ࡱ>  Z\Y_bd0m` bjbj 8(<(M<0###;;;;;;;$=hk?f;9$"#$$;0<?+?+?+$|L0;?+$;?+?+8|T:0 j6<( 9;<0M<9?)0?$:?:#"#?+# $###;;/+###M<$$$$(((((((((  MACROBUTTON MTEditEquationSection2 Equation Chapter 1 Section 1 SEQ MTEqn \r \h \* MERGEFORMAT  SEQ MTSec \r 1 \h \* MERGEFORMAT  SEQ MTChap \r 1 \h \* MERGEFORMAT PHYSICS 536 Experiment 2: Frequency Response of AC Circuits A. Introduction The concepts investigated in this experiment are reactance, impedance, and resonance circuits. Many features of the scope will be used: including dual traces; differential inputs; and external triggering. Since this is the first experiment in which you have used the oscilloscope so a little extra care is appropriate. Refer to the General Instructions for the Laboratory for additional information. A brief synopsis of the concepts and equations needed for this experiment are presented below. AC (alternating current) can be represented as a complex quantity. In the complex plane the vector representing current is referred to as a phasor. The magnitude of this vector is proportional to the current amplitude and the angle from the real axis is the phase. Voltage can also be represented in similar fashion. Reactance Capacitive and inductive reactance are analogous to resistance and have units of ohms. Ohms law,  EMBED Equation.DSMT4 , is an essential element in DC circuit analysis. If the concept of impedance is introduced, an analogous relationship still holds,  EMBED Equation.DSMT4 , where Z is impedance. Z is defined as  EMBED Equation.DSMT4 , where X is the reactance. Z is a complex quantity. Likewise, AC voltages and currents can be represented as complex quantities. As such, they have two independent components: amplitude and phase. Accordingly,  EMBED Equation.DSMT4  . The magnitude of the voltage is given by the relationship  EMBED Equation.DSMT4 . The reactance of a capacitor is given by the formula  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 1) Reactive inductance is  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 2) Since impedance, voltage, and current are represented as complex quantities rather than using the cumbersome standard notation we utilize a shorthand notation in which the first quantity is the magnitude of the phasor and the second is the angle relative to the real axis. This polar notation for phasors is commonplace in electronics. For example, for a capacitor,  EMBED Equation.DSMT4 . Likewise, for an inductor,  EMBED Equation.DSMT4 . Given an input voltage across a capacitor the current is given by  EMBED Equation.DSMT4  so current leads voltage in capacitor. Likewise, for an inductor  EMBED Equation.DSMT4  so current lags voltage in an inductor. 2. RLC Circuit. Phasors provide a simple way to visualize the relationship between current and voltage when R, C, and L are mixed in a circuit. The length of the vector represents the amplitude (i.e. peak-value) of the sine wave. Current and voltage, VR, are in phase for a resistor, hence the voltage across a capacitor, VC, and inductor, VL, are both 90 out of phase with VR in a series circuit. Kirchoffs voltage law requires that applied voltage, Vs is  EMBED Equation.DSMT4   equal to the sum of voltages across the three components. Although the three component voltages peak at different times the vector sum of the voltages across the three components sum to the input voltage  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 3) The phase angle is given by  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 4) At the resonance frequency fo, XC , and XL are equal.  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 5) Since the reactances sum to zero both VL and VC must also sum to zero. At the resonance frequency VR is maximum. The current in the circuit also is maximized since  EMBED Equation.DSMT4 . Since I is maximized, the magnitude of both VC and VL are individually at maximum, although 180o out of phase with each other. When the frequency is not near the resonant frequency, the phasors representing VC, and VL do not cancel, so VR is less for a given Vs. It is convenient to do calculations in terms of impedance. For this RLC circuit the impedance is given  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 6) and  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 7)  EMBED Equation.DSMT4  3. Parallel LC Circuit and practical inductors Long wires are used to wind real inductors, hence they have some resistance, rs, that spoils the ideal character of series LC resonance; the impedance is equal to rs, rather than zero, at resonance. The equivalent resistance in the parallel representation is,  EMBED Equation.DSMT4 . An AC source driving a parallel inductor and capacitor as a practical matter resembles a parallel RLC circuit, where rs and rp are the resistances of the current source and inductor, respectively. Figure 2. EMBED Visio.Drawing.11  The condition for resonant frequency is that the capacitive and reactive reactances sum to zero. This condition yields a resonant frequency of  EMBED Equation.DSMT4  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 8) At this frequency the impedance is resistive and the voltage is maximum and given by  EMBED Equation.DSMT4  The quality factor of any circuit can be defined as 2( times the energy stored per cycle divided by the energy lost or dissipated (usually heat) per cycle.  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 9) For the parallel RLC circuit in Fig. 2 the Q is given by  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 10) B is the bandwidth of the circuit, i.e. the difference between the upper and lower frequencies at which the gain in the output voltage, vo, has changed by 30%. In terms of the resonant frequency and the quality factory the bandwidth is given by  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 11) B. Inductors and Capacitors in Circuits The reactance of an inductor, L, or capacitor, C, are investigated in this section. Figure 3. EMBED Visio.Drawing.11  The signal generator applies a sinusoidal voltage, vs, across L or C. The 10 ( resistor is included to measure the current in the circuit. The current across this resistor is given by  EMBED Equation.DSMT4 . A small resistor is used so that the voltage, vB, across it is negligible compared to the voltage across the capacitor or inductor. Accordingly, vA is the voltage drop across the capacitor or inductor. 