PHY132 Labs (©P. Bennett, JCHS) -1- 08/11/04 PHY 132 LAB LRC circuit (phases) Introduction "Phase" means the relative advance or retardation (expressed in radians) of one wave with respect to another. Two waves whose crests coincide are said to be "in phase" - if the crests of one lie over the valleys of the other, they are pi out of phase. It is only simple to define phase for waves of the same frequency. In this lab we will measure the phase difference between voltage and current for each component in a series LRC resonant circuit. The purpose of the lab is teach you about the concept of phase and reactance in AC electrical circuits. Resonant circuits are used in oscillators to generate radio waves, and are studied with this same circuit in the next lab. In this experiment we compare the phase of the voltage across "reactive" components (inductance, capacitance) with that of the current, which is the reference. (The phase of the current is equal to that of the voltage across any resistor , so we actually measure voltage instead. Voltage and current are "in step" across a resistor, but not across a capacitor or inductor). Capacitors and inductors have "Reactance", which is a kind of frequency- dependant resistance, and is used for inductors and capacitors because these circuit elements change the phase relationship between current and voltage through them, unlike a resistor. We will see that Ohm's still holds in frequency-dependant form for reactances. Theory R C V_in L Fig. 1 Generic series LRC circuit. Consider the steady-state (sinewave) behavior of a series LRC circuit as shown in Fig. 1. By definition, the same current flows through each component of a series circuit (it has the same value everywhere in the circuit at any particular time) and can be written as i(t) = I 0 sin(ωt) eq. 1 PHY132 Labs (©P. Bennett, JCHS) -2- 08/11/04 Here ω = 2πf, where f is the frequency in cycles per second or Hz. This waveform has amplitude I 0 and phase angle zero. The voltage across each component is a sinewave at the same frequency given by v j (t) = V j sin(ωt + Φ j ). eq. 2 Note that in this context, v(t) is completely specified by amplitude and phase Φ j . Voltage amplitudes are given by Ohm’s law (using complex AC reactances X) as V j = I 0 X j eq. 3 and phase is given by tan(Φ j ) = (X/R) j eq. 4 Values for X and Φ are shown in table 1 below. Component(s) X = |Z| Φ R R 0 C 1/(wC) -π/2 L – ideal coil wL +π/2 Series (L+R+C) √(R 2 +(X L -X C ) 2 ) tan -1 (X L -X C )/(R) (L+r) - real coil √(r 2 +(wL) 2 ) tan -1 (X L /r) Table 1. Reactance and phase angle for various circuit components and combinations. Phase shift can be determined directly from your scope trace by finding the time between successive zero crossings (∆t) compared to the full period of the waves (T). The phase shift is then simply given by the ratios Φ/2π = ∆t/T eq. 5 PHY132 Labs (©P. Bennett, JCHS) -3- 08/11/04 An example is shown in Fig. 2 for the case of ω = 2000 rad/sec, Vout/Vin = 1/√2 and Φ ~ 2π(0.4msec/3.14msec) ~ -π/4 radians. Note that v out is lagging v in in accord with the negative phase shift. Fig. 2 Sample v(t) waveforms for RC filter. The time delay ∆t needed to find the phase difference between these two waves is the distance along the horizontal axis between points where the two wave cross it, close together. Note that the circuit has a resonant frequency where X L =X C or ω 0 L = 1/(ω 0 C) or ω 0 2 LC=1 eq. 6 At this frequency, the total impedance is minimum, and given by Z LRC (ω 0 ) = R. The current flow is maximized and the phase shift is zero. Resonance can usually be more accurately determined from phase shift than from amplitude, since a zero can be determined more accurately than a maximum. V(t) for RC filter -4 -3 -2 -1 0 1 2 3 4 01234 time (msec) V (vol ts) V_in V_out x PHY132 Labs (©P. Bennett, JCHS) -4- 08/11/04 Fig. 3 Circuit used for measuring LRC behavior. The circuit we will use is shown in Fig. 3. Note that the coil is non- superconducting and has an unavoidable non-zero internal resistance “r”. This circuit element is only experimentally accessible as the combination (L+r). The internal “r” contributes to the total circuit resistance R in the equations and table above. Thus we have R = R load +r eq. 7 Scope ch #2 senses the input voltage, while scope ch #1 senses the series current, given by voltage across the series resistor R load . This voltage is in phase with the current because the resistor impedance has zero phase shift. We will also connect the Pasco A/D voltage probes to various components in turn and “capture” the waveform into the computer. It is “triggered” on i(t), to guarantee that the captured wave is synchronized from one measurement to the next, so we can compare phases. Procedure 1. Connect the circuit of Fig. 3 using R = 20 ohms, C = 10 µfd and L = 85mH. 2. Input a sinewave about 100 Hz, 2V pp (peak-to-peak on scope ch #2). Get both waves showing on the scope simultaneously. Find and record the resonant R_load C L r Pasco #B DVM black red Scope ch#Scope ch#2 Sig. Gen. Ground Pasco #A PHY132 Labs (©P. Bennett, JCHS) -5- 08/11/04 frequency f 0 where the current (scope ch. #1) is maximum and phase shift is zero. 3. Sketch and describe the waveforms as you tune through resonance. How can you tell that the phase changes sign? 4. Set the frequency a little below resonance, where the output voltage across R is about 1/2 of its value at resonance. Does the voltage across the LRC circuit (scope ch #2) lead or lag the current (ch #1) under these conditions? 5. Using the DVM, measure voltages across each component around the circuit, including v in . 6. Next, digitize the waveform v(t) for each component in the circuit. Load the setup file “LRCphase.sws”. Connect the Pasco probe B to each component in turn around the circuit and capture the waveform. Be sure that probe A triggers properly, giving a sinewave with zero phase shift. Be careful to observe the current polarity when you connect the Pasco probes, using black on the clockwise side of each component, including the signal generator. Note that the scope cannot be moved around the circuit because one side is necessarily grounded, but the Pasco probe has “floating” inputs, so they can be connected anywhere. Copy/paste or export this data to EXCEL or GA for analysis. Note the sign of the phase shift - in table 1 the effect of a capacitor is a phase lag. Analysis 1. Compare your resonant frequency f 0 with theory. Component values are accurate to R(1%), C(10%) and L(5%). 2. Plot the waveforms for each component (Pasco-B) including the signal generator. You should find that the i(t) waveforms (Pasco-A) match for all components if triggering was done correctly. Find the voltage phase angle and amplitude for each component. This is illustrated in figure 2 above. Present these in a single table showing phase angle and reactance V j /I o = V j R/V r . 3. Add the V rms voltages around the circuit (v R , v L , v C) and compare with v in . Why do these values “not follow Kirchhoff’s law”? 4. Plot all v j (t) waveforms (v R , v L , v C and v in ) and their sum on a single chart. Do these waveforms follow Kirchhoff’s law? PHY132 Labs (©P. Bennett, JCHS) -6- 08/11/04 Pre-Lab Quiz: PHY132 Your name________________________Section day/time___________________ Consider a series LRC circuit with R = 200 ohms, L = 0.16Henry and C = 0.3µfd. The current is 10mA amplitude at 1kHz. Remember ω = 2πf, where f is the frequency in cycles per second or Hz. 1. Find the voltage amplitude for each component: V R , V L , V C . 2. Find the total circuit voltage, across the combination LRC. 3. Find the resonant frequency. rculber Microsoft Word - 11PHY132 LRC.doc