LRC Circuits. Purpose. Principles PHYS 2211L LAB 7

Transcription

1 Purpose This experiment is an introduction to alternating current (AC) circuits. Using the oscilloscope, we will examine the voltage response of inductors, resistors and capacitors in series circuits driven by an AC power source. We will also find the resonant frequency for each of two inductor-resistor-capacitor (LRC) circuits and compare these frequencies with those predicted by theory. Principles AC Circuits In alternating current (AC) circuits, the current varies with time, both in magnitude and direction. The voltage drop across circuit elements also varies with time, both in magnitude and polarity (sign). It has been found that a time-varying voltage is a more efficient way to deliver power to electric devices. The most common time-varying voltage has the form of a sine wave. Look at Diagram 1, which represents a typical sine wave voltage it might be the output of an ordinary electric socket. From the graph, we see that the voltage starts at zero (or ground), rises to a maximum value (called the peak voltage, V p ), then falls back to zero. The voltage then reverses polarity and falls to a maximum value below ground before rising again. As the voltage reverses polarity, the current reverses direction, so that charge oscillates back and forth in the wires hence the name alternating current. All of this takes place in a sinusoidal and cyclic fashion. The ever-changing voltage and current have effects that do not arise in DC circuits, some of which we will examine in this lab. First, some terminology: The amplitude of the signal is its maximum strength. It is the maximum value the voltage reaches, whether positive or negative. See Diagram 1. The period (T) of the signal is the time for the signal to go through one complete cycle from crest back to crest, for instance. The frequency (f) of the signal is the inverse of the period how many cycles 1 take place in one second: f =. T 87

2 With oscilloscopes, it is convenient to measure AC voltages peak-to-peak. That is, we measure the voltage difference between the top of a crest and the bottom of a trough. The peak-to-peak voltage is twice the amplitude of the signal, but if we measure all voltages and currents peak-to-peak, the factor of two will cancel in any calculations Note on DMMs: Digital multimeters measure the root-mean-square value of AC voltages: V rms. This is the square root of the average squared voltage over one cycle. For sine wave voltages this works out to be: V rms = V p / 2, where V p is the peak voltage. (The peak-to-peak voltage, V pp, is twice V p.) Inductors, Capacitors and Resistors in an AC Circuit A capacitor is a device that stores charge. It usually consists of two conducting electrodes separated by non-conducting material. Current does not actually flow through a capacitor in circuit. Instead, charge builds up on the plates when a potential is applied across the electrodes. The maximum amount of charge a capacitor can hold at a given potential depends on its capacitance. The unit of capacitance is the farad, which is dimensionally equivalent to seconds per ohm. The voltage across the capacitor at any time depends on the charge on its electrodes: V c = Q/C. It takes time for this charge to build up. If the frequency of the driving voltage is high, the capacitor will not have time to charge fully before the current in the circuit reverses direction and the capacitor starts discharging. For this reason, the voltage across a capacitor decreases at high frequencies and increases at low frequencies it is inversely proportional to the frequency of the current. For the same reason, there is a time lag between the voltage across the capacitor and the current in the circuit. We say that the current and the voltage are out of phase: they reach their maximum or minimum values at different times. We find that the current leads the capacitor voltage by a quarter of a cycle. If we associate 360 degrees with one full cycle, then the current and voltage across the capacitor are out of phase by 90 degrees. An inductor is simply a coil of conducting wire. When a time-varying current flows through the coil, a back-emf is induced which counters, to some extent, the applied EMF of the voltage source. (See the discussion of Faraday s Law of Induction in your Physics text.) This back-emf is proportional to how fast the current is changing in other words, it is proportional to the frequency of the alternating current. At high frequencies the effect is large; at low frequencies the effect is small. At zero frequency which is direct current there is no effect at all: the inductor is just another piece of wire. 88

