Impedance Measurements in Circuits: ECE 225L Exp. 3 at Boise State Univ., Lab Reports of Electrical Circuit Analysis

Download Impedance Measurements in Circuits: ECE 225L Exp. 3 at Boise State Univ. 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 #3: Impedance Measurements 1 Objectives The objectives of this laboratory experiment are: • to use phase and amplitude measurements in the determination of unknown series and parallel impedance connections. 2 Theory In this experiment, we will explore several ways of measuring the impedance of a “black box” whose impedance is unknown. There are two types of black boxes as shown in Table 1 below. Table 1: Types of Black Boxes Type Impedance Description SRL Series RL Circuit PRC Parallel RC Circuit In addition, type SRL can consist of a nonideal inductor having a small series resistance and type PRC can consist of a single, almost ideal, capacitor with no shunt resistor in parallel with it. 2.0.1 Series RL Circuit − V 1 ~ + − V 2 ~ (a) (b) R s R s R c R c j LL s s ω ++ − v (t) 1 + − v (t) 2 Figure 1: Series RL Impedance Determination Consider the series RL circuit shown in Figure 1 where Rs and Ls are unknown elements to be determined. This unknown series impedance is placed in series with a calibrating series resistor with known value Rc. Assume that the sinusoidal input and voltages, v1(t) and v2(t), respectively, v1(t) = √ 2V1 cos(ωt) v2(t) = √ 2V2 cos(ωt− φ) = √ 2AV1 cos(ωt− φ) (1) 1 are observed with a dual-trace oscilloscope. The transfer function between the output phasor voltage Ṽ2 and the input voltage phasor Ṽ1 is obtained as Ṽ2 Ṽ1 = Rc Rc + Rs + jωLs = Rc√ (Rc + Rs)2 + (ωLs)2 6 − tan−1 ωLs Rc + Rs (2) On the other hand, Ṽ2 Ṽ1 = A 6 − φ (3) Therefore, the amplitude ratio of the two waveforms can be determined from the rms or peak-to- peak voltage magnitudes as A = |Ṽ2| |Ṽ1| = V2,pp V1,pp = Rc√ (Rc + Rs)2 + (ωLs)2 (4) = Rc Rc + Rs 1√ 1 + (ωLs/(Rc + Rs))2 = Rc cosφ Rc + Rs (5) The unknown series resistance Rs can thus be deduced from the peak-to-peak amplitude ratio A and the phase shift φ as Rs = Rc ( cosφ A − 1 ) (6) The unknown inductance Ls is then determined from tanφ = ωLs Rc + Rs (7) Ls = (Rc + Rs) tan φ ω = Rc sinφ ωA (8) 2.0.2 Parallel RC Circuit c R p C p + − V 1 ~ R c R p 1 + − V 2 ~ (a) (b) ω p j C R + − v (t) 1 + − v (t) 2 Figure 2: Parallel RC Impedance Determination Consider the parallel RC circuit shown in Figure 2 where Rp and Cp are unknown elements to be determined. This unknown shunt impedance is placed in series with a calibrating series resistor with known value Rc. Assume that the sinusoidal input and voltages, v1(t) and v2(t), respectively, v1(t) = √ 2V1 cos(ωt) v2(t) = √ 2V2 cos(ωt− φ) = √ 2AV1 cos(ωt− φ) (9) 2 Part B: Parallel RC Circuit Select a value for the calibrating resistor Rc and measure it using an RLC meter at a frequency of 1 kHz. Construct the circuit of Figure 2 on a protoboard. Apply a 1-kHz sinusoidal waveform with 10 V peak-to-peak and no offset. Measure peak-to-peak voltage V1,pp, peak-to-peak voltage V2,pp, period T , and delay time ∆t using the oscilloscope: Raw Data Nominal Rc (Ω) 3.3k 5.1k Measured Rc (Ω) V2,pp (V) V1,pp (V) T (µs) |∆t| (µs) Computations A |φ| (deg) Rp (Ω) Cp (nF) Then compute the amplitude ratio A and the phase shift φ and deduce the values of the parallel resistance Rp and of the parallel capacitance Cp. Compare these values to those measured using an RLC meter at a frequency of 1 kHz. Select another value of the calibrating resistance Rc and repeat the above measurements. Nominal Rp (Ω) Measured Rp (Ω) Relative ∆Rp (%) Relative ∆Rp (%) 2k Nominal Cp (nF) Measured Cp (nF) Relative ∆Cp (%) Relative ∆Cp (%) 68 5 Report Questions 1. Summarize your calculations of Rs and Ls in the final report. Which value of the calibrating resistance Rc gives the best accuracy? 2. Summarize your calculations of Rp and Cp in the final report. Which value of the calibrating resistance Rc gives the best accuracy? 3. Suppose you determine the impedance of a block box in the form of a parallel impedance Z̄p = Rp ‖ jXp. If the block is actually a series impedance Z̄s = Rs + jXs, find expressions of Rs and Xs in terms of Rp and Xp. 4. Report any ambiguity encountered during the performance of this lab experiment and suggest ways to improve on it. 5 Boise State University Department of Electrical and Computer Engineering ECE 225L – Circuit Analysis and Design Lab Experiment #3: Impedance Measurements Date: Data Sheet Recorded by: Part A: Series RL-Circuit Raw Data Nominal Rc (Ω) 1k 2k Measured Rc (Ω) V2,pp (V) V1,pp (V) T (µs) |∆t| (µs) Computations A |φ| (deg) Rs (Ω) Ls (mH) Nominal Rs (Ω) Measured Rs (Ω) Relative ∆Rs (%) Relative ∆Rs (%) 100 Nominal Ls (mH) Measured Ls (mH) Relative ∆Ls (%) Relative ∆Ls (%) 330 Part B: Parallel RC-Circuit Raw Data Nominal Rc (Ω) 3.3k 5.1k Measured Rc (Ω) V2,pp (V) V1,pp (V) T (µs) |∆t| (µs) Computations A |φ| (deg) Rp (Ω) Cp (nF) Nominal Rp (Ω) Measured Rp (Ω) Relative ∆Rp (%) Relative ∆Rp (%) 2k Nominal Cp (nF) Measured Cp (nF) Relative ∆Cp (%) Relative ∆Cp (%) 68