Parallel Resonance

Objectives:

After completing this lab experiment using, you should be able to:
Observe the effect of frequency on impedance.
Observe the effect of Quality factor on parallel resonance.
Calculate and verify the resonant frequency in a parallel LC circuit.
Identify the phase relation between current and voltage in a parallel LC circuit.

Parts List:
Resistor (2) 1 Ω, (1) 100 Ω, (1) 500 Ω.
Inductor (1) 100 mH.
Capacitor (1) 47 nF.

Procedures:    

Part I:
Connect the following circuit in Multisim.

     Figure 1: Parallel LC Circuit


Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:

f_r=√(1-((R_W^2∙C)⁄L))/(2π√LC)=2.32kHz

Calculate the inductive reactance, capacitive reactance, total reactance (XL||XC) impedance magnitude, and phase angle for each frequency shown in Table 1. Ignore the winding resistance for your calculations.
Measure and record the resistor voltage for each of the frequencies listed in Table 1.

Frequency
(in Hz) Calculated Measured
XL XC XT VR(rms)
700 439.8 Ω 4837.5 Ω -4397.7 20.589 mV
900 565.5 Ω 3762.5 Ω -3197.0 14.961 mV
1k 628.3Ω 3386.3 Ω -2758 12.902 mV
2k 1256.6 Ω 1693.1 Ω -436.5 2.007 mV
Resonant freq. 2.32k (fr)
(from step 2) 1457.7Ω 1459.6 Ω -1.9 83.637 uV
3k 1885 Ω 1128.8 Ω 756.2 3.6 mV
5k 3141.6 Ω 677.3 Ω 2464.3 11.64 mV
7k 4398.2 Ω 483.8 Ω 3914.4 18.472 mV

Table 1: Calculated and measured values

Draw the frequency response curve from the above results on Plot 1.







Connect multimeters or current probes to measure total current or resistor current (IR), inductor current (IL) and capacitor current (IC). 
Measure and record the rms values for IR, IL, and IC in Table 2.

Frequency (in Hz) IC IL IR
700 2.074 mA 22.663 mA 20.589 mA
900 2.666 mA 17.627 mA 14.961 mA
1k 2.963 mA 15.864 mA 12.902 mA
2k 5.925 mA 7.932 mA 2.007 mA
Resonant freq. (from step 2) 2.32kHz 6.814 mA 6.897 mA 83.637 uA
3k 8.888 mA 5.288 mA 3.6 mA
5k 14.813 mA 3.173 mA 11.64 mA
7k 20.738 mA 2.266 mA 18.472 mA

                         Table 2: Measured voltage values   

Draw the current phasor on Plot 2.

Plot 2: Current Phasor

Disconnect the digital multimeters from the circuit.
Connect the Bode plotter as shown in Figure 2. 

Figure 2. Circuit with Bode Plotter

Measure the resonant frequency using the Bode plotter as show in Figure 3. 

Figure 3. Bode Plot Output Showing Resonant Frequency

Record the resonant frequency for the circuit in Table 3.
Calculate the Q factor for the circuit using the following equation.

Q= X_L/R_W
Replace the winding resistor RW with a 100 Ω resistor as shown in Figure 4.

Figure 4. Parallel Resonant Circuit with RW = 100 Ω

Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:

f_r=√(1-((R_W^2∙C)⁄L))/(2π√LC)

Measure and record the resonant frequency for the circuit in Table 3.
Calculate the Q factor for the circuit using the following equation.

Q= X_L/R_W
Replace the winding resistor RW with a 500 Ω resistor.
Calculate the exact resonant frequency, fr,
Measure and record the resonant frequency for the circuit in Table 3.
Calculate the Q factor for the circuit using the following equation.

Q= X_L/R_W

Resonant Frequency  

Winding Resistance Calculated Measured Q Factor
1 Ω 2.321 kHz 2.344 kHz 0.1
100 Ω 2.316 kHz 2.344 kHz 0.001
500 Ω 2.181 kHz 2.291 kHz 0.0002

Table 3. Resonant Frequency and Q Factor

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