3 [Diagram 1 here] 89

4 [Diagram 2 here] 90

5 Just as with the capacitor, there is a phase difference between the voltage across the inductor and the current in the circuit. In the inductor, however, current lags the voltage by a 90-degree phase angle. The unit of inductance is the henry, which is equivalent to an ohm-second. For future reference, note the product of an inductance and a capacitance is a time squared (L x C = T 2 ). Since resistance is not frequency dependent, there is no phase difference between the voltage across a resistor and the current in the circuit. That is, the current and the voltage across the resistors in an AC circuit are always in phase. See Diagram 2 for the phase relationship between these three circuit components. Impedance and Reactance Inductors and capacitors will impede the flow of current in a circuit, just as will resistors. The mechanism of this impedance, however, is much different than for a resistor. As described above, inductors and capacitors react to the flow of current, and their resistance to current flow is called reactance. Reactance is given the symbol X and is measure in ohms. For capacitors, we find that (1) 1 X C = 2πfC where X C is the reactance, f is the frequency of the voltage and C is the capacitance of the capacitor. At low frequencies the reactance is large the capacitor is, after all, essentially a gap in the circuit. At high frequencies, however, the current is oscillating so fast that this gap hardly matters, and the reactance is small. The voltage drop across a capacitor is given by: V C = IX C where I is the current at any time in the circuit. This is similar to Ohm s Law for a resistor in a DC circuit. But note that both I and V c. are functions of time. For inductors, we find that (2) = 2πfL X L 91

6 Here X L is the reactance of the inductor, f the frequency of the voltage and L the inductance of the inductor. The voltage drop across the inductor is: V L = IX L Again, this is similar to Ohm s Law. Again, the current and voltage are time-varying. Resistors have no frequency response. Their response in an AC circuit is given by just as in DC circuits. V R = IR The LRC Circuit When an inductor, a resistor and a capacitor are connected in series in an AC circuit, the resistance to current flow caused by the three together is called impedance. For an AC circuit, we rewrite Ohm s Law as: V p = where Z is the impedance, V p is the maximum (peak) voltage and I p is the maximum (peak) current. Z is measured in ohms. I The impedance depends on the reactances of the circuit elements. However, we cannot simply add the individual reactances as we did in the case of resistors in a series DC circuit. This is because the voltage drops across the different circuit elements are not in phase with each other they are not going through their cycles in step with each other. Examine Diagram 2, which illustrates the way the voltage varies with time in each of the circuit elements. In particular, note that when the voltage is a maximum across the resistor, it is zero across the inductors and capacitor, with the inductor ahead of the resistor and the capacitor behind it in phase. We say the inductor leads the resistor by 90 degrees and the capacitor lags the resistor by 90 degrees. At the same time, the inductor and capacitor are 180 degrees out of phase. p To add up the reactances we must take these phase differences into account. This is simplified by drawing a phasor diagram (see Diagram 3). The impedance Z of the circuit is the vector sum of the reactance phasors. From the diagram, we see that: Z 92

7 (3) Z = R 2 + ( X L X C ) 2 Note that R in the above should be the total resistance in the circuit. The expression also applies to RC and RL circuits, with X L = 0 or X C =0. 93

8 [Diagram 3 here] 94

9 The Resonant Frequency Since X L and Xc are frequency dependent, so is the impedance Z. From the expression above, we see that the impedance in the circuit will be a minimum when X L = X C When this occurs, the current in the circuit will have its maximum value for a given input voltage V 0. The circuit is said to be in resonance with the input voltage, and the frequency at which this occurs is called the resonant frequency, f 0. Using the above expressions for X L and Xc, we can derive: (4) f 0 = 1 2π LC Note that the square root in the denominator has units of time. This time is called the time constant for the circuit. We will use the oscilloscope to track the change in the voltage across the circuit as we vary the input frequency. At resonance, the voltage across the inductor and capacitor will be approximately zero. We can compare the frequency at which this occurs with the expression above. (Note: finding resonance in a circuit is what you are doing when you tune your radio. In that case, the radio station sends out a signal (radio waves) of a definite frequency. The signal is the input voltage for the circuit in the radio set; turning the tuning knob adjusts a variable capacitor in the set until resonance is found. At resonance, the amplitude of the signal becomes large.) 95

