LCR CIRCUITS
Circuits containing an inductor L, a capacitor C , and a resistor R, have special characteristics useful in many applications. Their frequency characteristics (impedance, voltage, or current vs. frequency) have a sharp maximum or minimum at certain frequencies. These circuits can hence be used for selecting or rejecting specific frequencies and are also called tuning circuits. These circuits are therefore very important in the operation of television receivers, radio receivers, and transmitters. In this section, we will present two types of LCR circuits, viz., series and parallel, and also discuss the formulae applicable for typical resonant circuits. A series LCR circuit includes a series combination of an inductor, resistor and capacitor whereas; a parallel LCR circuit contains a parallel combination of inductor and capacitor with the resistance placed in series with the inductor. Both series and parallel resonant circuits may be found in radio receivers and transmitters. The selectivity of a tuned circuit is its ability to select a signal at the resonant frequency and reject other signals that are close to this frequency. A measure of the selectivity is Q, or the quality factor . The study of these circuits is basically an application of alternating current circuit analysis. We make use of the complex number notation with sinusoidal varying quantities like alternating voltage and current. In general, the impedance Z is a sum of the real part called resistance R and the complex part called the reactance X , i.e., Z = R + jX . The magnitude and phase of the impedance are given by ⎛ X ⎞ and φ = tan −1 ⎜ ⎟ , respectively. ⎝ R ⎠
2 2 R + X
Since in an inductor, voltage leads the current by π/2, the reactance of L is j L , while in case of a capacitor, voltage lags behind the current by π/2, the reactance of C is 1 . If the current in the circuit is I , the relative voltage drops across the jω C inductor, capacitor and resistor can be represented in the phasor diagram as shown in Figure 1. We will study the property of resonance in context of series as well as parallel configurations of LCR circuit. It is a very useful property of reactive a.c. circuits and is employed in a variety of applications. One of the common applications of resonance effect is in radio and television transmissions, e.g., tuning a radio to a particular station by selecting a desired frequency (or band of frequencies). The series resonant circuit can be used for voltage magnification. A parallel resonant circuit provides current magnification and can be used in induction heating. Another
application of resonant circuit is screening certain frequencies out of a mix of different frequencies with the help of circuits called filters.
Figure 1: Phasor diagram
Learning Outcomes After performing this experiment you will be able to 1. 2. 3. 4. 5. 6.
explain why the series LCR circuit is called an acceptor circuit study the response of LCR circuit by varying the resistance in the circuit present graphically the variation of current with frequency in a series LCR circuit find the resonant frequency for a series LCR circuit and hence find the Quality factor explain why the parallel LCR circuit is called a rejector circuit. present graphically the variation of current with frequency in a parallel LCR circuit and hence find the anti-resonant frequency and Quality factor
1
SECTION A
LCR Series circuit
Let us consider the LCR circuit, which consists of an inductor, L, a capacitor, C , and a resistor, R, all connected in series with a source as shown in Figure 2. We will first derive the condition of resonance and then explain the methods of determination of the resonant frequency and hence the Quality factor.
Apparatus • • • • • • •
Function generator an inductance coil three capacitors a resistance box a.c. voltmeters / multimeter / Cathode Ray Oscilloscope (CRO) one a.c. milliammeter Connecting wires.
