EET 202
LAB EXPERIMENT INSTRUCTIONS
AC circuit analysis
Lab # 5: Resonance
Experiment: Parallel Resonance.
Objective:
After performing this experiment, you will be able to:
 Compute the resonant frequency, Q and the bandwidth of a parallel resonant circuit.
 Measure the frequency response of a parallel resonant circuit.
 Use the frequency response to determine the bandwidth of a parallel resonant circuit.
Materials Required:
Inductor: 100 mH – 1 Piece.
Resistor: 1 KΩ – 1 Piece.
Capacitor: 0.047 µ F – 1 Piece.
Summary of theory:
In an RLC parallel circuit, the current in each branch is determined by applied voltage and the impedance of that branch. For an “ideal” inductor (no resistance), the branch impedance is X_{L} and for a capacitor the branch impedance is X_{C}. Since X_{L} and X_{C} are functions of frequency, it is apparent that the currents in each branch are also dependent on frequency. For any given L and C, there is a frequency at which the currents in each are equal and of opposite phase. This frequency is the resonant frequency and is found using the same equation as was used for series resonance.
The circuit and the phasor diagram for an ideal parallel RLC circuit at resonance as illustrated in Figure 521. Some interesting points to be observed are: The total source current at resonance is equal to the current in the resistor. The total current is actually less than the current in either the inductor or the capacitor. This is because of the opposite phase shift which occurs between inductors and capacitors, causing the addition of currents to cancel. Also, the impedance of the circuit is solely determined by R, as the inductor and capacitor appear to be open. In a twobranch circuit consisting of only L and C, the source current would be zero, causing the impedance to be finite, of course, this does not happen with actual components that do have resistance and other effects.
Figure 521
In a practical twobranch parallel circuit consisting of an inductor and a capacitor, the only significant resistance is the winding resistance of the inductor. Figure 522 (a) illustrates a practical parallel LC circuit containing winding resistance. By network theorems, the practical LC circuit can be converted to an equivalent parallel RLC circuit, as in the Figure 522 (b). The equivalent circuit is easier to analyze. The phasor diagram for the ideal parallel RLC circuit can then be applied to the equivalent circuit as was illustrated in Figure 521. The equations to convert the inductance and its winding resistance to an equivalent parallel circuit are
Where R _{p (e q)} represents the parallel equivalent resistance and R_{W} represents the winding resistance of the inductor. The Q in the conversion equation is the Q for the inductor.
Q = X_{L }/ R_{W}
Figure 522
The selectivity of series circuits was discussed in the previous part in series resonance. Similarly the parallel resonant circuits also respond to a group of frequencies. In parallel resonant circuits, the impedance as a function of frequency has the same shape as the current versus frequency curve for series resonant circuits. The bandwidth of a parallel resonant circuit is the frequency range at which the circuit impedance is 70.7% of maximum impedance. The sharpness of the response to the frequencies is again measured by circuit Q. The circuit Q will be different from Q of the inductor if there is additional resistance in the circuit. If there is no additional resistance in parallel with L and C, then the Q for a parallel resonant circuit is equal to the Q of the inductor.
Procedure:
 Measure the value of a 100 mH inductor, 0.047 µF capacitor, and a 1.0 KΩ resistor. Enter the measured values in Table 521. If it is not possible to measure the inductor or capacitor, use listed values.
 Measure the resistance of the inductor. Enter the measured inductor resistance R_{W} in Table 521.

Listed Value 
Measured Value 
L_{1} 
100 mH 

C_{1} 
0.047 µF 

R_{S1} 
1.0 KΩ 

RW (L_{1} resistance) 

Table 521

Computed  Measured 
fr 

Q 


BW 

fi = BW/4 

Table 522
 Construct the circuit in Figure 523. The purpose of R_{S1} is to develop a voltage that can be used to sense the total current in the circuit. Compute the resonant frequency of the circuit using the equation
Enter the computed resonant frequency in Table 522. Set the generator to the f_{r} at 1.0 V_{PP} output as measured with your oscilloscope. Use peaktopeak values for all voltage measurements in this experiment.
Figure 523
 The Q of a parallel LC circuit with no resistance other than the inductor winding resistance is equal to the Q of the inductor. Compute the approximate Q of the
parallel LC circuit from; Enter the compounded Q in Table 522.
 Compute the bandwidth from the equation
Enter this as the compounded BW in the Table 522.
 Connect your oscilloscope across R_{S1} and tune for resonance by observing the voltage across the resistor, R_{S1}. Resonance occurs when the voltage across R_{S1} is minimum, since the impedance of the parallel LC circuit is highest. Measure the resonant frequency (f_{r}) and record the measured result in Table 522.
 Compute a frequency increment (f_{r}) by dividing the computed bandwidth by 4.
That is Enter the computed f_{r} value in Table 522.
 Use the measured resonant frequency (f_{r}) and the frequency increment (f_{i}) from the Table 522 to compute 11 frequencies according to the computed Frequency column of the Table 523. Enter the 11 frequencies in the column 1 of the Table 523.
Computed Frequency

VR_{S1} 
I 
Z 
f_{r} – 5 f_{i} =  
f_{r} – 4 f_{i} =  
f_{r} – 3 f_{i} =  
f_{r} – 2 f_{i} =  
f_{r} – 1 f_{i} =  
f_{r} =  
f_{r} + 1 f_{i} =  
f_{r} + 2 f_{i} =  
f_{r} + 3 f_{i} =  
f_{r} + 4 f_{i} =  
f_{r} + 5 f_{i} = 
Table 523
 Tune the generator to each of the computed frequencies listed in Table 523. At each frequency, check that the generator voltage is still at 1.0 V_{PP}; then measure the peaktopeak voltage across R_{S1}. Record the voltage VR_{S1} in the column 2 of the Table 523.
 Compute the total peaktopeak current, I, at each frequency by applying Ohm’s Law to the sense resistor R (That is, I = VR_{S1}/R_{S1}). Record the current I in column 3 of Table 523.
 Use Ohm’s law with the measured source voltage (1.0 V_{PP}) and the source current at each frequency to compute the impedance at each frequency.
Complete Table 523 by listing the computed impedance Z.
 On Plot 521, draw the impedance versus frequency curve. From your curve determine the bandwidth. Complete Table 522 with the measured bandwidth.
Plot 521