Experiment 2: Sinusoidal Steady State and Resonant Circuits | ECE 225, Lab Reports of Electrical Circuit Analysis

Download Experiment 2: Sinusoidal Steady State and Resonant Circuits | ECE 225 and more Lab Reports Electrical Circuit Analysis in PDF only on Docsity! Boise State University Department of Electrical and Computer Engineering ECE 225L – Circuit Analysis and Design Lab Experiment #2: Sinusoidal Steady State and Resonant Circuits 1 Objectives The objectives of this laboratory experiment are: • To investigate 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 waveforms. 2 Theory Electric circuits containing components like capacitors and inductors can introduce a phase shift between an exciting (input) sine waveform and a measured (output) sine waveform. This phase shift may be an important parameter to be measured in certain applications. This experiment investigates the timebase and Lissajous methods for measuring such a phase shift between two sine waveforms. 2.1 Timebase Method 2D1D v 21 v Figure 1: Timebase Method for Measuring the Phase Difference Between Two Sine Waveforms Figure 1 shows two sinusoidal waveforms, v1(t) = Vm1 cosωt = √ 2V1 cosωt (1) v2(t) = Vm2 cos(ωt + φ) = √ 2V2 cos(ωt + φ) (2) 1 where v1(t) is the reference waveform with peak magnitude Vm1 (and rms magnitude V1), and v2(t) is a secondary waveform with peak magnitude Vm2 (or rms magnitude V2) and shifted by an angle φ with respect to the first waveform. The secondary waveform v2(t) is said to be lagging the reference waveform v1(t) if it peaks later in time as shown in the above figure. In this case, the angle φ in Equation (2) is negative (−180o < φ < 0o). The waveform v2(t) is said to be lead- ing the reference waveform v1(t) if it peaks earlier in time with a positive phase φ (0o < φ < 180o). The timebase method of phase measurements consists of displaying both waveforms simultaneously on the screen and measuring the distance (in scale divisions) between two identical points on the two traces. In Figure 2(a), this phase shift in degrees is determined from the relation φ = 360o × D2 D1 (3) where D1 corresponds to the time interval of one complete cycle of the reference waveform and D2 is the phase shift between two identical zero crossings with rising edges on both waveforms. 2.2 Lissajous Method (b)(a) ABA 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 vertical 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 waveforms. The patterns shown above indicate phase relationships between the two waveforms. 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) : 0o ≤ φ ≤ 90o =⇒ φ = sin−1 A B (4) Pattern (b) : 90o ≤ φ ≤ 180o =⇒ φ = 180o − sin−1 A B (5) 2 Part B: Lissajous Patterns Using the same setup as in Part A and for the same frequencies recorded, set up the X-Y (or versus) mode on the infinium scope and observe 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 A. f (Hz) Vi,pp (V) Vo,pp (V) A B 2.00 0.75 2.00 1.00 2.00 1.25 2.00 1.50 2.00 1.75 2.00 2.00 1.75 2.00 1.50 2.00 1.25 2.00 1.00 2.00 0.75 Part C: Parameter Measurements Measure the four parameters above using a shared RLC meter at 1 kHz. f (kHz) R (kΩ) L (mH) C (µF) Nominal 1 5.1 330 0.068 Measured 1 5 Report Questions 1. Using the measured value of C at 1 kHz and the resonant frequency fo, use Equation (12) to find a value of L in mH. 2. Compute the voltage gain G(f) = |Ṽo|/|Ṽi| = Vo,pp/Vi,pp as a function of frequency f and plot G(f) as a function of frequency for Part A. 3. Compute the phase shifts φ(f) = 360o ×D2/D1 using the timebase method for Part A. 4. Compute the phase shifts φ(f) = sin−1 A/B using the Lissajous method for Part B. (Add a negative sign to these phase shifts according to your observations using the timebase method.) 5. Plot both phase shifts φ(f) = 360o×D2/D1 and φ(f) = sin−1 A/B on the same graph. 6. Use the gain and phase plots to find the phase shifts between the input and output waveforms when the gain is 1/ √ 2 = 0.707. 7. Discuss the accuracy of the timebase and Lissajous methods. 5 Boise State University Department of Electrical and Computer Engineering ECE 225L – Circuit Analysis and Design Lab Experiment #2: Sinusoidal Steady State and Resonant Circuits Date: Data Sheet Recorded by: Equipment List Equipment Description BSU Tag Number or Serial Number HP/Agilent 54810A Infinium Oscilloscope HP/Agilent 33120A Function/Arbitrary Waveform Generator Part A: Timebase Method f (Hz) Vi,pp (V) Vo,pp (V) D1 (ms) D2 (µs) 2.00 0.75 2.00 1.00 2.00 1.25 2.00 1.50 2.00 1.75 2.00 2.00 1.75 2.00 1.50 2.00 1.25 2.00 1.00 2.00 0.75 Part B: Lissajous Patterns f (Hz) Vi,pp (V) Vo,pp (V) A B 2.00 0.75 2.00 1.00 2.00 1.25 2.00 1.50 2.00 1.75 2.00 2.00 1.75 2.00 1.50 2.00 1.25 2.00 1.00 2.00 0.75 Part C: Parameter Measurements with an RLC Meter f (kHz) R (kΩ) L (mH) C (µF) Nominal 1 5.1 330 0.068 Measured