Theory Let an alternating voltage V 0sin t or V 0 e jω t be appl applie ied d to an indu inducto ctorr L , a resis resisto tor r R and a capacitor C all in series as shown in Figure 2. If I is the instantaneous current flowing through the circuit, the applied voltage in phasor form is given by I V = V R + V L + V C = RI + jω LI + jω C
⎡ 1 ⎤ = ⎢ R + jω L + ⎥ I ω j C ⎣ ⎦ ⎡ 1 ⎞⎤ ⎛ = ⎢ R + j ⎜ ω L − ⎟⎥ I ω C ⎝ ⎠⎦ ⎣ The impedance Z =
1 ⎞ ⎛ = R + j⎜ ω L − ⎟ ω C ⎠ I ⎝ V
2
1
If
we
write Z = Ze iφ = Z cos φ + jZ sin φ ,
then
2 2 ⎡ 2 ⎛ 1 ⎞ ⎤ Z = ⎢ R + ⎜ ω L − ⎟ ⎥ and ω C ⎝ ⎠ ⎥⎦ ⎢⎣
1 ⎞ ⎛ ⎜ ω L − ⎟ ω C ⎠ ⎝ tan φ = . R
Therefore, current. I =
V 0 e
jω t
Ze
jφ
=
V 0 Z
e
j (ω t −φ )
Figure 2: Series LCR circuit
Three cases thus arise: 1 1. ω L > , tan φ is positive and applied voltage leads current by phase angle φ . ω C 1 2. ω L < , tan φ is negative and applied voltage lags behind current by φ . ω C 1 3. , tan φ is zero zero and and applied applied voltage voltage and and current current are in phase. phase. This This ω L = ω C condition is known as resonance and frequency as resonant frequency ( 0 ). ω L =
1 ω C
or
3
⇒ ω 2 =
1 LC
ω = ω 0 =
1 LC
or f 0 =
ω L −
ω 0 2π
1 ω C
1
=
(1)
2π LC
= 0 and V L = V C
If L, R and υ (frequency of function generator) are fixed and the capacitance is I varied, then for lower values of C, > ω LI or V C > V L . As the capacitance is ω C increased in the circuit, the situation called resonance is achieved when V C = V L . If C is increased further, V C will decrease and we have V C < V L . The point of intersection of V C and V L versus 1
curves will give resonance condition. This is depicted in C Figure 3. At resonance V R is a maximum while V LC is minimum as shown in Figure 4. Corresponding to maximum value of V R , C is obtained. Similarly, for minimum
value of V LC , C is obtained. This value of C makes the given circuit resonant at the supply frequency with constant values of L and R.
Figure 3: Variation of V L and VC with 1
4
C
Figure 4: Variation of V LC and VR with 1
C
Theoretically at resonance V LC should be zero. This should be so if the inductor is of negligible resistance and there are no other losses. The minimum value of V LC is a measure of the effective resistance of inductor coil which is equal to the d.c. resistance plus a.c. resistance corresponding to iron and hysteresis losses. At resona resonant nt freque frequency ncy f 0 , the imped impedance ance of of circuit circuit is minimu minimum. m. Hence Hence frequen frequencies cies near f 0 are passed more readily than the other frequencies by the circuit. Due to this reason LCR-series circuit is called acceptor circuit. The band of frequencies which is allowed to pass readily is called pass-band. The band is arbitrarily chosen to be the I range of frequencies between which the current is equal to or greater than 0 . Let 2 f 1 and f 2 be these limiting values of frequency. Then the width of the band is (refer to Figure 5) BW = f 2 − f 1 . (2) The Quality factor is defined in the same way as for a mechanical oscillator and is given by f 0 resonant frequency . (3) = Q= bandwidth f 2 − f 1 Q-factor is also defined in terms of reactance and resistance of the circuit at resonance, i.e.,
5
Q=
X L R
=
L
0
R
.
I 0
0.7 I 0
I
BW
2
Figure 5: Bandwidth for a series LCR resonant circuit
Figure 6: Variation of current with frequency for different R values
Also,
6
(4)
Q=
X C R
=
1 ω 0 CR
.
(5)
The resonance condition is also evident from the resonance curves or the graphs V between I R = R and f for different values of R shown in Figure 6. The R bandwidth as well as Q-factor can be calculated.
