FREQUENCY RESPONSE OF R, L AND C ELEMENTS

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FREQUENCY RESPONSE OF R, L AND C ELEMENTS Marking scheme : Methods & diagrams : 3 Graph plotting : - Tables & analysis : 2 Questions & discussion : 3 Performance : 2 Aim: This experiment will

1 FREQUENCY RESPONSE OF R, L AND C ELEMENTS Marking scheme : Methods & diagrams : 3 Graph plotting : - Tables & analysis : 2 Questions & discussion : 3 Performance : 2 Aim: This experiment will investigate the frequency response of the various components in a circuit. Then the resonance behaviour of capacitors and inductors in a circuit will be studied, together with its Q-value property. Such a circuit set-up can be used as an analog filter, in applications such as noise reduction, signal enhancement in video or audio systems. READ the Preliminary Theory BEFORE attempting to use the RLC applet on the computer (see page 126) to further understand this topic. Equipment List: Tektronics TDS210 digital real time oscilloscope with two probes Stanford Research Systems DS335 Function Generator Circuit board with Load resistor, Capacitor and Inductor PC for tabling and plotting Preliminary Theory: The total resistance is independent of the frequency of the AC power supply in the circuit. This is true for resistors in any configuration (parallel and series). However, the reactance of an inductor and a capacitor will depend on the frequency and are determined by : X L = ωl (3) X C = 1 (4) ωc where X L and X C are the reactances for the inductor and capacitor respectively while ω = 2 π f and f is the frequency of the AC power supply at time t. The unit for reactance X is Ω. Consider a circuit with resistor R and element X of reactance X T in series (see the above diagram). The total impedance (in the unit of Ω) of the circuit is given by: Z = R 2 + XT 2 The reactance X T depends on the element: 43

2 X T X L X C X L X C (1/X C 1/X L ) 1 Element X inductor only capacitor only inductor and capacitor in series inductor and capacitor in parallel The AC analog to Ohm s law is V = IZ where V and I are the rms or effective values of the voltage and current. Because the capacitor reactance is dependent on frequency, the current and voltage outputs will also be frequency dependent. One of the characteristics of a series RC (where element X is capacitor C) circuit is the filtering capability of the input power supply. The resistor acts as a highpass filter, allowing only frequencies above a given critical frequency (breakpoint frequency) f c to pass while the capacitor acts as a lowpass filter which passes only frequencies below the critical frequency. This critical frequency is given by: f c = 1 2πRC and it occurs when the capacitive reactance X C = R. In an ideal case, the voltage gain for the resistor (V R /V ) in a highpass filter increases from 0 to a maximum value of 1 where it then plateaus at f c, as shown in the following curve. The voltage gain for the capacitor will decrease from a maximum plateau to a minimum value also at f c. V is the potential drop across the function generator (or the total output voltage) and is equal to the sum of V R and V C. Experimentally, some attenuation (or filtering) of the output voltage still occurs at high 44

3 frequencies and the gain does not achieve the maximum value of 1. The experimental value of f c occurs when the gain is If a capacitor is allowed to discharge through a resistor, the energy will exponentially dissipate away. However, if the capacitor is allowed to discharge through an inductor, the energy stored in the capacitor will simply be transferred to the inductor during the discharge, since an inductor stores energy. Initially, the electrical energy from the capacitor is transferred into the magnetic energy of the inductor. When the electrical energy of the capacitor becomes zero, the process is reversed. The result is an oscillation with the energy repeatedly passed back and forth between the capacitor and the inductor at a frequency of f. This harmonic process will continue indefinitely if both the inductor and capacitor do not have any internal resistance. Hence, an AC circuit which has both of these elements in parallel (element X consisting of L and C in parallel), known as a notch filter, will exhibit an output which will oscillate at that same frequency f. Moreover, this output will also oscillate between minimum and maximum as the input frequency is swept up and down the range. This leads to the phenomena of ringing and resonance. When an AC signal is applied at a special frequency the amplitude of the output voltage of the circuit increases dramatically to its maximum. This is the resonance frequency ω (= 2πf ) and it occurs when X C = X L. ω = 1 LC Resonance behaviour has widespread practical applications, and is important, for example, in generating radio signals and in tuning a radio receiver to respond only to signals in a given range (for example in television and radio channel selections). The sharpness of the peak in the resonance response curve is called the quality factor or the Q-value of the resonance. Q-value can be calculated from Q value = ω RC = R ω L = ω /BW where BW is the bandwidth or the difference between the two halfpower frequencies (or the 1 2 peak current or peak voltage frequencies) and is read from the resonance response curve as shown here. If small damping (ie low resistance) occurs in the system then the Q-value is very large and the amplitude of oscillation at resonance is illustrated by a sharply defined peak. In such a case, the oscillating system will lose energy slowly and the noise performance is enhanced thus, giving a better quality output. If the damping is larger, then the Q-value is much smaller so that the amplitude of oscillation at resonance is much lower and described by more of a flattened curve. A circuit with low R, for a given L and C, has a sharper resonance or higher Q-value. Increasing the resistance makes the resonance less sharp. The former circuit is more selective: it produces 45

