AC Circuits "Look for knowledge not in books but in things themselves." W. Gilbert (1540-1603) OBJECTIVES To study some circuit elements and a simple AC circuit. THEORY All useful circuits use varying voltages, changing magnitude or even completely reversing polarity. In the present exercise we will study the behavior of some basic components and two elementary electrical filters, of the sort that might be used in audio equipment, when stimulated with a varying voltage. To keep things simple, we will consider only a sinusoidal voltage which oscillates at a steady frequency. For purposes of analysis, circuits are usually considered to be made from resistors, capacitors and inductors. The components themselves are frequently characterized by the current that flows through them in response to a sinusoidal voltage at angular frequency !. As shown in your text, the current will then also be sinusoidal, with the ratio of peak voltage to peak current, the reactance, depending on frequency. There may also be a phase shift between the current and voltage, so that they peak at different times. The ideal relationships are given by Component Reactance resistor XR= R (1) capacitor XC= 1/!C (2) inductor XL=!L (3) Verifying the relations requires measuring the amplitudes of current and voltage to determine the reactance. A method for doing this will be described later. As you might expect, real components are more complicated, but it is often possible to manufacture a reasonable approximation to these ideals. In fact, commercial resistors and capacitors are rather good, but inductors are not. The resistance of the coil turns out to be significant in most practical cases (superconductors are not practical), so we need to consider a more complex model for inductors. One approximation is to assume that the coil resistance is effectively in series with an ideal inductor, as shown in Fig. 1. Using the fact that the current flow through both components is the same, we can draw a phasor diagram showing the voltageAC Circuits 2 drop across the resistor and across the ideal inductor. The total voltage drop across the model inductor is then the vector sum VLm= Ip(!2L2+ RL2)1/ 2 (4) from which it is easy to predict the reactance for this model inductor XLm= (!2L2+ RL2)1/ 2 (5) At low frequencies, where !L << RL, Eq. 5 suggests that the inductor behaves like a resistor, in that the reactance is almost equal to RL. At high frequencies, ! >> RL/L the reactance increases to !L, as expected for an ideal inductance. Later we will see if this accounts for the properties of a real inductor. The first circuit example is an RC filter, shown in Fig. 2. We are interested in finding the fraction of the input signal voltage Vs that appears across the capacitor as a function of frequency. Figure 2 provides the phasor diagram for the circuit, from which we see Vs= IpR2+1!2C2" # $ % & ' 1/ 2 (6) RLLpI RI !LpLLmV"Lm Fig. 1 Model of a real inductor and corresponding phasor diagram used to obtain reactance. RpI RsVφVVVRCCsI pωCAC Circuits 3 Solving for the current and using Eq. 2 to relate VC to Ip, we get VC= Vs1(1 +!2"2)1/ 2 (7) where ! = RC is the time constant of the circuit. The circuit in Fig. 2 is called a low-pass filter because low input frequencies are passed to the output essentially unaffected but high frequencies are strongly attenuated, as graphed in Fig. 3. Low-pass filters are often used in audio devices to remove high-frequency digital artifacts from the desired audio signal. Our other circuit example is a bandpass filter, which attenuates signals outside a certain frequency range. Bandpass filters are useful, for example, in tuning a radio receiver to a desired 0 2 4 6 8 100.01.00.5ωτVCVs/ Fig. 3 Ratio of output to input voltage for a low-pass RC filter, as a function of scaled frequency. LRLCRVLC Fig. 4 Bandpass filter using a parallel LC resonant circuit.AC Circuits 4 station. This function can be accomplished with a parallel LC circuit, as shown in Fig. 4. A detailed analysis is complicated, but it can be shown that the reactance of the LC circuit has a maximum at the angular resonant frequency !0= 1/ LC . We therefore expect the voltage VLC to have a maximum near that frequency. EXPERIMENTAL PROCEDURE The experimental work will consist of measuring the current-voltage characteristic of a typical commercial capacitor and an inductor, and observing the filtering action of two different circuits. The required procedures are presented separately, following a description of the AC signal generator. Function generator operation Figure 5 shows the control panel of the function generator most often used in this lab. The instrument is turned on with the power switch, item (1) in the figure. A sine, square or triangle waveform is selected with the buttons labeled FUNCTION (3). The frequency range is determined by pushbuttons (2), and the exact frequency is then set using the coarse and fine FREQUENCY knobs (12, 11). The output amplitude is determined by the AMPL knob (4). Pulling gently on the AMPL knob decreases the output amplitude by a factor of 10. The display (13) shows the output frequency numerically in Hz or kHz, according to the indicator (14) and decimal point on the display. The selected output appears as a voltage at OUTPUT (5). The other controls and outputs are more specialized, and will not be used here. Fig. 5 The front panel of the function generator, with controls marked.AC Circuits 5 Component characteristics Figure 6 specifies the circuit used to measure the current and voltage for the component, Z, to be tested. The voltage VZ and VR can be measured with the DMM set for AC voltage. The current is found by applying Ohm’s Law with the known value of R. The voltage source is a function generator, set to a convenient amplitude and the desired frequency. As a test, wire the circuit as shown, using R = 150 !, and another resistor for Z. Measure VR and VZ at any convenient frequency below 1000 Hz, and deduce the reactance of the resistor. You should get a value reasonably close to the marked resistance of the component. When you are