Pre-lab Assessment Choose the correct answer
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Which of the following are applications of resonant circuits? a) radio and television transmission b) voltage magnification and current magnification c) induction heating d) all of the above. The real part of impedance is called a) resistance b) inductive reactance c) capacitive reactance d) none of the above. The imaginary part of impedance is called a) resistance b) inductive reactance c) capacitive reactance d) reactance. Capacitors and inductors oppose an alternating current. This is known as a) resistance b) resonance c) reactance d) impedance In the case of an inductor a) voltage leads the current by π/2 b) voltage lags behind the current by π/2 c) current leads the voltage by π/2 d) voltage leads the current by π. In the case of a capacitor a) voltage leads the current by π/2 b) voltage lags behind the current by π/2 c) current leads the voltage by π/2 d) voltage lags behind the current by π. The reactance of a capacitor increases as the: a) frequency increases b) frequency decreases c) applied voltage increases d) applied voltage decreases
7
(8)
(9)
The reactance of an inductor increases as the: a) frequency increases b) frequency decreases c) applied voltage increases d) applied voltage decreases The minimum value of V LC is a measure of the effective resistance of inductor coil. (True/False)
Answer the following question
(10) Why is a series LCR resonant circuit called an acceptor circuit?
Procedure 1. 2. 3. 4. 5. 6.
Connect the circuit as shown in Figure 2. Switch on the a.c. source and set its output voltage V i to a value (say, 3V rms) and frequency to a known value. Record the voltages across the known resistor, capacitor, inductor and the series combination of the inductor and capacitor in Table 1. Repeat step 3 for different values of C. Increase the frequency gradually in steps and record the voltage across resistor in Table2. Repeat step 5 for two different R values.
Observations Table 1: Variation of various voltages with 1 f = ……Hz,
S. No.
C
R = …… Ω, L = …… mH
1 C -1/2
V C C (volts)
V R (volts)
V L (volts)
V LC (volts)
(μF) 1 2 3 4 5 6
Table 2: Variation of voltage across resistor with frequency for different values L = …… mH, C = …… μF
8
R
S. No.
Frequency f (Hz)
V R1
V R2
V R3
(volts)
(volts)
(volts)
I 1 =
V R1
I 2 =
R1
(mA)
V R2 R2
(mA)
I 3 =
V R3 R3
(mA)
1 2 3 4 5 6
Precautions • • •
The connecting wires should be straight and short. If the amplitude of the output voltage of the oscillator changes with frequency, it must be adjusted. The values of inductance and capacitance are so selected that the natural frequency of the circuit lies almost in the middle of the available frequency range.
Calculations Plot the following graphs: 1.
Graph no.1: VL and VC on y-axis and 1
2.
Graph no. 2: V R and VLC on y-axis and 1
C
on x-axis
3.
on x-axis C Graph no. 3: Current I on y-axis and frequency f on x-axis for different sets sets R corresponding to different values of R
•
The point of intersection of V C and V L versus 1
C
curves in graph no. 1
gives C = …… μF. The point where V R is a maximum in graph no. 2 gives C = …… μF.
• • • • •
Mean value of C = …… μF. The minima of the curves in graph no. 3 give f 0 = …… Hz. Theoretical value of f f 0 (using Equation (1)) =............. Hz.
• • •
Bandwidth for R 1 (using Equation (2)), f 2 - f 1 = …… Hz. Quality factor for R R1 (using Equation (3)), Q = ……. Theoretical value of Q for R R1 (using Equation (4) or (5)) = …….
• •
Bandwidth for R R2, f 2 - f 1 = …… Hz. Quality factor for R R2, Q = …….
The point where V LC is minimum in graph no. 2 gives C = …… μF.
.
9
•
R2 = ……. Theoretical value of Q for R
• • •
Bandwidth for R R3, f 2 - f 1 = …… Hz. R3, Q = ……. Quality factor for R Theoretical value of Q for R R3 = …….
.
