lcr circuits

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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 2 2 X R + and = R X 1 tan φ , respectively. Since in an inductor, voltage leads the current by π/2, the reactance of is L L jω , C while in case of a capacitor, voltage lags behind the current by π/2, the reactance of is C jω 1 . If the current in the circuit is I, the relative voltage drops across the 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

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Page 1: Lcr Circuits

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 22 XR +

and ⎟⎠⎞

⎜⎝⎛= −

RX1tanφ , respectively.

Since in an inductor, voltage leads the current by π/2, the reactance of isL Ljω ,

Cwhile in case of a capacitor, voltage lags behind the current by π/2, the reactance of is

Cjω1 . If the current in the circuit is I, the relative voltage drops across the

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

Page 2: Lcr Circuits

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. explain why the series LCR circuit is called an acceptor circuit 2. study the response of LCR circuit by varying the resistance in the circuit 3. present graphically the variation of current with frequency in a series LCR

circuit 4. find the resonant frequency for a series LCR circuit and hence find the Quality

factor 5. explain why the parallel LCR circuit is called a rejector circuit. 6. present graphically the variation of current with frequency in a parallel LCR

circuit and hence find the anti-resonant frequency and Quality factor

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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 tV ωsin0 or be applied to an inductor , a resistor tjeV ω0 L

R and a capacitor C all in series as shown in Figure 2. If I is neous current flowing through the circuit, the applied voltage in phasor form is given by

the instanta

CjILIjRIVVVV CLR ω

ω ++=++=

IC

LjR

ICj

LjR

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+=

⎥⎦

⎤⎢⎣

⎡++=

ωω

ωω

1

1

The impedance

⎟⎠⎞

⎜⎝⎛ −+==

CLjR

IVZ

ωω 1

2

Page 4: Lcr Circuits

If we write , then φφφ sincos jZZZeZ i +==2

12

2 1

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+=

CLRZ

ωω and

RC⎟⎠⎞

ω1L⎜

⎝⎛ −

φtan .

Therefore, current. ( )φωφ

ω−== tj

j

tj

eZV

ZeeV

I 00

Figure 2: Series LCR circuit

Three cases thus arise:

1. C

ω 1> , φtan is positive and applied voltage leads current by phase angleφ .

2. C

ω 1< , φtan is negative and applied voltage lags behind current byφ .

3. C

ω 1= , φtan is zero and applied voltage and current are in phase. This

condition is known as resonance and frequency as resonant frequency ( 0ω ).

LCCL 11 2 =⇒= ω

ωω

or

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Page 5: Lcr Circuits

4

LC1

0 == ωω

or

LC

fππ

ω2

12

00 == (1)

01=−

CL

ωω and CL VV =

If L, R and υ (frequency of function generator) are fixed and the capacitance is

varied, then for lower values of C, LICI ωω

> or . As the capacitance is

increased in the circuit, the situation called resonance is achieved whenV . If C is increased further, will decrease and we have V

LC VV >

LC VLC V=

CV < . The point of intersection

of and versus CV LVC

1 curves will give resonance condition. This is depicted in

Figure 3. At resonance is a maximum while V is minimum as shown in Figure 4. Corresponding to maximum value of , C is obtained. Similarly, for minimum value of V , C is obtained. This value of C makes the given circuit resonant at the supply frequency with constant values of L and R.

RV LC

RV

LC

Figure 3: Variation of VL and VC with C

1

Page 6: Lcr Circuits

Figure 4: Variation of VLC and VR with C

1

Theoretically at resonance should be zero. This should be so if the inductor is of negligible resistance and there are no other losses. The minimum value of 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.

LCV

LCV

At resonant frequency , the impedance of circuit is minimum. Hence frequencies near 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

range of frequencies between which the current is equal to or greater than

0f

0f

20I . Let

and be these limiting values of frequency. Then the width of the band is (refer to Figure 5)

1f 2f

12 ffBW −= . (2) The Quality factor is defined in the same way as for a mechanical oscillator and is given by

12

0

fff

bandwidthfrequencyresonantQ

−== . (3)

Q-factor is also defined in terms of reactance and resistance of the circuit at resonance, i.e.,

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Page 7: Lcr Circuits

6

RL

RX

Q L 0ω== . (4)

Figure 5: Bandwidth for a series LCR resonant circuit

I

Figure 6: Variation of current with frequency for different R values

Also,

0

0.7 I0

f1 f2BWf

I

Page 8: Lcr Circuits

CRRX

Q C

0

== . (5)

The resonance condition is also evident from the resonance curves or the graphs

between R

VI R

R = and for different values of f R shown in Figure 6. The

bandwidth as well as Q-factor can be calculated.

