Sirindhorn International Institute of Technology Thammasat University School of Information, Computer and Communication Technology COURSE : ECS 34 Basic Electrical Engineering Lab INSTRUCTOR : Dr. Prapun Suksompong (prapun@siit.tu.ac.th) WEB SITE : http://www.siit.tu.ac.th/prapun/ecs34/ EXPERIMENT : 5 RC Circuit and Resonant RLC Circuit I. OBJECTIVES 1. To investigate the discharging of a capacitor, and the time constant of an RC circuit. 2. To determine the resonant frequency and frequency response of a series RLC circuit. II. BASIC INFORMATION 1. A function of capacitor in electric circuits is to restore energy by charge charging. Rate of charging represented by capacitor voltage V(t) can be expressed by the following equation. t V ( t) V (1 e ) in where V in is the voltage applied to the capacitor, and is the time constant which is defined, in the case of RC circuits, as RC When the capacitor discharges energy, the voltage across the capacitor will be decreased according to the following equation. t V () t V e where V is the initial voltage. When a capacitor of C farads is charged through a resistor of R ohms, the time constant in seconds of the charging circuit is = RC. In one time constant, a capacitor charges to approximately 63.2 % of the applied voltage across its terminals. If we apply a squarewave signal to an RC circuit, we will get an output signal as shown in Figure 5-1. At the rising edge of the square-wave, the capacitor will be charged; and at the trailing edge, the capacitor will be discharged.
In part A of this experiment, the value of are found using three different methods (see Figure 5-1-2): 1) Measure t.37. Then, t.37. 2) Measure t half. Then, calculate thalf ln2. 3) Measure R and C. Then, calculate RC. 2. Resistors limit the amount of current in dc as well as in ac circuits. In addition to resistors, reactive components, such as inductors and capacitors, impede currents in ac circuits. 3. Inductive reactance of a coil is given by X L = 2 f L ohms, where L is the inductance in henries, f is the frequency of signal in the circuit. The ac voltage across a coil, V L is equal to the product of the alternating current in the coil and the inductive reactance of the coil; that is V L = I jx L where j = 1. Voltage Charging period Discharging period time pulse input voltage voltage across capacitor Figure 5-1-1: Pulse input voltage and the voltage across a capacitor. 4. The capacitive reactance is given by X C = 1/(2 f C) ohms. It is inversely proportional to frequency and capacitance. 2
4 V 2 V 2 1.47 V e t half t.37 Figure 5-1-2: measurement. 5. A series RLC circuit, as well as its impedance phasor diagram is shown in Figure 5-2. The reactance in the series RLC circuit is X = X L X C, and the impedance is found by using the formula Z = R + j(x L X C ). R X L V C L -9 R Figure 5-2: A series RLC circuit and its impedance phasor diagram. X C We can understand the behavior of an RLC circuit according to the change of frequency from the graph in Figure 5-3. This graph illustrates how impedance changes with frequency. At frequency below f o, X C > X L, and the circuit is capacitive. At the resonant frequency f o, X C = X L, so the circuit is purely resistive. At frequencies above f o, X C < X L, and the circuit is thus inductive. The impedance magnitude is minimum at resonance, and increases in value above and below the resonant point. At zero frequency, both X C and Z are infinitely large, and X L is zero. This is because the capacitor looks like an open circuit at Hz, while the inductor looks like a short. As the frequency increases, X C decreases, and X L increases. Since X C is larger than X L at frequencies below f o, Z decreases along with X C. At f o, X C = X L and Z = R. At frequencies above f o, X L becomes increasingly larger than X C, causing Z to increase. 3
Z( ) X C Z X L f o Z = R f Figure 5-3: Impedance of a series RLC circuit as a function of frequency. V m R V m 2 R I m Figure 5-4-1: Frequency Response curve of a series RLC circuit. Figure 5-4-1 shows the frequency response curve of a series RLC circuit. Three significant points have been marked on the curve. These are, the resonant frequency, 1 and 1 and 2. For series RLC circuits, we have. Points 1 and 2 are LC located at 7.7 % of the maximum (at o ) on the curve. They are called the half power points, and the frequency separation between them is called the bandwidth (BW) of the circuit. For the series RLC circuit, BW is defined as R BW = 2 1 L For a circuit intended to be frequency selective, the sharpness of the selectivity is a measure of the circuit quality. The quality of the frequency response is described quantitatively in terms of the ratio of the resonance frequency to the bandwidth. This ratio is called the quality factor (Q) of the circuit. Then we have Q = o /BW 4
For the series RLC circuit, the current in the circuit is maximum at o, as can be seen in Figure 5-4-1. At resonant frequency, the voltages V L and V C across L and C, respectively, are equal. Thus, at o IX L = IX C I = V R V L = V C = VQ where Q = X L /R = X C /R = 1. L R C = /BW The equation V L = V C = VQ becomes significant for the values of Q > 1. For such values, V C and V L are greater than the applied voltage V. Remark: In general, the maximum values for frequency. (See Figure 5-4-2.) V L and V C will NOT occur at resonant V C V L V R f Figure 5-4-2: Voltage values across the resistor L V in a series RLC circuit. V R, the capacitor V C f, and the inductor 5