Result The value of C which makes the given circuit resonant at the supply frequency with given values of L L and R is C = …… μF. The resonant frequency f 0 with L = …… mH and C = …… μF is …… Hz Theoretical value of f f 0 = …… Hz % Error = ……. A comparison of the experimental and theoretical Quality factor for L = …… mH and C = …… μF for different R values is given below in tabular form:
S. No. 1 2 3
R (Ω)
Experimental Q-factor
Theoretical Q-factor
Post-lab Assessment Choose the correct answer
(1)
An inductor and a capacitor are connected in series. At the resonant resonant frequency the resulting impedance is a) maximum b) minimum c) totally reactive d) totally inductive (2) An inductor and a capacitor form a series resonant circuit. The capacitor capacitor value is increased by four times. The resonant frequency will a) increase by four times b) double c) decrease to half d) decrease to one quarter (3) An inductor and a capacitor form a series resonant circuit. If the value of the inductor is decreased by a factor of four, the resonant frequency will a) increase by a factor of four b) increase by a factor of two
10
c) decrease by a factor of two d) decrease by a factor of four (4) The resonant resonant frequency for a series LCR LCR circuit with L = 100 mH, C = 0.01 μF is approximately a) 250 Hz b) 255 Hz c) 5033 Hz d) 5000 KHz (5) The point of intersection of V C and V LC versus 1 curves will give resonance C condition. (True/False) (6) In the V R and V LC versus 1 curves, V R is a maximum while V LC is C minimum at resonance. (True/False) (7) A "high Q" resonant circuit is one which a) has a wide bandwidth b) is highly selective c) uses a high value inductance d) uses a high value capacitance Answer the following question
(8)
When are the voltage and current in a series LCR circuit in phase? phase?
Answers to Pre-lab Assessment 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
d a d c a b b a True At resona resonant nt frequen frequency cy f 0 , the impedan impedance ce of circuit circuit is minimu minimum. m. Hence Hence frequencies frequencies near f 0 are passed more readily than the other frequencies frequencies by the circuit. Due to this reason LCR-series circuit is called acceptor circuit.
Answers to Post-lab Questions 1. 2. 3. 4. 5.
b b c b c
11
6. 7. 8.
False True When inductive and capacitive reactances are equal.
12
SECTION B
LCR parallel circuit
The parallel resonant circuit obeys the same formula for resonant frequency as the series resonant one, but at resonance the parallel resonant circuit has very high impedance. The resistance at resonance offered by the parallel resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance. We now discuss how a series LCR circuit is different than a parallel LCR circuit. The condition of resonance in this case is known as anti-resonance. We will derive the condition of anti-resonance of a parallel LCR circuit. The laboratory method of determination of the anti-resonant frequency and hence the Quality factor is explained.
Apparatus • • • • • • •
An audio oscillator an inductance coil three capacitors a resistance box a.c. voltmeters / multimeter / Cathode Ray Oscilloscope (CRO) one a.c. milliammeter Connecting wires
Theory Consider a circuit containing an inductor L and a capacitor C connected in parallel to an a.c. source (Figure 7). The resistance R is connected in series with the inductor L and includes its resistance. The total admittance of the LCR combination is given by 1 1 1
=
Z X C
+
X L + R
Therefore 1 Z
=
1
+
1
1 jω C jω L + R
= jω C +
13
R − jω L R 2 + ω 2 L2
=
ω L ⎡ ⎤ + j ⎢ω C − 2 2 2⎥ R + ω L R + ω L ⎦ ⎣ R
2
2
2
R
V 0 sin ω t
V R
C
L
Figure 7: Parallel LCR circuit
For the condition of resonance, current and voltage are in phase and the coefficient of j, i.e., the reactive term which brings about a phase change is zero, hence L ω 0C − 2 0 2 2 = 0 R + ω 0 L 2π f 0 C =
2π f 0 L R 2 + 4π 2 f 02 L2
which gives f 0 =
1
1
−
R 2
2π LC L2 At resonance, the impedance of the circuit is maximum and is given by R 2 + L2ω 02 Z = R
(6)
L2 ⎛ 1
R 2 ⎞ ⎜ = R + ⎜ − 2 ⎟⎟ R ⎝ LC L ⎠
or Z =
L
RC The impedance at resonance is called dynamic resistance . The current I = V / Z has minimum value (Figure 8). It is for this reason that the condition of resonance for a
14
parallel LCR circuit is known as anti-resonance and the corresponding frequency as the anti-resonance frequency.