10 LAB 7 Procedures There are three parts to this lab: 1. RC Circuit a. Observe the phase relationship between the resistor and capacitor. b. Measure and calculate the impedance. 2. RL Circuit a. Observe the phase relationship between the resistor and inductor. b. Measure and calculate the impedance. 3. RLC Circuits (2 component sets) a. At a given frequency, measure the voltages across each component and determine their reactances. b. Find the resonant frequency for the circuit. Equipment Resistors (2) Inductors (2) Capacitors (2) Oscilloscope Signal Generator LRC meter Banana wires Alligator clips Set Note: A suggested set of circuit components are: Inductor (mh) Capacitor (µf) Resistor Ω Test frequency (hz) , , Set up the oscilloscope and the signal generator. Turn on the oscilloscope and set all switches as detailed in the oscilloscope section in the introduction. 96

11 LAB 7 Procedures Turn on the signal generator (SG). Select sine wave output. Set the frequency range at 1K 10K hz. Turn the frequency knob to mid-range. Using a BNC connector and a banana lead, connect the low impedance output of the SG to channel 1 of the scope and set the SG s output (voltage) to half maximum. Connect the ground terminals of the SG and the scope. (If your SG has only a high impedance output, set the output to maximum and remember to include the output impedance in the resistance of your circuit in your calculations.) Adjust the frequency and voltage knobs on the scope until you have about two cycles displayed. Usually a setting of 1 volt/division on the voltage (amplitude) knob and.1 ms/division on the frequency knob is a good place to start. Also adjust the trace-position knobs so that the trace is centered. 2. RC circuit: Examine voltage across resistor and capacitor Using alligator clips and banana wires, connect the resistor and capacitor from set 1 in series with the SG. Put the capacitor nearest to ground. Refer to the circuit diagram below. Connect channel 1 of the scope across the resistor. Connect channel two of the scope backward across the capacitor. That is, connect the channel 2 terminal to the ground end of the capacitor. The scope s ground terminal should be connected between the resistor and the capacitor. This means the capacitor display will be reversed in polarity: crests will be displayed as troughs and vice versa. Select the CHOP or Dual button on the scope. This will display both traces at once. Note that you have an amplitude knob and a vertical adjust for each channel. Set both channels at 1 volt/division so that the scale is the same for both. Adjust the frequency knob (one knob for both traces) so that there are about two cycles of each trace on the screen. 97

12 LAB 7 Procedures With this set-up, we are looking at the voltage drops across the resistor and the capacitor separately. Notice that the voltage drops are out of phase with each other. Find the frequency at which the voltage drops across the resistor and the capacitor are equal. Measure this frequency with the scope and record the value. Sketch the resistor and capacitor traces, with the correct phase difference between them. Measure the voltages across each element: V R and V C. Record these. Also measure and record the output voltage of the SG, V 0. Using the LRC multimeter, measure the resistance of the resistor and the capacitance of the capacitor. Also record the output impedance of the SG (if known). Include this in the total resistance in the circuit in calculations. Analysis Measure the phase difference between the two traces: ϕ = 2πt / T, where t is the time difference between the two peaks and T is the period of the signal. Also indicate whether φ is positive or negative. (Remember, since we are looking backward across the capacitor, a lag on the display is actually a lead in the circuit and vice versa.) Determine the current in the circuit by using the voltage drop across the resistor: I pp = V R /R. Determine the reactance of the capacitor by X C = V C /I pp. Calculate X C by equation (1) above and compare these two values for X C. Calculate the impedance Z in the circuit from equation (3) above and compare this to the value determined by Z = V 0 /I pp. 3. RL circuit: Examine voltage across resistor and inductor Connect a resistor-inductor circuit using the resistor and inductor from set 1 and the signal generator as voltage source. The inductor goes closest to ground. Refer to the circuit diagram below. 98