III. MATERIALS REQUIRED - Function generator, multi-meter, and oscilloscope. - Resistor: 1, 1 k. - Inductor: 22 mh. - Capacitors:.1 F,.47 F, and.1 F. IV. PROCEDURE Part A: To investigate the discharging of a capacitor oscilloscpe 1k + 4Vp-p square wave.1 F Figure 5-5: The RC circuit for investigating the discharging of a capacitor. 1. Connect the circuit of Figure 5-5, where the oscilloscope is in DC mode. 2. Set the output of the function generator to be a square wave with amplitude of 4 Vp-p and frequency of 5 Hz. Then set the DC offset level to 2 V, such that the square-wave waveform has its lower peak at V and its higher peak at 4 V. 3. Adjust volts/div and time/div until the waveforms can be seen on the oscilloscope. Then, draw the voltage waveforms across the function generator (shown by dash line in Figure 5-1), and across the capacitor (shown by solid line) in Table 5-1. 4. Next, find the time constant by measuring the time elapse t.37 during the capacitor discharges energy, while the voltage across the capacitor drops to.37 times of the maximum voltage as shown in Figure 5-6. Then, put the value in Table 5-2. Figure 5-6: The discharging behavior of a capacitor. 6
Q: Why.37 times of the maximum voltage? A: The voltage value after one time constant is equal to.37 times of the maximum voltage. From the following equation, t V () t V e At t =, V () t V e 1 V( t).37v Hence, we can find when the voltage is.37 times of the maximum voltage. Note that V in the above equations is equivalent to V PS in Figure 5-6. 5. Measure the time elapse t half that the voltage across the capacitor drops to half of the maximum voltage. 6. Calculate by the following equation, and record the value in Table 5-2. t half ln 2 7. Calculate the product of R and C, and record the value in Table 5-2. Part B: Series resonant RLC circuit Oscilloscope Ch- A Ch- B V Sine-wave generator R2 1 L C Figure 5-7: The series RLC circuit. 1. You are supplied with one inductor of the value 22 mh, and three capacitors with values of.1,.47 and.1 F. Measure all values of the inductor and capacitors, and 7
1 calculate the resonant frequency f o = o /2 = Hz for each of the capacitors. 2 LC Record the calculated resonant frequencies in Table 5-3 under Calculated resonant frequency. Connect the circuit of Figure 5-7 with C =.1 F. 2. Turn on the sine-wave generator. 3. Turn on the oscilloscope, and adjust it to view the sine wave output of the generator. 4. Increase the output of the generator until the scope indicates an 8 Vp-p voltage. Maintain this voltage throughout this experiment. 5. Observe the rms value of V R across the resistor as the frequency of the generator is varied. Record the resonant frequency f o at which V R is maximum. Measure f o and record the value in the.1 F row of Table 5-3. Caution: As you adjust the frequency f, the voltage across the generator output will change. Readjust it back to 8 V p-p. 6. Replace the.1 F with a.47 F capacitor. Check the generator output voltage to verify that it is 8 Vp-p; adjust the voltage if necessary. Find fo and record the value in the.47 F row of Table 5-3. 7. Replace the.47 F with a.1 F capacitor. Check the generator output voltage and adjust to maintain 8 Vp-p. Find f and record the value in the.1 F row of Table 5-3. 8. Check the generator output voltage, and adjust the value to maintain at 8 Vp-p. With C =.1 F, vary and record the frequency according to Table 5-4. Record the corresponding values of V R, V L, and V C in Table 5-4, where V L and V C are the voltage across L and C, respectively. 8
Table 5-1. Square wave input voltage and voltage across capacitor volts/div =, time/div =. TA Signature: Table 5-2. Time constant of an RC circuit from t.37 from t half / ln 2 from RC Table 5-3. Resonant frequency of a series RLC circuit TA Signature: Inductor L, mh Capacitor C, F Resonant Frequency, Hz Calculated Measured 22.1 22.47 22.1 TA Signature: 9
Table 5-4. Frequency response of a series RLC circuit Frequency deviation Frequency f, Hz Voltage across R, V R, V Voltage across L, V L, V Voltage across C, V C, V 1 5 2 1 + 1 + 2 + 5 + 1 TA Signature: 1
QUESTIONS True or False 1. In a circuit containing impedance, the current and voltage are 9 degrees out of phase. 2. The value of reactance in a series RLC circuit can exceed the value of impedance. 3. The value of resistance in a series RLC circuit can exceed the value of impedance. 4. In a series RLC circuit, the capacitor voltage can be greater than the source voltage. 5. A series RLC circuit operating above its resonant frequency is inductive. 6. There is only one combination of L and C for each resonant frequency. Fill in the blanks. 7. At resonance, the power factor of a circuit is. 8. Impedance is a combination of and. 9. Give the definition of bandwidth. Bandwidth is 1. The series RLC circuit in Figure 5-2 has an inductance of 1 H, a capacitance of 2 pf, and a resistance of 1. Determine its resonant frequency and bandwidth. Also determine X C, X L, I L, and V L at 1. MHz if the source voltage V = 1. Vpeak. Resonant frequency = Bandwidth = At 1. MHz X C = X L = I L = V L = 11
11. The series RLC circuit in Figure 5-2 has a capacitance of 25 F, and a resistance of 1. We then adjust the inductance until V R is maximum. Determine V R, V L, and V C at = 2 rad/sec if the source voltage V = 5 Vpeak. V R = V L = V C = 12. What are the characteristics of series resonant circuits? 13. Suppose you set the voltage across the output of the signal generator at 2V rms. You then connect the generator output across a 1Ω resistor. Now, you measure the voltage across the generator again but you get a value which is significantly less than 2 V rms. Why? 14. How can you to adjust the DC offset of the signal generator? 15. Is changing the DC offset on the signal generator the same as changing the vertical position of the trace in oscilloscope? Explain. 16. What are the differences between the DC and AC mode of the oscilloscope? 12