Series Resonance AC 2 Fundamentals Exercise 2: Q and Bandwidth of a Series RLC Circuit EXERCISE OBJECTIVE When you have completed this exercise, you will be able to calculate the bandwidth and Q of a series RLC circuit by using standard formulas. You will verify your results with an oscilloscope. DISCUSSION I RESON ) at the resonant frequency (f r A sharply, or highly, selective circuit is responsive to a limited range of frequencies. A broadly, or poorly, selective circuit is responsive to a wide range of frequencies. Would a response curve of a highly selective series RLC circuit have a narrow (sharp) bandwidth (B) or broad (wide) bandwidth? a. narrow b. broad 60 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance The response curve of a series RLC tuning circuit has a broad (wide) bandwidth. Would this type of response curve make a highly selective tuning circuit for a radio? a. yes b. no The selectivity of a resonant circuit is determined by its bandwidth (B). The bandwidth is determined by the upper and lower cutoff frequencies of the circuit. FACET by Lab-Volt 61
Series Resonance AC 2 Fundamentals In the response curve of a series RLC circuit, I RESON is the maximum circuit current at the resonant frequency (f r ). What is I RESON of the circuit shown? I RESON V = R1 GEN I RESON = ma pk-pk (Recall Value 1) The lower and upper cutoff frequencies are 70.7% of, or 3 db down from, the maximum current at resonance (I RESON ). 62 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Determine the 3 db point when I RESON is 15 ma pk-pk. I 3dB = 0.707 x I RESON I 3dB = ma pk-pk (Recall Value 2) The bandwidth (B) is determined from the upper and lower cutoff frequencies. B = f 2 f 1 The bandwidth (B) of this response curve is a. 8.5 khz. b. 34.5 khz. c. 17.0 khz. A characteristic of a resonant circuit is its sharpness of resonance. This sharpness is indicated by a factor called Q. X L Q = R where X L is the inductive reactance at the resonant frequency and R is the total resistance in series with X L. The higher the Q of a circuit, the better its selectivity. Less than 10 is a low Q, while more than 250 is a very high Q. The higher the Q, the smaller the bandwidth (B). f r B = Q FACET by Lab-Volt 63
Series Resonance AC 2 Fundamentals Which curve displays the lower circuit Q? a. A b. B The Q of the resonant circuit also determines how much step-up voltage is across L1 or C1. Compute the resonant rise across C1 or L1 with the following expression. V C1 or V L1 = Q x V GEN With a circuit Q of 3, the resonant step-up voltage across either L1 or C1 equals a. 15 V pk-pk. b. 45 V pk-pk. c. 3 V pk-pk. 64 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance PROCEDURE Adjust the generator output voltage (V GEN ) for a 15 V pk-pk, 20 khz sine wave. 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. FACET by Lab-Volt 65
Series Resonance AC 2 Fundamentals Determine the resonant frequency (f r ) by using an oscilloscope (CH 1) to measure the period (T) at V GEN. 1 fr = T f r = khz (Recall Value 1) With the circuit tuned at resonance, determine the maximum circuit current (I RESON ). I RESON = V R3 R3 I RESON = ma pk-pk (Recall Value 2) 66 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Use your measured value of I RESON ( ma pk-pk [Step 5, Recall Value 2]) to determine the 3 db point. I 3dB = 0.707 x I RESON I 3dB = ma pk-pk (Recall Value 3) To determine the lower cutoff frequency ( 3 db point), slowly decrease the generator frequency as you observe a decrease in voltage across current-sensing resistor R3. Stop when the current equals your calculated value of I 3dB ( ma pk-pk [Step 6, Recall Value 3]). FACET by Lab-Volt 67
