Boise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab

Transcription

1 Objecties Boise State Uniersity Department of Electrical and Computer Engineering ECE 22L Circuit Analysis and Design Lab Experiment #2: Sinusoidal Steady State and Resonant Circuits The objecties of this laboratory experiment are: To inestigate the sinusoidal steady-state response of a resonant circuit in the phasor domain. To compare the timebase and the Lissajous methods for measuring the phase shift between two sinusoidal waeforms. 2 Theory Electric circuits containing components like capacitors and inductors can introduce a phase shift between an exciting (input) sine waeform and a measured (output) sine waeform. This phase shift may be an important parameter to be measured in certain applications. This experiment inestigates the timebase and Lissajous methods for measuring such a phase shift between two sine waeforms. 2. Timebase Method 2 D D 2 Figure : Timebase Method for Measuring the Phase Difference Between Two Sine Waeforms Figure shows two sinusoidal waeforms, (t) = V m cos(ωt θ ) = 2V cos(ωt θ ) () 2 (t) = V m2 cos(ωt θ 2 ) = 2V 2 cos(ωt θ 2 ) (2)

2 where (t) is the reference waeform with peak magnitude V m (and rms magnitude V ), and 2 (t) is a secondary waeform with peak magnitude V m2 (or rms magnitude V 2 ) and shifted by an angle θ = θ 2 θ with respect to the first waeform. The secondary waeform 2 (t) is said to be lagging the reference waeform (t) if it peaks later in time as shown in the aboe figure. In this case, the phase shift θ (80 o < θ = θ 2 θ < 0 o ). The waeform 2 (t) is said to be leading the reference waeform (t) if it peaks earlier in time with a positie phase φ (0 o < θ = θ 2 θ 2 < 80 o ). The timebase method of phase measurements consists of displaying both waeforms simultaneously on the screen and measuring the distance (in scale diisions) between two identical points on the two traces. In Figure 2(a), this phase shift in degrees is determined from the relation θ = 360 o D 2 = 360 o t D T where D = T is the common period for both waeforms and D 2 = t is the time delay between two zero crossings with rising (or falling) edges on both waeforms. 2.2 Lissajous Method (3) A B A (a) (b) Figure 2: Phase Shift Computation Using Lissajous Patterns The Lissajous-pattern method of phase measurement is also called the X-Y phase measurement. To use this method, both signals are applied to two channels and the scope is then switched to the X-Y mode whereby the reference signal is applied to the horizontal input and the secondary signal is applied to the ertical input. A pattern known as a Lissajous pattern will appear on the screen. This pattern can be used to compute the phase shift between the two waeforms. The patterns shown aboe indicate phase relationships between the two waeforms. In order to calculate the phase shift φ, it is necessary to center the pattern on the X-Y axis as shown in Figure 2. The phase angle is obtained as follows for each pattern. Pattern (a) : 0 o θ 90 o = θ = sin A B Pattern (b) : 90 o θ 80 o = θ = 80 o sin A B (4) (5) 2

3 i(t) R ~ I R i (t) L C o (t) ~ V i j j ωc ~ V o (a) (b) Figure 3: Resonant RLC Circuit in (a) Time Domain and (b) Phasor Domain 2.3 Resonant Circuit Consider the aboe RLC circuit which is excited by a sinusoidal input i (t) = 2V i cos(ωt θ ) The output measurement is taken as the oltage across the LC parallel combination o (t) = 2V o cos(ωt θ 2 ) The oltage gain or transfer function is obtained as the ratio of the output oltage to the input oltage in the phasor domain Ṽ o Ṽ i = Ṽo θ 2 Ṽi θ = = () 2 R 2 ( ω 2 LC) 2 j /jωc R j /jωc = jr( ω 2 LC) tan R( ω2 LC) Identifying Equations (6) and (7), the oltage gain and phase shift between the output and input waeforms are Ṽo Ṽi = () 2 R 2 ( ω 2 LC) 2 (8) θ = θ 2 θ = tan R( ω2 LC) The output oltage reaches a maximum with zero phase shift (6) (7) (9) Ṽo = Ṽi θ = 0 o (0) () at the resonant frequency ω o = LC = f o = 2π LC This resonance property of circuits with inductors and capacitors can be used for component measurement. At resonance, the impedance of the parallel LC combination reaches its highest impedance (ideally an open circuit). The loading effect of the LC circuit is minimized at this frequency and the output oltage is maximum. 3 (2)

