Application of Series Resonance - Lab Report Example

Electrical & Electronic Principles 2 Experiment Series Resonance Experiment Number: Section: Experiment: Lab Partners: Instructor Name: Abstract Series resonance is mainly applied in tuning devices such as in AM radio receivers to selectively respond to specific frequency while discriminating other frequencies. It’s achieved in when an RLC circuit is arranged in series and at a point when the magnitude of inductive reactance and capacitive reactance is equal. This experiment was done with the aim of obtaining the resonant frequency and the half power frequencies of a series RLC circuit. It was also aimed at determining the effects of resistance on resonant bandwidth. A resistor capacitor and inductor were arranged in series with a signal generator which was used to generate different frequencies below an above the calculated resonant frequency, ƒo = 9188.8149Hz. The changes in both the capacitive and inductive reactance were observed and tabulated. It was noted that an increase in frequency caused an increase in inductive reactance while a decrease in capacitive reactance. Introduction and Background Series resonance occurs in a series RLC circuit when the magnitude of capacitive is equal to the magnitude of inductive reactance. It is a special frequency that highly depends on the values of capacitor, resistor and the inductor connected in series. Resonance in series RLC occurs at minimum resistance when both the capacitive and inductive resistances are 180o out of phase. There exists sharp minimum impedance at series resonance which is mainly applicable in tuning devices for selectivity. The value of the resistor, R, highly determines how sharp the minimum impedance is for tuning. A major application of series resonance exists in AM receivers where the series RLC is used to produce series resonance for selectivity and tuning of AM radio stations. The series resonance is used to selectively respond to a signal of a given frequency while discriminating the signals of different frequencies. An RLC circuit is said to have high selectivity when its response around the given frequency is narrowly peaked and a low selectivity when its band is widely peaked. Basically, selectivity defines how well an RLC (resonant) circuit responds to a certain frequency while discriminating against all other frequencies. In a series RLC circuit the capacitive resistance, XC and the inductive resistance, XL have opposite effects on the phase angle thus the total their total impedance is less than the individual resistances. The circuit is inductive when the inductive resistance, XL, is more than the capacitive resistance, XC, while it is capacitive when the capacitive resistance is more than the inductive resistance. Resonance occurs when the magnitude of capacitive resistance equals inductive resistance, a point characterized by minimum impedance and maximum current. The resultant impedance of the circuit at resonance equals the value of R, purely resistive impedance. At resonance; XL = XC and Zr = R ƒo = 1 /( 2ӆ√LC) The current is maximum at resonance Imax= Vs/ R, but decreases above and below resonance due to the increase in impedance. The voltages across the capacitor and inductor are at their peak at resonance but at 180o out of phase hence they cancel out. The range of lower and upper cut off frequencies, for which the current is 70.7% of the resonant value, make up the bandwidth of the resonant circuit. At half-power frequencies, the true power delivered from the source is usually half the power delivered to the devices at resonance. BW= ƒ2- ƒ1 ƒo = ½ (ƒ1+ƒ2) The bandwidth to a resonant circuit is inversely proportional to the quality factor, Q, of the circuit and directly proportional to its resonance frequency, ƒo. A narrow bandwidth results from a high value of Q. Bandwidth in terms of quality factor is therefore given by; BW= ƒo /Q Objectives To obtain the resonant frequency of a RLC circuit through both experimental procedure and theoretical means To obtain the half power frequencies and observe the amplitude of the current approaching either side of the resonant frequency, ƒo To observe the effects of resistance on resonant bandwidth Experimental Procedure Apparatus Inductor, decade box Capacitor, decade box Resistor Philips analogue meter Oscilloscope plus probes