R1 > R2 > R
R R2 R1
I
Figure 8: Variation of current with frequency for different R values
The shape of the impedance versus frequency curve in a parallel LCR circuit is the same as the shape of the current versus frequency curve in a series LCR circuit. In other words, the circuit has very high impedance at the anti-resonant frequency. The parallel tuned circuit is used to select one particular signal frequency from among others. It does this by rejecting the resonant frequency because of its high impedance. This is the reason why this type of circuit is also known as a rejector circuit. The circuit is more selective if it offers high impedance at resonance and much lower impedance at other frequencies. The Q-factor is defined in the same way as for a series LCR circuit. As in series circuit, Q can also be written as Q=
ω 0 L R
=
1 ω 0CR
=
1 L R
C
Pre-lab Assessment Choose the correct answer
(1)
An inductor and a capacitor are connected in parallel. At the resonant frequency the resulting impedance is a) maximum b) minimum c) totally reactive d) totally inductive
15
(7)
(2)
The anti-resonant frequency for a parallel LCR circuit does not depend on the value of resistance used in the circuit. (True/False)
(3)
A parallel LCR circuit is more selective if it offers a) high impedance at resonance b) low impedance at resonance c) high impedance at frequencies other than the resonant frequency d) b and c.
Answer the following questions
(4) (5) (6)
What is the impedance at anti-resonance for a parallel LCR circuit? Why is the condition of resonance for a parallel LCR circuit known as antiresonance? Why is a parallel LCR resonant circuit called a rejector circuit?
Procedure 1. 2.
Connect the circuit as shown in Figure 6. Switch on the a.c. source and set its output voltage V i to a value, say, 3V rms and frequency to a known value. Increase the frequency gradually in steps and record the voltage across resistor in Table 2. Repeat step 3 for two different R values.
3. 4.
Observations Table 3: Variation of voltage across resistor with frequency for different values
R
L = …… mH, C = …… μF
S. No.
Frequency f (Hz)
V R1
V R2
V R3
(volts)
(volts)
(volts)
I 1 =
V R1 R1
(mA) 1 2 3 4 5 6
16
I 2 =
V R2 R2
(mA)
I 3 =
V R3 R3
(mA)
Calculations Plot a graph with current I on y-axis and frequency f on x-axis for different sets corresponding to different values of R R.
• • • • •
The maxima of the curve for R R1 in the graph gives f 0 = …… Hz. Theoretical value of f f 0 for R R1 (using Equation (6)) =............. Hz. Bandwidth for R R1 (using Equation (2)), f 2 - f 1 = …… Hz. Quality factor for R R1 (using Equation (3)), Q = ……. Theoretical value of Q for R R1 (using Equation (7)) = …….
• • • • •
The maxima of the curve for R R2 in the graph gives f 0 = …… Hz. Theoretical value of f f 0 for R R2 =............. Hz. Bandwidth for R R2, f 2 - f 1 = …… Hz. Quality factor for R . R2, Q = ……. Theoretical value of Q for R R2 = …….
• • • • •
The maxima of the curve for R R3 in the graph gives f 0 = …… Hz. Theoretical value of f f 0 for R R3 =............. Hz. Bandwidth for R R3, f 2 - f 1 = …… Hz. Quality factor for R . R3, Q = ……. Theoretical value of Q for R R3 = …….