Series Resonance AC 2 Fundamentals Determine the lower cutoff frequency (f 1 ) by using an oscilloscope (CH 1) to measure the period (T) at V GEN. 1 f1 = T f 1 = khz (Recall Value 4) To determine the upper cutoff frequency ( 3 db point), slowly increase the generator frequency from the lower cutoff frequency (f 1 ) past resonance (f r ). As the current decreases from resonance, stop increasing generator frequency when the current equals your calculated value of I 3db ( ma pk-pk [Step 6, Recall Value 3]). 68 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Determine the upper cutoff frequency (f 2 ) by using an oscilloscope (CH 1) to measure the period (T) at V GEN. f 2 1 = T f 2 = khz (Recall Value 5) Using your values of lower (f 1 ) and upper (f 2 ) cutoff frequencies, compute the bandwidth (B) of the circuit. B = f 2 f 1 B = khz (Recall Value 6) f 1 = khz (Step 8, Recall Value 4) f r = khz (Step 4, Recall Value 1) f 2 = khz (Step 10, Recall Value 5) Is the selectivity of this resonant circuit determined by the bandwidth? a. yes b. no B = khz (Step 11, Recall Value 6) FACET by Lab-Volt 69
Series Resonance AC 2 Fundamentals Compute the circuit Q from your values of resonant frequency and bandwidth. f r Q = B Q = (Recall Value 7) B = khz (Step 11, Recall Value 6) f r = khz (Step 4, Recall Value 1) Place CM switch 9 in the ON position to increase the value of R2 to 3.3 k. Tune for series resonance, as determined by a voltage null across series combination L1 and C1 (CH 2). 70 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance With the series circuit at resonance and R2 equal to 3.3 k, connect the oscilloscope as shown, and measure the voltage drop across C1 (V C1 ). V C1 = V pk-pk (Recall Value 8) Determine the circuit Q with R2 equal to 3.3 k. V Q = V C1 GEN Q = (Recall Value 9) V C1 = V pk-pk (Step 15, Recall Value 8) Compare your values of circuit Q. In a series RLC circuit, does the bandwidth increase or decrease with an increase of series resistance? a. increase b. decrease Q = (Step 13, Recall Value 7) with R2 equal to 1 k Q = (Step 16, Recall Value 9) with R2 equal to 3.3 k Make sure all CMs are cleared (turned off) before proceeding to the next section. FACET by Lab-Volt 71
Series Resonance AC 2 Fundamentals CONCLUSION The lower and upper cutoff frequencies are determined by the 3 db points. The bandwidth is determined from the cutoff frequencies (B = f 2 f 1 ). Q can be computed from resonant frequency and bandwidth (Q = f r Q equals the voltage across either the capacitive or inductive component divided by the source voltage (Q = V C GEN or V L GEN ). In a series RLC circuit, an increase in resistance decreases Q. The higher the Q, the narrower the bandwidth (B) and the better the selectivity. The lower the Q, the wider the bandwidth and the poorer the selectivity. REVIEW QUESTIONS 1. GEN for a 15 V pk-pk, 20 khz sine wave. Place the CM switch 10 in the ON position to increase the value of C1. Increase the frequency of the generator to tune for series resonance, as determined by a voltage null across series combination L1 and C1. 72 FACET by Lab-Volt
AC 2 Fundamentals Series Resonance Measure the voltage across C1 (V C1 ) and determine the circuit Q from the equation Q = V C1 GEN. a. 1.3 b. 6.0 c. less than 0.5 d. greater than 10 2. The selectivity of a resonant RLC circuit is a function of the circuit Q. Which of the following Q values has the best selectivity? a. 3.5 b. 6.0 c. 1.0 d. less than 1.0 3. The peak current (I RESON ) in a series RLC circuit is 8.5 ma pk-pk. The 3 db (0.707) points occur at a. 4.25 ma pk-pk. b. 7.50 ma pk-pk. c. 6.00 ma pk-pk. d. 8.50 ma pk-pk. 4. The Q of an RLC series resonant circuit determines the a. resonant frequency. b. step-up voltage across the inductor. c. step-down voltage across the capacitor. d. generator voltage setting. 5. The bandwidth (B) of a series RLC resonant circuit is 20 khz with a lower cutoff frequency of 50 khz. The circuit s upper cutoff frequency is a. 20 khz. b. 70 khz. c. 30 khz. d. 60 khz. NOTE: Make sure all CMs are cleared (turned off) before proceeding to the next section. FACET by Lab-Volt 73