When the inductive and the capacitive reactance in an electrical circuit are an equal magnitude, and electrical energy will be produced by oscillating between the electrical and the magnetic field of the capacitor and the phenomena is known as the electrical resonance. In series, the resonance of an RLC circuit will occur when the reactors are both equal and 180 degrees apart, therefore. they will cancel each other. In impedance, the sharp minimum will occur, and it will be useful in the turning applications. A series resonant circuit has a minimum impedance where Z=R which is measured at a resonance frequency and putting the phase angle to be zero.
Theory
Under certain conditions, a series or parallel RLC circuit behaves as a pure resistance. This happens when XC=XL. For parallel resonant circuits, the impedance is maximum (in theory, infinite) at the resonant frequency. Total current is minimum at the resonant frequency, and the Bandwidth is the same as for the series resonant circuit; the critical frequency impedances are at 0.707Zmax.
Circuits of this type are widely employed in filter circuits, and radio and television tuning circuits.
Objectives
Apply circuit theory techniques to the solution of circuit problems
Use circuit theory to solve problems relating to series and parallel tuned circuits
Instructions
Part 1: Resonance in a series RLC circuit
1. Initially measured the coil resistance of the inductor and connected the circuit as shown in Figure 1.
2. Set the signal generator to give a 1 V RMS sine wave signal and connected the AC voltmeter to measure the voltage across the 47Ω resistor.
3. Adjusted the signal generator frequency until the measured voltage, and hence circuit current rose to a maximum. This is the resonant frequency of the circuit.
4. Calculated and recorded the current flowing in the circuit and measured the voltage across the capacitor at resonance.
Results
Resonance in a series RLC circuit
Frequency
Voltage
Current
4
37.79
0.000771
4.1
43.88
0.000881
4.2
50.63
0.00102
4.3
60.72
0.00122
4.4
75.45
0.00149
4.5
98.44
0.00191
4.6
135.881
0.0026
4.7
184.213
0.00361
4.8
196.983
0.0042
4.9
168.751
0.00375
5
132.82
0.003
5.1
106
0.00241
5.2
87.150
0.00197
5.3
73.419
0.00165
5.4
63.241
0.00141
5.5
55.511
0.00123
5.6
49.438
0.00109
5.7
44.568
0.000984
5.8
40.580
0.000893
5.9
37.275
0.000819
6
34.448
0.000756
Circuit Design with Multisim
The circuit below was modeled with Multisim software.
Plot from Lab results
Results from simulation
Calculations
For the plot of load current against frequency, the current increases with frequency till a maximum value then starts decreasing. The frequency at which the maximum current occurs is called the Resonant Frequency, and it is given by:
Resistance for a series circuit
Quality factor. This refers to the sharpness of the resonance in a series resonance circuit. It relates to the maximum energy stored in the circuit to the energy dissipated during each cycle of oscillation.
Bandwidth
Discussion
Theoretically, the presence of a resistor will cause the oscillations to die out over a period. It is seen from the experiment that the resonant frequency and the capacitive reactance are almost equal which caused electrical energy oscillated between the capacitor electrical field and the inductor magnetic field. The result is similar to the theoretical value where capacitive reactance and the inductive reactance of the circuit are equal. The inductive reactance is directly proportional to the frequency XL = 2πfL. When the inductive reactance is zero, the circuit acted as a short circuit just as the theoretical function but the inductive reactance increased as the frequency increased. When the frequency was increased further, the circuit acted as an open circuit.
The resonant current curve of theoretical value and that of the experimental value is similar. This is because both the capacitance and the inductive reactance cancelled each other since the current flowing through each other is the same in all elements. This makes the voltage across the capacitor and the inductor to be equal in magnitude, and they will be flowing in opposite directions cancelling each other. It is evident that the RLC circuit current in series is given by I = V / Z but the resonance current will be given by I = V / Z. As shown in the above graph, the maximum frequency is at a maximum at the resonance, and the impedance is at the minimum since it is only resistance.
From the graph, when the frequency increases at the start, there is a corresponding decrease in the impedance thereby increasing the current of the circuit. The frequency then becomes equal to the resonant frequency at the point where the capacitive reactance becomes equal to the inductive reactance. It causes the impedance of the circuit to reduce which is equal to the circuit’s resistance only and the current at this point is at its maximum. This is similar to the theoretical function of resonance in RLC circuits that are arranged in series.
The results from the simulation can be compared to the practical results obtained from practical results. The above bandwidth is the theoretical value of the simulation. The practical bandwidth is obtained by taking the frequency between two points. It gives the frequencies over which at least half of the maximum power and current is provided.
Conclusion
From the results obtained above, it is observed that the theoretical and practical bandwidth varies.
From the graphs shown above it can, therefore, be seen that a series resonant circuit can act as a bandpass filter. This is because, at low frequencies, the capacitor acts as an open circuit, and limits the amount of current that flows. At high frequencies, the inductor L will be an open circuit and virtually no current will flow again. However, at the resonant frequency, the inductance and capacitance of the circuit are equal. The difference between them is zero. Therefore, maximum current will flow in the circuit. At this point, only Resistance is limiting current. This explains the graph above. For the phase plot against the frequency, at point of resonance that is when the capacitance and the inductance are same, the angular difference is 1800 as shown above.
References
Constantinovici, L. D., " Govindsamy, M. (1999). Basic circuit analysis for electrical engineering. Kenwyn: Juta.
Hughes, E. (2008). Electrical and Electronic Technology. Essex: Pearson Education Limited .
Sadiku, M. N., " Alexander, C. K. (2013). Engineering Circuit Analysis. New York: Tata McGrawhill .