13 LAB 7 Procedures Connect channel 1 of the scope across the resistor and channel 2 backward across the inductor. Start with an SG output of 20,000 hz (select the highest frequency range button). Adjust the scope so that there are about two cycles of each trace displayed. Find the frequency at which the voltage drops across the resistor and the inductor are equal. Measure this frequency with the scope and record the value. Sketch the resistor and inductor traces, with the correct phase difference between them. Measure the voltages across each element: V R and V C. Record these. Also measure and record the output voltage of the SG, V 0. Using the LRC multimeter, measure the resistance of the resistor and the inductance of the inductor. Also record the output impedance of the SG (if known). Include this in the total resistance in the circuit in calculations. Analysis Measure the phase difference between the two traces: ϕ = 2πt / T, including whether the inductor leads or lags the resistor. Determine the current in the circuit by using the voltage drop across the resistor: I pp = V R /R. Determine the reactance of the inductor by X C = V L /I pp. Calculate X L by equation (2) above and compare these two values for X L. Calculate the impedance Z in the circuit from equation (3) above and compare this to the value determined by Z = V 0 /I pp. 5. RLC circuits: Determine inductor and capacitor reactances. Now we will set up two RLC circuits and determine the reactances of each circuit element. In step 6 we will look for the resonant frequency of each circuit. The procedures are the same for each component set. Refer to the circuit diagram below: 99

14 LAB 7 Procedures Measure and record the resistor, capacitor and inductor values using the LCR multimeter. Connect a resistor-inductor-capacitor circuit with signal generator as voltage source. Set the generator s output to 4.0 volts peak-to-peak. Set the frequency output to about: Set 1 Set 2 1,200 hz 12,000 hz Be sure to record the actual frequencies that you use. Measure and record the voltage drops across the resistor (V R ), inductor (V L ) and capacitor (V C ). We need only one channel for this, so set the scope display to channel 1 only. Analysis Calculate the (peak-to-peak) current in the circuit using the resistor and its voltage V drop: I = R pp R. Calculate the experimental values for the inductor reactance (X L ) and the capacitor reactance (X C ) from the measured voltages and the current: X = L V I L pp X = C V I C pp Calculate the theoretical reactances for the inductor (X L ) and the capacitor (X C ) at this frequency. Take the percent differences between the theoretical values and the experimental values. 6. Find the resonant frequency of the circuit Connect channel 1 of the scope across the resistor. Connect channel 2 backward across the inductor and capacitor in series. Find the frequency at which the voltage drop across the capacitor-inductor combination minimizes. This is the resonant frequency. 100

15 LAB 7 Procedures Note: For set two it may be difficult to determine where the minimum is (the components are not ideal; internal resistance in the inductor will prevent the inductor and capacitor from being exactly out of phase so that they completely cancel each other out). If you have trouble finding the minimum, try this alternate set-up: Look at the inductor on channel 1 and the capacitor on channel 2. When the voltages across these two components equalize, their combined voltage drop will be a minimum and this will be the resonant frequency. Compare your experimental results with that given by theory: f 0 = 1 2π LC 101

16 LAB 7 Data RC Circuit Data Frequency Phase Shift: Resistor Capacitor V 0 V R V C RL Circuit Data Frequency Phase Shift: Resistor Capacitor V 0 V R V L 102

17 LAB 7 Data LRC Circuit Source Voltage (V 0 ) Set Resistor Inductor Capacitor V R V L V C 1 2 Current & Reactances 1 Set Frequency I pp X L Theory Exp. % diff X C Theory Exp. % diff 2 Resonant Frequency Set Theory Experiment % diff