LCR CIRCUITS Institute of Lifelong Learning, University of Delhi

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1 L UTS nstitute of Lifelong Learning, University of Delhi

2 L UTS PHYSS (LAB MANUAL) nstitute of Lifelong Learning, University of Delhi

3 PHYSS (LAB MANUAL) L UTS ntroduction ircuits containing an inductor L, a capacitor, and a resistor, 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 called resonant frequencies. Such circuits are 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. n this section, we will present two types of L circuits, viz., series and parallel, and also discuss the formulae applicable for typical resonant circuits. A series L circuit includes a series combination of an inductor, resistor and capacitor whereas; a parallel L 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 defined as 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 the resonant 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. n general, the impedance Z is a sum of the real part called resistance and the complex part called the reactance X, i.e., Z = + jx. The magnitude and phase of the impedance are given by X X and tan, respectively. Since in an inductor, voltage leads the current by π/, the reactance of L is j L, while in case of a capacitor, voltage lags behind the current by π/, the reactance of is. f the current in the circuit is, the relative voltage drops across j the inductor, capacitor and resistor can be represented in the phasor diagram as shown in Figure. We will study the property of resonance in context of series as well as parallel configurations of L circuit. t 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. nstitute of Lifelong Learning, University of Delhi 3

4 L UTS PHYSS (LAB MANUAL) Figure : Phasor diagram Learning Outcomes After performing this experiment you will be able to. explain why the series L circuit is called an acceptor circuit. study the response of L circuit by varying the resistance in the circuit 3. present graphically the variation of current with frequency in a series L circuit 4. find the resonant frequency for a series L circuit and hence find the Quality factor 5. explain why the parallel L circuit is called a rejecter circuit. 6. present graphically the variation of current with frequency in a parallel L circuit and hence find the anti-resonant frequency and Quality factor 4 nstitute of Lifelong Learning, University of Delhi

5 PHYSS (LAB MANUAL) L UTS SETON A L Series circuit Let us consider the L circuit, which consists of an inductor, L, a capacitor,, and a resistor,, all connected in series with a source as shown in Figure. 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 / athode ay Oscilloscope (O) one a.c. milliammeter onnecting wires. Theory Let an alternating voltage sin t or e jt be applied to an inductor L, a resistor and a capacitor all in series as shown in Figure. f is the instantaneous current flowing through the circuit, the applied voltage in phasor form is given by L jl j jl j jl The impedance Z jl nstitute of Lifelong Learning, University of Delhi 5

6 L UTS PHYSS (LAB MANUAL) Figure (a): Series L circuit Figure (b): Laboratory picture of series L circuit 6 nstitute of Lifelong Learning, University of Delhi

7 PHYSS (LAB MANUAL) L UTS i f we write Z Ze Z cos jz sin, then L tan. e Therefore, current. Ze Three cases thus arise:. L, jt j e Z t j Z L and tan is positive and applied voltage leads current by phase. 3. or or angle. L, tan is negative and applied voltage lags behind current by. L, tan is zero and applied voltage and current are in phase. This condition is known as resonance and frequency as resonant frequency ( ). L L f L () L L and L f L, and υ (frequency of function generator) are fixed and the capacitance is varied, then for lower values of, L or L. As the capacitance is. increased in the circuit, the situation called resonance is achieved when f is increased further, intersection of and will decrease and we have L versus is depicted in Figure 3. At resonance. The point of curves will give resonance condition. This is a maximum while shown in Figure 4. orresponding to maximum value of Similarly, for minimum value of L L L is minimum as, is obtained., is obtained. This value of makes the given circuit resonant at the supply frequency with constant values of L and. L nstitute of Lifelong Learning, University of Delhi 7

8 L UTS PHYSS (LAB MANUAL) Figure 3: ariation of L and with Figure 4: ariation of L and with 8 nstitute of Lifelong Learning, University of Delhi

9 PHYSS (LAB MANUAL) L UTS Theoretically at resonance L 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. At resonant frequency f, the impedance of circuit is minimum. Hence frequencies near f are passed more readily than the other frequencies by the circuit. Due to this reason L-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 L. Let f and f be these limiting values of frequency. Then the width of the band is (refer to Figure 5) BW. () f f The Quality factor is defined in the same way as for a mechanical oscillator and is given by Q resonant frequency bandwidth. (3) Q-factor is also defined in terms of reactance and resistance of the circuit at resonance, i.e., Q X f f L f L. (4).7 f f f BW Figure 5: Bandwidth for a series L resonant circuit nstitute of Lifelong Learning, University of Delhi 9

10 L UTS PHYSS (LAB MANUAL) Figure 6: ariation of current with frequency for different values Also, Q X. (5) The resonance condition is also evident from the resonance curves or the graphs between and f for different values of shown in Figure 6. The bandwidth as well as Q-factor can be calculated. Pre-lab Assessment Now to know whether you are ready to carry out the experiment in the lab, answer the following questions. f you score at least 8%, you are ready, otherwise read the preceding text again. (Answers are given at the end of this experiment.) Answer the following question: hoose the correct answer: () 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. nstitute of Lifelong Learning, University of Delhi

