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Problem Solving
Part 2
Resonance
Parallel and series circuits do not behave
the same way at and around resonance.
However, the method for determining their
resonant frequency is the same.
Resonant Frequency
We calculate the resonant frequency of
the circuit in Fig. 1-A.
fr 

1
2 LC
1

6
6.28 (0.150 H )(0.05 X 10 F )
1
6
Hz
6.28 0.0075 X 10
1

Hz
3
(6.28)(0.866 X 10 )
 1.839 X 10 Hz  1839 Hz
3
A parallel circuit at resonance has the following
characteristics:
• A phase shift of 00
• An impedance equal to resistance
• A resistor current that equals the total current
• Equal and opposite currents through the
inductive and capacitive branches
• Maximum impedance and minimum current
Effects of Resonance
The best way to consider the effects of
resonance on a parallel circuit is to
calculate all values and construct a current
graph. We first calculate the reactance.
X L  2fL
3
 (6.28)(1839 Hz )(150 X 10 H )
 1732
1
XC 
2fC
1

6
(6.28)(1839 Hz )( 0.05 X 10 F )
3
1X 10


0.5774
 1732
The calculations show that one
characteristic of resonance exists:
Inductive reactance equals capacitive
reactance.
We now calculate current. Since both
reactances are equal, both currents will be
equal. However, the phase angles will be
in opposition.
ET
IL 
XL
80

0
173290 
0
 0.00462  90 A
0
ET
IC 
XC
80 V

0
1732  90 
0
 0.0046290 A
0
I XT  I L  I C
 (0.00462  90 A)  (0.0046290 A)
0
 0A
0
ET
IR 
R
0
80 V

0
22000 
0
 0.003640 A
I T  I R  I XT A
2
2
 0.00364  0 A
2
2
 0.00001325 A
 0.00364 A
As shown in Fig. 1-3, the currents of the two
reactors are equal and opposite, so their
vector sum is zero. The net total current in
the circuit is that in the resistor. Current
flows through one inductor in one direction
while flowing through the other inductor in
the opposite direction. At resonance,
energy transfers back and forth between
the two inductors, with any energy loss
replaced by the voltage source.
In a parallel resonance, the total current
equals the resistor current.
IC= 0.00462 A
IXT= 0 A
IR = IT = 0.00364 A
IL = 0.00462 A
Fig. 1-3. Circuit Currents at Resonance
The current through the inductor lags
the applied voltage by 900, while the
capacitor current leads it by 900. The
inductor and capacitor branch
currents are 1800 out of phase with
each other. When one has a great
supply of electrons, the other has a
demand, and vice versa. The net
result is that once resonance occurs,
these two reactors transfer electrons
back and forth. Refer to Fig. 1-4.
1
IX
IR
!X
2
Fig. 1-4. Current Paths at Resonance
Conditions that need to exist before
resonance can occur:
• The reactances must be equal for the
timing of their fields to be the same.
• The frequency of the applied source must
be the same as the resonant frequency for
oscillation to occur.
We now calculate impedance. Since little
energy is taken from the source at
resonance, we expect impedance to be
high. The impedance of a parallel RLC
circuit is at its highest level during
resonance. The phase angle at
resonance is 00.
ET
Z
IT
80 V

0
0.003640 A
0
 21980 
0
The frequency characteristics of the circuit
(Fig. 1-A) which we have evaluated at
frequencies above, below and at
resonance are summarized in the graph in
Fig. 1-5.
38.3
0
-55.3
2198
1724
1252
0.00639
0,00464
0.00364
Fig. 1-5. Relationships in a Parallel RLC Circuit
Quality Factor and Bandwidth
As with series circuits, the quality factor Q is
an indication of the relationship between
reactance and resistance in an RLC
parallel circuit. Q for a parallel circuit is
determined by the reactance of the
inductor and its internal resistance.
Refer to Fig. 1-6 where RL represents the
100-ohm internal resistance of the
inductor, while XL is its 2400-ohm
reactance. XL and XC are equal since
the circuit is resonant. Q is calculated
as:
X L 2400
Q

 24
RL
100
4
2400 
18V
1200 Hz
2400 
3
100 
2
Fig. 1-6. Coil Resistance
Bandwidth (BW) is inversely related to Q and
gives an indication of the frequency
response of a circuit. High-Q circuits have
a narrow bandwidth, while low-Q circuits
have a wide bandwidth. The bandwidth for
the circuit is:
f r 1200 Hz
BW  
 50 Hz
Q
24
The half-power points of a circuit are
the two frequencies where the voltage
drops to 0.707 times the peak, or
resonant frequency, value. Since
voltage causes current in a circuit, the
current also drops to 0.707 of its peak
value. Since power equals voltage
times current, then the power is now
0.5 of what it was at resonance. The
phase shift is normally 450 at the halfpower points.
The half-power points for this circuit are:
(1200 Hz) – (50/2 Hz) = 1175 Hz
(1200 Hz) + (50/2 Hz) = 1225 Hz
If a 100-ohm resistor RS was placed in series with
the 100 ohms of internal resistance that this
inductor has, Q and BW would change, as
follows:
XL
2400
Q