4 high currents only for a narrow bandwidth, ie a small range of f. The circuit with larger R responds to a wider range of frequencies and so has a larger BW bandwidth. Experimental Tasks: 1. Highpass/Lowpass Filter Connect up the circuit as shown below. With the switch S1 open, both the resistor R and capacitor C are in series. Connect CH1 of the oscilloscope across points AC and CH2 across AB. The peak-to-peak total output voltage V pp (or the potential drop across the generator) will be measured on CH1 while the V Rpp will be measured on CH2. Set it such that both CH1 and CH2 display the V pp values. It is important to ensure that the probe earths for both the channels are connected to 0V of the function generator and all the polarities are connected as shown in the diagram. Make sure that the Impedance is set to High-Z (SHIFT 6) on the Function generator. Set the amplitude of the output voltage to 20V, and set the frequency to sweep from 1Hz to 100kHz in steps of 0.1Hz by doing the following on the front panel buttons: (i) Select the AMPL button. (ii) Type 20 in the data entry box and select V pp for an output of 20V peak-to-peak sinusoidal waveform. (iii) Select START FREQ button. (iv) Type 1 and select Hz for 1Hz. (v) Select STOP FREQ button. (vi) Type 100 and select khz for 100kHz. (vii) Select SWEEP RATE button. (viii) Type 0.1 and select Hz for a sweep rate of 0.1Hz. (ix) To begin the sweep, select SHIFT and then START FREQ. (x) To stop the sweep, select SHIFT and then START FREQ again. 46

5 Describe what happens to the amplitude of V Rpp and compare it to that of V pp as a function of frequency. The frequency will now be set as single input, with V pp, V Cpp and V Rpp being recorded for each input. (i) Open up the workbook RLC template.xls on Excel and enter in the data as you go along. (ii) Select FREQ button, enter the desired numerical values and select the appropriate units (Hz or khz). (iii) For each frequency entry, Autoset the oscilloscope before obtaining the V pp and V Rpp values. Take the required measurements (V pp and V Rpp ) for the frequencies listed in the table. The impedances, current and voltage ratio values will automatically be calculated in the table. Due to the importance of the earthing connection for the oscilloscope, turn off the function generator and reconnect the circuit as shown in the following diagram for the next set of measurements. Turn off the power supply to the circuit and reconnect CH1 across AC and CH2 across BC. The peak-to-peak output voltage V pp for the circuit will again be measured on CH1 while V Cpp will be now be measured on CH2 of the oscilloscope. Record V Cpp for the same frequencies as the previous step. Print out the table when all data has been collected. Go to the last sheet of the workbook (ie click on the Plotting sheet). All the data from the first ( RC ) sheet will automatically be transfered here. The ratios of V Rpp /V pp and V Cpp /V pp will provide the gain values for the resistor and the capacitor. From Table 1, plot the voltage gains V Rpp /V pp and V Cpp /V pp as a function of frequency f on the same graph. To plot, highlight the necessary columns, copy and paste into Scidavis. Plot. Use the cursor function when obtaining values of the curves. Describe the relationship of V Rpp and V Cpp as a function of frequency. What is the experimental breakpoint frequency f c? Compare it with the calculated value. 47

6 At what voltage gain value does f c occurs? How does that compare with the value of 0.707? Compare the experimental and calculated reactance X C values. X Ccalc is calculated from Eq Resonance Close the switch S1 in the circuit and the inductor L is now parallel with C. CH2 measures the output V LCpp across the points BC. Go to the RLC sheet of the workbook and obtain the voltage outputs across R and LC together with the total output voltage V pp on CH1 at the frequencies listed in the table. Print the table when all data has been collected. As like in the previos section, go to Plotting worksheet and plot from Table 2 the circuit current I pp vs ω. Plot a point-graph and NOT a line-graph. Draw, by hand, a curve that would best-fit the data points. Label on the curve the resonance frequency ω. Obtain the value with the help of the cursor function. Compare the experimental and theoretical ω. What is the Q-value? Compare it with the expected value. Conclusions: Summarize your results and conclusions for all parts of this experiment. 48

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