11 PHYSS (LAB MANUAL) L UTS () 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) apacitors and inductors oppose an alternating current. This is known as a) resistance b) resonance c) reactance d) impedance (5) n the case of an inductor a) voltage leads the current by π/ b) voltage lags behind the current by π/ c) current leads the voltage by π/ d) voltage leads the current by π. (6) n the case of a capacitor a) voltage leads the current by π/ b) voltage lags behind the current by π/ c) current leads the voltage by π/ 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 (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 L is a measure of the effective resistance of inductor coil. (True/False) () Why is a series L resonant circuit called an acceptor circuit? Procedure. onnect the circuit as shown in Figure.[ The box in Figure 7 shows knobs to change the values of L, and for making different series combinations.]. Switch on the a.c. source and set its output voltage i to a value (say, 3 rms) and frequency to a known value. 3. ecord the voltages across the known resistor, capacitor, inductor and the series combination of the inductor and capacitor in Table.Figure 8 shows voltage across inductor L i.e. L. 4. epeat step 3 for different values of. 5. ncrease the frequency gradually in steps and record the voltage across resistor in Table. 6. epeat step 5 for two different values. nstitute of Lifelong Learning, University of Delhi

12 L UTS PHYSS (LAB MANUAL) Figure 7 : L series combinations Figure 8: Measuring voltage L across L nstitute of Lifelong Learning, University of Delhi

13 PHYSS (LAB MANUAL) L UTS Observations Table : ariation of various voltages with f = Hz, = Ω, L = mh S. No (μf) -/ (volts) (volts) L (volts) L (volts) Table : ariation of voltage across resistor with frequency for different values L = mh, = μf S. No Frequency f (Hz) (volts) (volts) 3 (volts) (ma) (ma) 3 (ma) 3 3 Precautions The connecting wires should be straight and short. f 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. nstitute of Lifelong Learning, University of Delhi 3

14 L UTS PHYSS (LAB MANUAL) alculations Plot the following graphs:. Graph no.: L and on y-axis and. Graph no. : and L on y-axis and on x-axis on x-axis 3. Graph no. 3: urrent on y-axis and frequency f on x-axis for different sets corresponding to different values of The point of intersection of and L versus curves in graph no. gives = μf. The point where a maximum in graph no. is gives = μf. The point where L a minimum in graph no. gives = μf. Mean value of = μf. The minima of the curves in graph no. 3 give f = Hz. Theoretical value of f (using Equation ()) =... Hz. Bandwidth for (using Equation ()), f - f = Hz. Quality factor for (using Equation (3)), Q =. Theoretical value of Q for (using Equation (4) or (5)) =. Bandwidth for, f - f = Hz. Quality factor for, Q =. Theoretical value of Q for =. Bandwidth for 3, f - f = Hz. Quality factor for 3, Q =. Theoretical value of Q for 3 =. esult The value of which makes the given circuit resonant at the supply frequency with given values of L and is = μf. The resonant frequency f with L = mh and = μf is Hz Theoretical value of f = Hz % Error =. A comparison of the experimental and theoretical Quality factor for L = mh and = μf for different values is given below in tabular form: S. No. (Ω) Experimental Q-factor Theoretical Q-factor 3 4 nstitute of Lifelong Learning, University of Delhi

15 PHYSS (LAB MANUAL) L UTS Post-lab Assessment hoose the correct answer () 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 () 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. f 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 c) decrease by a factor of two d) decrease by a factor of four (4) The resonant frequency for a series L circuit with L = mh, =. μf is approximately a) 5 Hz b) 55 Hz c) 533 Hz d) 5 KHz (5) The point of intersection of and L versus curves will give resonance condition. (True/False) (6) n the and L versus curves, is a maximum while 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 L is Answer the following question (8) When are the voltage and current in a series L circuit in phase? nstitute of Lifelong Learning, University of Delhi 5

16 L UTS PHYSS (LAB MANUAL) Answers to Pre-lab Assessment. d. a 3. d 4. c 5. a 6. b 7. b 8. a 9. True. At resonant frequency f, the impedance of circuit is minimum. Hence frequencies near f are passed more readily than the other frequencies by the circuit. Due to this reason L-series circuit is called acceptor circuit. Answers to Post-lab Questions. b. b 3. c 4. b 5. c 6. False 7. True 8. When inductive and capacitive reactances are equal. 6 nstitute of Lifelong Learning, University of Delhi

17 PHYSS (LAB MANUAL) L UTS SETON B L 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 L circuit is different than a parallel L circuit. The condition of resonance in this case is known as anti-resonance. We will derive the condition of anti-resonance of a parallel L 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 / athode ay Oscilloscope (O) one a.c. milliammeter onnecting wires Theory onsider a circuit containing an inductor L and a capacitor connected in parallel to an a.c. source (Figure 9). The resistance is connected in series with the inductor L and includes its resistance. The total admittance of the L combination is given by Therefore Z Z X X L j jl jl j L j L L L nstitute of Lifelong Learning, University of Delhi 7

18 L UTS PHYSS (LAB MANUAL) sin t L Figure 9 (a): Parallel L circuit Figure 9(b): Laboratory picture of parallel L circuit 8 nstitute of Lifelong Learning, University of Delhi