1 - Using f = 32 kHz, vs = 10 V (p-p), C = 5 nf, measure vB. Refer to GIL sections 5.4, 5.6, 5.7, and 5.8 for additional details. Use vB to calculate the current, i, and use i and vA to calculate XC. Verify that the resistance is small compared the the reactance. Measure the relative phase between current and voltage (refer to GIL section 5.11 for further details). Internal triggering can be used. Recall that the scope probe attenuates the signal by a factor of 10. 2 - Repeat step 2 for an inductor with the following measurements f = 1 MHz, vs = 10 V(p-p), L = 100 H. C. Practice Using Oscilloscope Triggering. Connect the  output of the signal generator to input 1 of the scope and the  1 volt signal to input 2 of the scope. Set the generator for 10 kHz. Press the 1V amplitude button on the generator and turn the amplitude dial to 10. The output signal will be 3 V(p-p) sine wave because the buttons are labeled with the effective value. Press the channel 1 and 2 menu buttons to display both signals at the same time. The two traces should be separate on the face to face to avoid confusion. The traces are labeled with an arrow on the left side of the display. Set the trigger for rising slope on cannel 2. Press TRIGGER menu button. Select EDGE, RISING slope, CH2 source, AUTO mode, DC coupling. The trigger time is marked by an arrow at the top of the display. The trigger level is marked with an arrow on the right side of the display. Set the trigger level approximately half way between the peak values of the sine wave. Notice that the output and 1-volt sine waves are in phase. In this step you observe the advantage of triggering on a constant signal when you want to observe changes in a signal. The output signal can be switched between sine wave and square wave. The 1-volt signal in channel 2 that is used to trigger the scope is not affected by this switch. The output square wave has the opposite polarity of the sine wave, i.e. the square wave is low when the sine wave is high. Switch between the sine and square wave because the trigger signal in channel 2 is not changing. Next you will trigger from the signal that is changing to see that the phase relationship is not displayed correctly. Change the trigger source to channel 1. Then switch between the square and sine wave while observing channel 1. The sine and square wave appear to be in phase, i.e. both are high in the same part of the cycle. This is not the true relationship, as observed in part a). Since we are triggering on channel 1, both wave forms have a positive slope at the trigger point. Notice that the 1-volt signal in channel 2 is displayed in phase with the sine wave and out of phase with the square wave. Change the trigger between channels 1 and 2 until you understand the relationships. Ask the instructor for help if this illustration is not clear. When you are observing changing signals on both input channels, you can not get a constant trigger from either of them. The scope provides a third external trigger. Move the 1-volt signal from channel 2 to external-trigger. Press TRIGGER menu and select EXT source. Switch between square and sine wave. The display in channel 1 will be the same observed in part a). D. Representation of a Simple RC Circuit In the following experiments use the vector representations of AC signals to help you visualize the effect of phase difference in the following circuit. Figure 4 EMBED Visio.Drawing.11  3 As an exercise assume that the signal current is 1.67 mA (p-p). Calculate the voltage (p-p) across the capacitor, vAB, and resistor, vB. What is the phase between vB and vAB. Calculate the voltage, vA (p-p), of the applied signal and the phase relative to vB. 4 - Use a dual trace mode. Adjust the p-p value of the generator signal to be equal to the vA calculated in exercise 2. Measure vB and the phase between vA and vB. Remember the location of vertical grid mark where vB reaches a peak because vB will not be seen when you switch to differential mode. Switch the scope to differential horizontal sweep or triggering. The scope displays the difference between the A and B channels, i.e., vAB. 5 - Measure vAB and the phase between vAB and vB. E. Series RLC Circuit In this section you will investigate the behavior of a series RLC circuit at the resonance frequency and away from resonance. Refer to Fig. 5.   6 - Measure the resonance frequency by observing when vB in minimum. Measure the amplitude of vC at this frequency and the phase between vA and vC. Explain how vB can be minimum and how vC can be very large at resonance. As part of the laboratory report calculate the resonant frequency of the circuit using vS = 5 V (p-p). Calculate vB, vBC, and vC at resonance. 7 Using vS = 5 V (p-p), measure the peak-to-peak value of vB and vC, and the phase shifts between vA and vB, and between vA and vC. 8 Set up a spreadsheet that calculates the reactance and impedance of this circuit, as a function of frequency, for the component values shown in the figure and vS given in step 7. Graph the results, showing clearly the resonant frequency. Include this graph in the laboratory report. Which component dominates for f = f0, f < f0, and f > f0? Explain your interpretation of the spreadsheet calculation. F. Parallel RLC Circuit In this section you will consider the impedance of a parallel RLC circuit as a function of frequency. The signal generator with a resistor in series is used as a current source.  Resistance 9 - Apply Nortons theorem to the generator and resistor to obtain the equivalent current and resistance. Include these with your laboratory report. 10 - Observe the resonance frequency at which vB is maximum. Use the observed vB at resonance to calculate the resistance (rs, rp). Using this resistance calculate the effective quality factor, Q, of the circuit. 11 - Vary the frequency to measure the bandwidth, i.e., the points at which vB has dropped by 30% from the value at fo. Use Q from step 10 to calculate the bandwidth and compare to that observed. These calculations and measurements should demonstrate that the equivalent resistances (rs and rp) limit the maximum signal amplitude, vB, that can be obtained from a parallel LC circuit for a given current. rs and rp also set a lower limit on the bandwidth that the circuit can provide. Components Resistors: 10(, 510(, 5.1(, 20K. Capacitor: 5fn Inductor: 100(H.    %ABbd  8 :  c f w } ' ( ) * 3 4 5 6 < ? 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