4 3 Equipment Agilent DSO504A Digital Storage Oscilloscope Agilent 33220A Function/Arbitrary Waeform Generator Keysight E4980AL Precision LCR Meter 330-mH Inductor, µF Capacitor, 5.-kΩ Resistor, Protoboard Procedure Part A: Parameter Measurements Measure the three RLC parameters using a shared RLC meter set at a testing frequency of khz. Part B: Timebase Method f (khz) R (kω) L (mh) C (µf) Nominal Measured Build the resonant RLC circuit of Figure 3(a) using R = 5. kω, L = 330 mh, and C = µf. Hook up i (t) to channel and o (t) to channel 2 of the oscilloscope. Energize your circuit with a 2-Vpp sine wae with ariable frequency and zero offset. First, find the resonant frequency where the output oltage is maximum and in phase with the input oltage using the XY mode. Record the maximum amplitude of the output oltage at the resonant frequency and complete the table below by recording the measurements in the third and fourth columns at frequencies below and aboe the resonant frequency. When, all measurements are recorded, compute the alues of the phase φ in degrees in the last column of this table using the gien formula and make sure these alues are negatie in the last fie rows. f (Hz) V i,pp (V) V o,pp (V) t (µs) φ = 360f t (deg)

5 Part C: Lissajous Patterns Using the same setup as in Part B and for the same frequencies recorded, set up the X-Y (or ersus) mode on the infinium scope and obsere a Lissajous pattern. Turn on the two pairs of scope cursors and measure the quantities A and B for each of the frequencies recorded in Part B. Make sure to manually add a negatie sign to the angle φ in the last fie rows of the last column. f (Hz) V i,pp (V) V o,pp (V) A B φ = sin A/B (deg) Data Analysis and Interpretation. Using the measured alue of L and C at khz, use Equation (2) to compute a alue of the resonant frequency f o (Hz). Compare this alue of the resonant frequency to the one obtained by measurement and summarize their alues in the table below: Resonant Frequency f o (Hz) Measured Computed Relatie Error (%) 2. Compute the oltage gain G(f) = Ṽo / Ṽi = V o,pp /V i,pp as a function of frequency f and plot G(f) as a function of frequency for Part B. 3. Plot both phase shifts φ(f) = 360f t from Part B and φ(f) = sin A/B from Part C on the same graph. 4. Use the gain plot to find the two frequencies in Hz where the gain is equal to / 2 = of the maximum gain. Then, using the phase plot from the timebase method, find the phase shifts between the input and output waeforms at these two frequencies. 6 Discussion Reiew the lab manual on how to write a discussion regarding the sources of error when there are discrepancies between the predicted and measured results. Discuss the accuracy of the timebase and Lissajous methods in your lab report. 5

6 Boise State Uniersity Department of Electrical and Computer Engineering ECE 22L Circuit Analysis and Design Lab Experiment #2: Sinusoidal Steady State and Resonant Circuits Date: Data Sheet Recorded by: Equipment List Equipment Description Agilent DSO504A Digital Storage Oscilloscope Agilent 33220A Function/Arbitrary Waeform Generator BSU Tag Number or Serial Number Part A: Parameter Measurements with an RLC Meter Part B: Timebase Method f (khz) R (kω) L (mh) C (µf) Nominal Measured f (Hz) V i,pp (V) V o,pp (V) t (µs) φ = 360f t (deg) Part C: Lissajous Patterns f (Hz) V i,pp (V) V o,pp (V) A B φ = sin A/B (deg)