Pre-lab Assessment

Choose the correct answer

(1) 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.

(2) The real part of impedance is called a) resistance b) inductive reactance c) capacitive reactance d) none of the above.

(3) The imaginary part of impedance is called a) resistance b) inductive reactance c) capacitive reactance d) reactance.

(4) Capacitors and inductors oppose an alternating current. This is known as a) resistance b) resonance c) reactance d) impedance

(5) 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.

(6) 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.

(7) The reactance of a capacitor increases as the: a) frequency increases b) frequency decreases c) applied voltage increases d) applied voltage decreases

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Page 9: Lcr Circuits

(8) The reactance of an inductor increases as the: a) frequency increases b) frequency decreases c) applied voltage increases d) applied voltage decreases

(9) The minimum value of is a measure of the effective resistance of inductor coil. (True/False)

LCV

Answer the following question

(10) Why is a series LCR resonant circuit called an acceptor circuit?

Procedure

1. Connect the circuit as shown in Figure 2. 2. Switch on the a.c. source and set its output voltage Vi to a value (say, 3V rms)

and frequency to a known value. 3. Record the voltages across the known resistor, capacitor, inductor and the series

combination of the inductor and capacitor in Table 1. 4. Repeat step 3 for different values of C. 5. Increase the frequency gradually in steps and record the voltage across resistor in

Table2. 6. Repeat step 5 for two different R values.

Observations

Table 1: Variation of various voltages with C

1

f = ……Hz, R = …… Ω, L = …… mH

S. No. C

1

(μF)-1/2

VC (volts)

VR (volts)

VL (volts)

VLC (volts)

1 2 3 4 5 6

Table 2: Variation of voltage across resistor with frequency for different R values

L = …… mH, C = …… μF

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Page 10: Lcr Circuits

S. No.

Frequency f

(Hz) 1RV

(volts) 2RV

(volts) 3RV

(volts) 11

1

RV

I R=

(mA) 2

22

RV

I R=

(mA) 3

33

RV

I R=

(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

C1 on x-axis

2. Graph no. 2: VR and VLC on y-axis and C

1 on x-axis

3. Graph no. 3: Current I on y-axis and frequency f on x-axis for different sets corresponding to different values of R

• The point of intersection of CV and LV versus

C1 curves in graph no. 1

gives C = …… μF. • The point where RV is a maximum in graph no. 2 gives C = …… μF. • The point where LCV is minimum in graph no. 2 gives C = …… μF. • Mean value of C = …… μF. • The minima of the curves in graph no. 3 give f0 = …… Hz. • Theoretical value of f0 (using Equation (1)) =............. Hz. • Bandwidth for R1 (using Equation (2)), f2 - f1 = …… Hz. • Quality factor for R1 (using Equation (3)), Q = ……. • Theoretical value of Q for R1 (using Equation (4) or (5)) = ……. • Bandwidth for R2, f2 - f1 = …… Hz. • Quality factor for R2, Q = ……. .

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• Theoretical value of Q for R2 = ……. • Bandwidth for R3, f2 - f1 = …… Hz. • Quality factor for R3, Q = ……. . • Theoretical value of Q for R3 = …….

Result

The value of C which makes the given circuit resonant at the supply frequency with given values of L and R is C = …… μF. The resonant frequency f0 with L = …… mH and C = …… μF is …… Hz Theoretical value of f0 = …… 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 Q-factor Theoretical Q-factor 1 2 3

Post-lab Assessment

Choose the correct answer

(1) An inductor and a capacitor are connected in series. At the 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 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

Page 12: Lcr Circuits

c) decrease by a factor of two d) decrease by a factor of four

(4) The resonant frequency for a series 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 and versus CV LCVC

1 curves will give resonance

condition. (True/False) versus

C1(6) In the and RV LCV curves, is a maximum while is

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?

Answers to Pre-lab Assessment

1. d 2. a 3. d 4. c 5. a 6. b 7. b 8. a 9. True 10. At resonant frequency , the impedance of circuit is minimum. Hence

frequencies near are p ssed more readily than the other frequencies by the circuit. Due to this reason LCR-series circuit is called acceptor circuit.

