Series Resonance AC 2 Fundamentals Exercise 1: Series Resonant Circuits EXERCISE OBJECTIVE When you have completed this exercise, you will be able to compute the resonant frequency, total current, and impedance in a series RLC circuit by using standard formulas and procedures. You will verify your results with an oscilloscope. DISCUSSION At what frequency are the inductive and capacitive reactances equal in a series RLC circuit? a. cutoff frequency b. resonant frequency Resonant frequency is calculated from the following formula: 1 fr = 2 π LC where 2 = 6.28 L = the total value of inductance in henries. C = the total value of capacitance in farads f r = the resonant frequency in hertz. Calculate the resonant frequency (f r ) of this circuit. f r = khz (Recall Value 1) 46 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance When the resonant frequency of this circuit is 33.952 khz, the inductive reactance of L1 (X L1 ) and the capacitive reactance of C1 (X C1 ) are equal. At this frequency (33.952 khz), the inductive reactance (X L1 ) equals 2132. X L1 = 2 fl = (6.28) x (33.952 x 10 3 ) x (10 x 10 3 ) = 2132 What is the capacitive reactance of C1 (X C1 )? X C1 fc) X C1 = (Recall Value 2) Since X L1 and X C1 are equal, their effects cancel one another. The total circuit impedance is simply the circuit resistance. What is the circuit impedance (Z)? Z = (Recall Value 3) FACET by Lab-Volt 47
Series Resonance AC 2 Fundamentals At resonance, the total circuit current (I T ) can be determined from the applied voltage (V GEN ) and the total circuit resistance. I T V = R1 GEN I T 15 V = 1000 Ω pk pk I T = 15 ma pk-pk Knowing the resistance (1000 ), reactances (2132 ), and the total circuit current (15 ma pk-pk ), you can determine the individual component voltage drops. V R1 = I T x R1 = 15 ma pk-pk x 1000 = 15 V V L1 = I T x X L1 = 15 ma pk-pk x 2132 V L1 = 31.98 V 48 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Calculate the voltage drop across capacitor C1. V R1 = I T x R1 = 15 ma pk-pk x 1000 = 15 V V L1 = I T x X L1 = 15 ma pk-pk x 2132 = 31.98 V V C1 = I T x X C1 V C1 = V (Recall Value 4) The voltage drop across the resistance (R1) equals the applied voltage (V GEN ). At the resonant frequency, the circuit acts as if only the 1000 resistor exists in the circuit. Because the circuit appears totally resistive at resonance, the circuit current (I T ) is in phase with the applied voltage (V GEN ). FACET by Lab-Volt 49
Series Resonance AC 2 Fundamentals The voltage drops across the inductor and capacitor are equal, since X L1 equals X C1, and they share a common current (I T ). (V GEN )? a. yes b. no PROCEDURE Adjust the generator output voltage (V GEN ) for a 15 V pk-pk, 20 khz sine wave. 50 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Connect channel 2 of the oscilloscope across series combination L1 and C1, as shown. Increase the frequency of the generator to tune for series resonance as determined by a voltage null across series combination L1 and C1. Determine the resonant frequency (f r ) by using an oscilloscope (CH 1) to measure the period (T) at V GEN. f r f r = khz (Recall Value 1) FACET by Lab-Volt 51
Series Resonance AC 2 Fundamentals With the circuit tuned at resonance, determine the total circuit current (I T ). I T VR3 = R3 I T = ma pk-pk (Recall Value 2) Vary the frequency above and below resonance while observing the total circuit current (I T ). Is I T maximum at resonance? a. no b. yes While observing I T with the oscilloscope, retune the series circuit by varying the generator frequency several kilohertz (around 33.6 khz) until a current peak occurs. 52 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Connect the oscilloscope channels as shown. At resonance, observe the phase angle ( ) between the circuit current and the applied generator voltage (V GEN ). NOTE: Use V GEN (CH 1) as the reference. At resonance, is the circuit current in phase or out of phase with V GEN? a. in phase b. out of phase Vary the generator frequency several kilohertz above the resonant frequency while observing the phase relationship between the circuit current and applied voltage. Above resonance, circuit current lags V GEN, so the circuit acts inductively. Vary the generator frequency several kilohertz below the resonant frequency while observing the phase relationship between the circuit current and applied voltage. Below resonance, does circuit current lag or lead the applied voltage? a. lag b. lead FACET by Lab-Volt 53