19 PHYSS (LAB MANUAL) L UTS 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 which gives L L f f f L 4 f L (6) L L At resonance, the impedance of the circuit is maximum and is given by or L Z L L L Z The impedance at resonance is called dynamic resistance. The current has minimum value (Figure ). t is for this reason that the condition of resonance for a parallel L circuit is known as anti-resonance and the corresponding frequency as the anti-resonance frequency. L / Z > > 3 3 f Figure : ariation of current with frequency for different values The shape of the impedance versus frequency curve in a parallel L circuit is the same as the shape of the current versus frequency curve in a series L circuit. nstitute of Lifelong Learning, University of Delhi 9

20 L UTS PHYSS (LAB MANUAL) n 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. t 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 rejecter 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 L circuit. As in series circuit, Q can also be written as L Q L (7) Pre-lab Assessment Now to know whether you are ready to carry out the experiment in the lab, answer the following questions. f you score at least 8%, you are ready, otherwise read the preceding text again. (Answers are given at the end of this experiment.) hoose the correct answer: () 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 () The anti-resonant frequency for a parallel L circuit does not depend on the value of resistance used in the circuit. (True/False) (3) A parallel L 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 L circuit? (5) Why is the condition of resonance for a parallel L circuit known as antiresonance? (6) Why is a parallel L resonant circuit called a rejecter circuit? nstitute of Lifelong Learning, University of Delhi

21 PHYSS (LAB MANUAL) L UTS Procedure. onnect the circuit as shown in Figure 9.. Switch on the a.c. source and set its output voltage i to a value, say, 3 rms and frequency to a known value. 3. ncrease the frequency gradually in steps and record the voltage across resistor in Table. 4. epeat step 3 for two different values. Observations Table 3: ariation of voltage across resistor with frequency for different values L = mh, = μf S. No Frequency f (Hz) (volts) (volts) 3 (volts) (ma) (ma) 3 (ma) 3 3 alculations Plot a graph with current on y-axis and frequency f on x-axis for different sets corresponding to different values of. The minimum of the curve for in the graph gives f = Hz. Theoretical value of f for (using Equation (6)) =... Hz. Bandwidth for (using Equation ()), f - f = Hz. Quality factor for (using Equation (3)), Q =. Theoretical value of Q for (using Equation (7)) =. The minimum of the curve for in the graph gives f = Hz. Theoretical value of f for =... Hz. Bandwidth for, f - f = Hz. Quality factor for, Q =. Theoretical value of Q for =. The minimum of the curve for 3 in the graph gives f = Hz. Theoretical value of f for 3 =... Hz. Bandwidth for 3, f - f = Hz. Quality factor for 3, Q =. Theoretical value of Q for 3 =. nstitute of Lifelong Learning, University of Delhi

22 L UTS PHYSS (LAB MANUAL) esult The resonant frequency f with L = mh and = μf is Hz Theoretical value of f = Hz % Error =. A comparison of the experimental and theoretical Quality factor for L = mh and = μf for different values is given below in tabular form: S. No. (Ω) 3 Experimental f (Hz) Theoretical f (Hz) Experimental Q-factor Theoretical Q-factor Glossary Alternating voltage: An alternating voltage is a sinusoidally varying voltage, where the peak is value and is the angular frequency of the voltage. Anti-esonance: The condition in a parallel L circuit when the impedance of the circuit has maximum value and the current minimum is termed as antiresonance. Anti-esonant Frequency: For a parallel L 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.77 times the voltage value at resonance is called bandwidth. t 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. apacitance: The property of a conductor that describes its ability to store electric charge is called capacitance and is given by Q/ where Q is the charge stored on the conductor and is the potential difference between the conductor and earth. olor code: Dynamic esistance: The frequency-dependent resistance of a parallel L circuit at resonance is known as the dynamic resistance. mpedance: A measure of the total opposition that a circuit or a part of a circuit offers to electric current. t includes both resistance and reactance. nductance: t 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 L 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 times the maximum value. The a.c. ammeters and voltmeters measure the root mean square (rms) value of the current and potential difference respectively. nstitute of Lifelong Learning, University of Delhi

23 PHYSS (LAB MANUAL) L UTS Quality factor: t 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. eactance: 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 X for an inductor and capacitor respectively. esistance: t is a measure of the opposition offered by an electric circuit to the flow of electric current. esonance: The condition in a series L circuit when the impedance is purely resistive and hence minimum and current maximum is called resonance. esonance urve: 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. esonant Frequency: For a series L 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, X, 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 hoose the correct answer () 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 () An inductor and a capacitor form a parallel resonant circuit. f 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 L circuit with L = 9 mh, =.3 μf and = KΩ is approximately a) 476 Hz b) 95 Hz c) 94 Hz d) KHz nstitute of Lifelong Learning, University of Delhi 3

24 L UTS PHYSS (LAB MANUAL) Answers to Pre-lab Assessment. a. a 3. Z L 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. t 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. Answers to Post-lab Assessment. d. b 3. b 4 nstitute of Lifelong Learning, University of Delhi