Answers to Post-lab Questions

1. b 2. b 3. c 4. b 5. c

RV LCV

0fa0f

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Page 13: Lcr Circuits

6. False 7. True 8. When inductive and capacitive reactances are equal.

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Page 14: Lcr Circuits

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

RXXZ LC ++=

111

Therefore

RLjCjZ ++=

ωω1

111

222 LRLjRCj

ωωω

+−

+=

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Page 15: Lcr Circuits

⎥⎦⎤

⎢⎣⎡

+−+

+= 222222 LR

LCjLR

ωωω

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

0220

20

0 =+

−LR

LCω

ωω

220

220

0 42

2LfR

LfCf

ππ

π+

=

which gives

2

2

01

21

LR

LCf −=

π (6)

At resonance, the impedance of the circuit is maximum and is given by

RLRZ

20

22 ω+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 2

22 1LR

LCRLR

or

RCLZ =

The impedance at resonance is called dynamic resistance. The current has minimum value (Figure 8). It is for this reason that the condition of resonance for a

ZVI /=

tV ωsin0 C

VR R

L

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Page 16: Lcr Circuits

parallel LCR circuit is known as anti-resonance and the corresponding frequency as the anti-resonance frequency.

I

f

RR1 > R2 > R3 3

R2

R1

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

CL

RCRRLQ 11

0

0 ===ω

ω (7)

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

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Page 17: Lcr Circuits

(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) What is the impedance at anti-resonance for a parallel LCR circuit? (5) Why is the condition of resonance for a parallel LCR circuit known as anti-

resonance? (6) Why is a parallel LCR resonant circuit called a rejector circuit?

Procedure

1. Connect the circuit as shown in Figure 6. 2. Switch on the a.c. source and set its output voltage Vi to a value, say, 3V rms

and frequency to a known value. 3. Increase the frequency gradually in steps and record the voltage across resistor

in Table 2. 4. Repeat step 3 for two different R values.

Observations

Table 3: Variation of voltage across resistor with frequency for different R values

L = …… mH, C = …… μF

S. No.

Frequency f

(Hz) 1RV

(volts) 2RV

(volts) 3RV

(volts) 11

1

RV

I R=

(mA) 2

22

RV

I R=

(mA) 3

33

RV

I R=

(mA) 1 2 3 4 5 6

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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. • The maxima of the curve for R1 in the graph gives f0 = …… Hz. • Theoretical value of f0 for R1 (using Equation (6)) =............. Hz. • Bandwidth for R1 (using Equation (2)), f2 - f1 = …… Hz. • Quality factor for R1 (using Equation (3)), Q = ……. • Theoretical value of Q for R1 (using Equation (7)) = …….

• The maxima of the curve for R2 in the graph gives f0 = …… Hz. • Theoretical value of f0 for R2 =............. Hz. • Bandwidth for R2, f2 - f1 = …… Hz. • Quality factor for R2, Q = ……. . • Theoretical value of Q for R2 = …….

• The maxima of the curve for R3 in the graph gives f0 = …… Hz. • Theoretical value of f0 for R3 =............. Hz. • Bandwidth for R3, f2 - f1 = …… Hz. • Quality factor for R3, Q = ……. . • Theoretical value of Q for R3 = …….

Result

The resonant frequency f0 with L = …… mH and C = …… μF is …… Hz Theoretical value of f0 = …… 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 f0 (Hz)

Theoretical f0 (Hz)

Experimental Q-factor

Theoretical Q-factor

1 2 3

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Glossary

Alternating voltage: An alternating voltage is a sinusoidally varying 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

21 times the maximum value. The a.c.

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 XL 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.

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Resonant Frequency: For a series LCR circuit the frequency at which the reactance due to the inductor, XL, 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 circuit. The capacitor value is increased by four times. The resonant frequency will a) increase by four times b) double c) increase d) decrease

(2) An inductor and a capacitor form a parallel 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 KΩ is approximately a) 476 Hz b) 952 Hz c) 1904 Hz d) 1 KHz

Answers to Pre-lab Assessment

1. a 2. a

3. RCLZ =

4. False 5. Because the current at resonance is minimum. 6. 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.

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Answers to Post-lab Assessment

1. d 2. b 3. b