 12
RL  RS
200
f r 1200 Hz
BW 

 100 Hz
Q
12
The effects of this change can be seen in
Fig. 1-7. A lower Q results in a broader
bandwidth. Values of Q can range from 20
to 100 in a typical RLC circuit.
1200 Hz
Q = 24
Q = 12
1175
1 250
1175
1 225
Fig. 1-7. Circuit Q and BW
Uses for Parallel Resonant Circuits
The most common uses for parallel resonant
circuits are in radio and television
equipment. Their ability to discriminate
among different frequencies makes them
useful in the signal selection and rejection
process. Inductors and capacitors can be
placed in parallel to form a network that
allows most frequencies to pass except for
those that are close to the resonant
frequency.
Refer to Fig. 1-8, which shows a transformer
that is tuned to resonate at a particular
frequency.
7
6
Signal In
Signal Out
8
3
Fig. 1-8. Adjustable Transformer in Radio Circuit
A complex signal of many frequencies is
applied to the transformer primary. The
LC circuit on the primary side resonates
only at the desired frequency, with an
amplitude proportional to the size of the
desired portion of the incoming signal.
Transformer action causes this signal to
be reproduced at the secondary side of
the transformer.
The desired signal is then developed across
the high impedance of the secondary
parallel resonant circuit. Undesired
signals at frequencies beyond the halfpower points of the bandwidth are further
suppressed. This selection-discrimination
process allows a radio or television circuit
to select the desired station and block all
others.
Evaluating RLC Circuits
Three parameters are of interest in testing
RLC circuits:
• Resonant frequency
• Impedance
• bandwidth
Series Connection
Real inductors have internal series resistance
because of their windings.
Real capacitors have internal parallel resistance
because of their leakage.
Capacitor leads have inductance.
Inductor leads have capacitance.
The three components in an actual series RLC
circuit do not each have only on form of
opposition but each component has complex
oppositions.
The circuit we will evaluate is shown in Fig. 1-9.
The objective is to determine the resonant
frequency, impedance, and bandwidth.
4
L1
2.5mH
1
C1
0050uF
5
R1
47k
2
Fig. 1-9. Series RLC Circuit for Evaluation
The following results will be obtained if exact
values were used and if the components were
ideal.
1
fr 
2 LC
1

6.28 ( 2.5 X 10 3 H )( 0.05 X 10  6 F )

1
8
Hz
6.28 0.0125 X 10
1

Hz
4
(6.28)( 0.118 X 10
 14,243Hz
ET ET
IT 

Z
R
10V

 0.2128 A
47
XL
Q
R
(6.28)(14.243 X 103 Hz )( 2.5 X 10 3 H )

47
223.6

 4.757
47
f r 14,243Hz
BW 

 2,944 Hz
Q
4.757
The half-power points indicating the bandwidth
limits are:
(14,283 Hz) - (2944/2 Hz) = 12,771 Hz
(14,243 Hz) + (2944/2 Hz) = 15,715 Hz
Bandwidth and impedance are affected by
resistance, and the resistor is not the only
source of resistance. Inductor winding
resistance will be a significant factor. Do
not exceed the current capacity of the
inductor because excess current will
saturate an inductor and cause incorrect
results.
Impedance can be determined by dividing the
measured value of total voltage by the
measured value of total current. If you know
the actual value of the resistor you can use
resistor voltage to calculate circuit current.
The best way to determine resonance is by
measuring the resistor voltage which is
equal to its maximum at the resonant
frequency.
ET
ET
Z

IT ER / R
Bandwidth can also be determined by
measuring the resistor voltage.
BW limits at ER = (0.707)ERmax
Parallel Connection
In evaluating an RLC circuit for the parallel
connection, the resistance branch should be
larger, and a series resistance should be added
to limit inductor current.
Fig. 1-10. Parallel RLC Circuit for Evaluation
The resonant frequency of the circuit should
be the same as the resonant frequency of
the series circuit. The impedance of the
parallel section of this circuit (if ideal)
would be equal to the resistive branch of
1000 ohms.
One way to determine resonance is to
measure the parallel circuit voltage, the
voltage across the parallel network. The
network voltage is 0.707 times the resonant
value at the upper and lower half-power
points.