Series Resonance AC 2 Fundamentals In the next few steps, you will measure the voltage drops across R2 and C1 at resonance. You will compare V GEN with your measured voltage drops across R2 and C1. Adjust V GEN for a 15 V pk-pk, 20 khz sine wave. Increase the frequency of the generator to tune for series resonance, as determined by a voltage null across series combination L1 and C1. With the oscilloscope probes connected as shown, use the ADD-INVERT method to measure the voltage drop across R2 (V R2 ). V R2 = V pk-pk (Recall Value 3) 54 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Compare your measured value of V R2 ( V pk-pk [Step 14, Recall Value 3]) with the applied generator voltage (15 V pk-pk ). At resonance, does the circuit act as if only R2 existed in the circuit? a. yes b. no With the series circuit at resonance, connect the oscilloscope as shown, and measure the voltage drop across C1 (V C1 ). V C1 = V pk-pk (Recall Value 4) Compare your measured value of V C1 ( V pk-pk [Step 16, Recall Value 4]) with the applied generator voltage (15 V pk-pk ). Is V C1 larger or smaller than V GEN? a. smaller b. larger In the next few steps, you will change the values of R2 and C1. You will observe how changes in these components affect the series resonant circuit. Adjust the generator output (V GEN ) for a 15 V pk-pk, 20 khz sine wave. FACET by Lab-Volt 55
Series Resonance AC 2 Fundamentals Increase the frequency of the generator to tune for series resonance, as determined by a voltage null across series combination L1 and C1. Place CM switch 9 in the ON position to increase the value of R2 to 3.2 k. Tune for series resonance, as determined by a voltage null across series combination L1 and C1 (CH 2). Determine the circuit s new resonant frequency (f r ) and compare it to your previously measured value of f r ( khz [Step 4, Recall Value 1]). Did changing the value of R2 affect f r? a. yes b. no Place CM switch 10 in the ON position to increase the value of C1 to 0.0044 F. Tune for series resonance, as determined by a voltage null across series combination L1 and C1 (CH 2). 56 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Determine the circuit s new resonant frequency and compare it to your previously measured value of f r ( khz [Step 4, Recall Value 1]). Did changing the value of C1 affect f r? a. yes b. no Make sure all CMs are cleared (turned off) before proceeding to the next section. CONCLUSION At the resonant frequency, X L equals X C, and they cancel one another, leaving only the circuit resistance to control current. At series resonance, circuit current is maximum and in phase with the applied voltage. Inductance and capacitance values affect the resonant frequency. Circuit resistance has no effect on resonant frequency (f r ), but it does affect impedance and current. The voltage drops across the reactive components (X L and X C applied voltage and peak at resonance. FACET by Lab-Volt 57
Series Resonance AC 2 Fundamentals REVIEW QUESTIONS 1. GEN for a 15 V pk-pk, 10 khz sine wave. Place the CM switch 11 in the ON position to decrease the value of C1. This decrease changes the value of resonance (33.9 khz). Increase the frequency of the generator to tune for series resonance, as determined by a voltage null across series combination L1 and C1. The new resonant frequency (f r ) you determine at V GEN is approximately a. 20 khz. b. 24 khz. c. 34 khz. d. 48 khz. 58 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance 2. The step-up voltage across a coil or capacitor in a series RLC circuit peaks a. at twice the resonant frequency. b. at the resonant frequency. c. below the resonant frequency. d. above the resonant frequency. 3. At series resonance, the circuit current is a. minimum and in phase with the applied voltage. b. maximum and in phase with the applied voltage. c. maximum and out of phase with the applied voltage. d. minimum and out of phase with the applied voltage. 4. At the resonant frequency (f r ) of a series RLC circuit, X L is a. greater than X C, and the circuit acts inductively. b. less than X C, and the circuit acts capacitively. c. equal to X C, and the circuit acts resistively. d. equal to R, and the circuit acts resistively. 5. When the value of resistance in an RLC series circuit increases, the a. resonant frequency remains the same. b. circuit current increases at resonance. c. circuit current remains the same at resonance. d. resonant frequency increases. NOTE: Make sure all CMs are cleared (turned off) before proceeding to the next section. FACET by Lab-Volt 59