
I  14.9 A
IT
I  14.9 A
CAPACITIVE CURRENT
I  IC I L
RESISTIVE CURRENT
I T  I 2 R  I C  I L 
INDUCTIVE CURRENT
I T  3.5 A2  14.9 A2
COMBINED INDUCTIVE
AND CAPACITIVE CURRENT
2
I T  15.3 A
TO FIND THE TOTAL IMPEDANCE FOR THIS CIRCUIT USING OHM’S LAW
Z
VS 115V

 7.5
I T 15.3 A
FOR A CIRCUIT WITH INDUCTANCE
FOR A CIRCUIT WITH CAPACITANCE
RESONANCE P.347
RESONANT OCCURS WHEN
X L  XC
CAN OCCUR IN SERIES OR PARALLEL CIRCUITS WITH RCL OR LC COMPOENTS.
FOR ANY VALVE OF L AND C THERE IS ONLY ONE FREQUENCY WHERE,
X L  XC
fR 
THIS IS CALLED THE RESONANT FREQUENCY:
1
6.28  LC
DO EX. 13-11 p.348
fR
SERIES RESONANT CIRCUITS F.13-26
AT RESONANT
X L  X C ALSO
VL  VC
PARALLEL RESONANT CIRCUITS P.348
X L  X C ALSO
I L  IC
For resonance to occur in any circuit it must have at least one inductor and one
capacitor.
Resonance is the result of oscillations in a circuit as stored energy is passed from
the inductor to the capacitor.
Resonance occurs when XL = XC
At resonance the impedance of the circuit is equal to the resistance value as Z = R.
At low frequencies the circuit is capacitive as XC > XL.
At low frequencies the circuit is inductive as XL > XC.
The high value of current at resonance produces very high values of voltage across
the inductor and capacitor.
Series resonance circuits are useful for constructing highly frequency selective
filters. However, its high current and very high component voltage values can cause
damage to the circuit.
Resonant Circuits
XL3
XL1
XC3
XL2
XC1
XC2
Frequency
fr
Resonance occurs when XL equals XC.
There is only one resonant frequency for each LC combination.
However, an infinite number of LC combinations have the same fr .
PARALLEL RESONANT TANK CIRCUIT
THIS CIRCUIT WOULD PRODUCE A SINE WAVE FOREVER IF L AND C
WERE IDEAL COMPOENTS.
WITH REAL WORLD L AND C THE WAVEFORM WILL
DAMP OUT WITH TIME. YOU MUST FEED ENERGY
INTO THE TANK CIRCUIT TO KEEP THE SINE WAVE
PROPOGATING.
BANDWIDTH : RANGE OF f OF A CIRCUIT WHICH
PROVIDES 70.7% OR MORE OF THE MAX. RESPONSE.
Bandwidth, (BW) is the range of frequencies over which at
least half of the maximum power and current is provided
The selectivity of a circuit is dependent upon
the amount of resistance in the circuit. The
variations on a series resonant circuit are
drawn below. The smaller the resistance, the
higher the "Q" for given values of L and C.
The parallel resonant circuit is more
commonly used in electronics, but the
algebra necessary to characterize the
resonance is much more involved.
SERIES LC CIRCUIT
RESPONSE CURVE FOR LC CIRCUIT
ARE PLOTS OF EITHER VOLTAGE, CURRENT OR
INPEDANCE vs. FREQUENCY ABOVE AND BELOW
RESONANCE
Series Resonance
The resonance of a series RLC circuit occurs when the inductive and capacitive
reactance are equal in magnitude but cancel each other because they are 180
degrees apart in phase. The sharp minimum in impedance which occurs is useful in
tuning applications. The sharpness of the minimum depends on the value of R and
is characterized by the "Q" of the circuit.
XL
X L  XC  0
XC
An example of the application of resonant circuits is the selection of AM radio stations by the radio
receiver. The selectivity of the tuning must be high enough to discriminate strongly against stations
above and below in carrier frequency.
FILTERS: USE RC, RL, LC, AND RCL CIRCUITS TO FILTER ONE GROUP OF FREQUENCIES
FROM ANOTHER GROUP OF FREQUENCIES.
4 CLASSES OF FILTERS
1.LOW PASS
2.HIGH PASS
3.BAND PASS
4.BAND-REJECT 0R BAND STOP,
YOU TUBE: Passive RC low pass filters
http://www.youtube.com/watch?v=OBM5T5_kgdI
YOU TUBE: Passive RC high pass filters http://www.youtube.com/watch?v=4CcIFycCnxU
LOW PASS FILTER
THE HIGH PASS FILTER
BAND PASS FILTER