Signal generator The resistor, capacitor and the inductor were set up and their nominal values checked using the electronic bridge/ multi meter. The resulting values were then recorded and used to calculate the resonant frequency, ƒo The capacitance reactance and inductance reactance were tested by connecting the devices to a signal generator at a time. The frequency of the signal generator was adjusted below and above the calculated resonant frequency and the corresponding values of current and voltages observed and recorded differently for both the capacitor and the inductor. The observed values of current and voltages were used to calculate and tabulate the respective reactance of both the capacitor and the inductor. The capacitor, inductor, resistor and the ammeter were connected to the signal generator in series in order to make up a series RLC circuit for the measure of series resonance. The signal generator was connected to the circuit with the oscilloscope across the resistor. With the circuit connected, the signal generator was set to generate a sine wave form at a frequency below the calculated value of resonant frequency, ƒo . The circuit supply voltage was measured and recorded. The frequency of the of the waveform Vr on the oscilloscope was observed as well as the value of current on the ammeter while adjusting the frequency of the supply upwards from below the calculated value of ƒo until the maximum value for the current was observed. The waveform and values at resonant frequency were recorded including the Volt/div, Time/div, voltage across the resistor VR , maximum current Imax, voltage across capacitor VC and inductor VL While keeping the value of the supply voltage constant, the frequency of the supply was adjusted and the values of the lower and upper half power point frequencies obtained. The step was repeated using different resistor values and observations recorded. Experimental Results and Calculations Theoretical resonant frequency The nominal values of resistor, capacitor and the inductor were recorded as follows and used to calculate the theoretical resonant frequency, ƒo R= 470Ω; C= 10nF; L= 30mH; ƒo = 1 /( 2ӆ√LC) ƒo = 1 /{2ӆ√(30x10-3 )10x10-9} ƒo = 9188.8149Hz ƒo = 9.188K Hz Capacitor and Inductor Reactance Capacitor Frequency(Hz) Voltage (V) Current (mA) Reactance(Ω) 50 7.05 0.022 318K 1000 7.05 0.3 15.9 K 11000 6.07 9 1.45K 45000 0.164 7.87 353.7 75000 0.005 5.05 353.7 Inductor Frequency (Hz) Voltage (V) Current(mA) Reactance 50 0.804 121 6.28 1000 6.40 50.9 125.6 11000 6.02 4. 86 1.38 K 45000 0.14 0 5.65 K 75000 0.16 0 9.42 K The maximum value were recorded as follows VR = 0.16 Imax = 0.88 mA VL = 0.497 VC = 0.539 Value of current at half power frequencies I= 0.707 Imax I = 0.707x0.88 mA I = 622.16 x10-6 A VR (at ƒ1) = 0.2924 ƒ1 = 12.77 KHz ƒ2 = 9.7225 KHz BW= ƒ1-ƒ2 = (12.77 KHz - 9.7225 KHz) = 3.049 KHz (a) Quality factor, Q Q= ƒo / BW Q=9.188 KHz/ 3.049 KHz Q= 3.0134 (b) Quality factor, Q (Using Measured resistance) Q= XL/ R But XL = 2ӆ ƒo L XL = 2ӆ {9.188 KHz x 30mH} XL = 1732.121 Q= 1732.121/ 470 Q= 3.685 (c) Quality factor, Q (Using Vs and VL) Q= VL / Vs Q= 0.497/ 1.5 Q= 0.331 (f) (i) Maximum current Imax Imax = Vs/ Rt Imax =1.5v / 470 Imax = 3.19 mA (ii) Circuit resistance R= V/ Imax R= 1.5/ 0.88 mA R= 1.704 K Ω Discussion Series resonance occurs at a point when the inductive reactance is equal to the capacitive reactance in a series RLC circuit. At this point the capacitor and the inductor are out of phase by 180o hence their impedance cancels out leaving only the resistance from the resistor at play. The circuit is purely of resistive impedance. Increasing or decreasing the magnitude of the resistance from the resistor therefore greatly affects the bandwidth of the circuit. A higher magnitude of R will result into a narrow bandwidth while lower magnitudes of R will results into a wide bandwidth. The capacitive reactance increases with increase in frequency while the inductive reactance reduces with increase in frequency. Before the resonant frequency, the circuit is said to be capacitive given that XC>XL . The circuit achieves a series resonance at the point when XC=XL and becomes a inductive after the resonant frequency since XL > XC References Bakshi, A. V. Bakshi, U. A. (2008) Electric Circuits.1st Edition, Technical Publications Pune, Amit Recidency 412, India Read More