Result The resonant frequency f 0 with L = …… mH and C = …… μF is …… Hz Theoretical value of f f 0 = …… Hz % Error = ……. A comparison of the experimental and theoretical Quality factor for L = …… mH and C = …… μF for different R values is given below in tabular form:
S. No.
R (Ω)
Experimental f 0 (Hz)
Theoretical f 0 (Hz)
1 2 3
17
Experimental Q-factor
Theoretical Q-factor
Glossary Alternating voltage : An alternating voltage is a sinusoidally varying voltage, voltage, where is the peak value and is the angular frequency of the voltage. Anti-Resonance: The condition in a parallel LCR circuit when the impedance of the circuit is maximum and the current minimum is termed as anti-resonance. Anti-Resonant Frequency : For a parallel LCR circuit the frequency at which the current has minimum value, is called anti-resonant frequency. Bandwidth: The range of frequencies lying within the upper and lower cut-off frequencies which correspond to 0.707 times the voltage value at resonance is called bandwidth. It is also defined as the difference between the two half power frequencies which correspond to the points where the power has been reduced to one half of its value at resonance. Capacitance: The property of a conductor that describes its ability to store electric charge is called capacitance C and is given by Q/V where Q is the charge stored on the conductor and V is the potential difference between the conductor and earth. Color code: Dynamic Resistance : The frequency-dependent resistance of a parallel LCR circuit at resonance is known as the dynamic resistance. Impedance: A measure of the total opposition that a circuit or a part of a circuit offers to electric current. It includes both resistance and reactance. Inductance: It is the property of a conductor, often in the shape of a coil, defined as the electromotive force induced in a conductor per unit rate of change of current flowing through it. Pass-band: The electric waves lying within a certain range, or band, of frequencies allowed to pass, all other frequencies being blocked by the series LCR circuit. rms: An alternating potential difference has a value of one volt rms (root mean square) if it produces the same heating effect when applied to the ends of a resistance as is done by a steady potential difference of one volt applied to the same resistance in the same time. Numerically, rms value is 1 times the maximum value. The a.c. 2 ammeters and voltmeters measure the root mean square (rms) value of the current and potential difference respectively. Quality factor: It is a measure of the selectivity or the sharpness of the resonance curve and is denoted by Q. A low value of resistance in the circuit leads to a high Q. Quality factor is given by the ratio of the voltage across the inductor to the input voltage and is hence a dimensionless quantity. Since Q is ordinarily greater than unity, it is termed as the magnification factor of the circuit. Reactance: The frequency-dependent opposition to current flow, which results from energy storage rather than energy loss, is called reactance and is denoted by X L and XC for an inductor and capacitor respectively. Resistance: It is a measure of the opposition offered by an electric circuit to the flow of electric current. Resonance: The condition in a series LCR circuit when the impedance is purely resistive and hence minimum and current maximum is called resonance. Resonance Curve : A graph showing the variation of the voltage across a circuit (or a part of it) with frequency in the vicinity of resonance is the response curve or the resonance curve.
18
Resonant Frequency : For a series LCR circuit the frequency at which the reactance due to the inductor, X L, is exactly equal and opposite to the reactance due to the capacitor, XC, resulting in the impedance of the circuit being purely resistive, is called the resonant frequency. Selectivity: The selectivity of a tuned circuit is its ability to select a signal at resonant frequency and reject other signals that are close to that frequency.
Post-lab Assessment Choose the correct answer
(1)
An inductor and a capacitor form a parallel resonant resonant circuit. The capacitor value value is increased by four times. The resonant frequency will a) increase by four times b) double c) increase d) decrease (2) An inductor inductor and a capacitor form a parallel resonant resonant circuit. If the value of the inductor is decreased by a factor of four, the resonant frequency will a) increase by a factor of four b) increase c) decrease by a factor of two d) decrease by a factor of four (3) The anti-resonant frequency for a parallel LCR circuit with L = 900 mH, C = 0.03 μF and R = 1 K Ω is approximately a) 476 Hz b) 952 Hz c) 1904 Hz d) 1 KHz
Answers to Pre-lab Assessment 1. 2.
a a
3.
Z =
4. 5. 6.
L
RC False Because the current at resonance is minimum. The parallel tuned circuit is used to select one particular signal frequency from among others. It does this by rejecting the resonant frequency because of its high impedance. This is the reason why this type of circuit is also known as a rejector circuit.
19
Answers to Post-lab Assessment 1. 2. 3.
d b b
20