Transformers, Mutual Inductance and Coupled Coils - Lab Experiment 6 | ECE 225, Lab Reports of Electrical Circuit Analysis

Download Transformers, Mutual Inductance and Coupled Coils - Lab Experiment 6 | ECE 225 and more Lab Reports Electrical Circuit Analysis in PDF only on Docsity! Boise State University Department of Electrical and Computer Engineering ECE225L – Circuit Analysis and Design Lab Experiment #6: Transformers, Mutual Inductance, and Coupled Coils 1 Objectives The objectives of this laboratory experiment are: • To observe the phenomenon of inductive coupling • To perform polarity and turns ratio tests on a transformer • To measure the self and mutual inductances of a transformer • To measure the inductance of two coupled series-aiding or series-opposing coils 2 Theory Consider the two-winding shown in Figure 1(a) and assume that the primary winding is energized with a current i1 while the secondary winding is open-circuited. The primary current i1 creates a magnetic flux φ1 in the iron core with two components: • A large portion of this primary flux, the mutual flux φm1, is channeled through the magnetic core and links all turns of the secondary winding; • A small portion of this primary flux, the leakage flux φl1, links all turns of the primary winding but none of the turns of the secondary winding; These two fluxes result in the following flux linkages and inductances in each winding: λ1 = N1φ1 = N1(φl1 + φm1) = L1i1 (1) λ2 = N2φm1 = L21i1 (2) Similarly, assume that the secondary winding is energized with a current i2 while the primary winding is open-circuited as shown in Figure 1(b). The secondary current i2 creates a magnetic flux φ2 in the iron core with two components: • A large portion of this secondary flux, the mutual flux φm2, is channeled through the magnetic core and links all turns of the primary winding; • A small portion of this secondary flux, the leakage flux φl2, links all turns of the secondary winding but none of the turns of the primary winding; These two fluxes result in the following flux linkages and inductances in each winding: λ1 = N1φm2 = L12i2 (3) λ2 = N2φ2 = N2(φl2 + φm2) = L2i2 (4) 1 v 1 N − 1 i * + * l1 m1 φ φ 1 + * φ 2 N 1 N (a) 2 − + v 2 N − * 1 N 2 i1 i − v 1 − + * 2 v (d)(c) l2 l1 m2 m1 φ φ φ φ 2 N + 2 L * (b) 1 v 2 i 2 v − + l2 φ m2 * * 1 L M 2 i 1 i 2 v 1 v −− ++ Figure 1: Tests for Measuring Transformer Self and Mutual Inductances (a)-(c) and Transformer Symbol (d) If both windings are energized simultaneously as in Figure 1(c), the resulting flux linkages of each coil are given by: λ1 = N1(φl1 + φm1 + φm2) = L1i1 + L12i2 (5) λ2 = N2(φm1 + φl2 + φm2) = L21i1 + L2i2 (6) According to Faraday’s law, the voltages induced in each coil are v1(t) = R1i1 + dλ1 dt = R1i1 + L1 di1 dt + L12 di2 dt (7) v2(t) = R2i2 + dλ2 dt = R2i2 + L21 di1 dt + L2 di2 dt (8) where R1 and R2 are the resistances of the primary and secondary windings, respectively. 2 4 Procedure 1. Set up a step-down two-winding transformer with the 400-turn coil on the primary side connected to the 12.6-V AC power supply and leave the 200-turn coil open-circuited on the secondary side as shown in Figure 3(a). Turn the AC power switch ON and apply 12.6 V to the 400-turn coil. Make sure that the labels indicating the number of turns (200 and 400 turns, respectively) are showing on the top faces of each coil. Label the input primary terminals as 1-2 with terminal 1 being the “positive” terminal and terminal 2 being the negative terminal. Assume mentally that terminal 1 is marked with a dot. Similarly, label the terminals on the secondary side as 3-4. Assume that terminal 3 is the positive terminal and terminal 4 the negative one. Observe the voltages v12(t) and v34(t) on the oscilloscope and record the peak-to-peak amplitudes of both voltages as well as their phase shift. (This phase shift is either 0o or 180o.) Deduce where the second polarity dot should be placed (terminal 3 or terminal 4). V12,pp (V) V34,pp (V) ∆θ (deg) Dotted Terminals 1 and 2. Hook up the two-winding transformer as shown in Figure 3(b). Measure and record the DC resistance of coil 1-2 using a handheld multimeter. Apply a 12.6 VAC to the primary side and record the primary rms current and the primary and secondary rms voltages. The benchtop multimeter is used to read the rms current in the primary (400-turn) winding. R1 (Ω) I1 (Arms) V1 (Vrms) V2 (Vrms) 3. Turn off the AC power supply. Unhook the two-winding transformer. Measure and record the DC resistance of coil 3-4 using a handheld multimeter. Connect the secondary side to the AC power supply by applying 6.3 VAC to the secondary winding as shown in Figure 3(c). Record the secondary rms current and the secondary and primary rms voltages. R2 (Ω) I2 (Arms) V2 (Vrms) V1 (Vrms) 4. Connect terminals 2 and 3 of the two-winding transformer as shown in Figure 4(a). Apply 6.3 V to terminals 1 and 4. Record the series rms current in both coils, the series rms voltage of both coils, and the individual voltages across each coil. I1 = I2 (Arms) V14 (VAC) V12 (Vrms) V34 (Vrms) 5. Connect terminals 2 and 4 of the two-winding transformer as shown in Figure 4(b). Apply 12.6 V to terminals 1 and 3. Record the series rms current in both coils, the series rms voltage of both coils, and the individual voltages across each coil. I1 = I2 (Arms) V13 (Vrms) V12 (Vrms) V34 (Vrms) 5 + − V 1 V A V + − + − V 1 400 t. 200 t. 6.3 V I 2 + − V 2 (a) (b) (c) 1 + − + − V =12.6 V 12 V 34 400 t. 200 t. 400 t. 200 t.A V + + V 2 − V − 12.6 V I Figure 3: Polarity and Turns Ratio Tests (a) and Self and Mutual Inductance Measurements (b)-(c) 6 1 2 3 4 3 4 1 2 V = 6.3 V 14 V = 12.6 V 13 (b) (a) * 400 t. 200 t.+ − 400 t. 200 t. + − * * * Figure 4: Inductance measurements of (a) series-opposing and (b) series-aiding coils 5 Report Questions 1. Compute the voltage ratio v12/v34 of the 400-turn and 200-turn coils in Part 1 and compare it to the theoretical value corresponding to the turns ratio 400/200 = 2. Explain why it is greater or smaller than the theoretical value. 2. Calculate the self and mutual reactances X1 and X21 from the measurements of Part 2. Deduce the self and mutual inductances L1 and L21 in mH, respectively. 3. Calculate the self and mutual reactances X2 and X12 from the measurements of Part 3. Deduce the self and mutual inductances L2 and L12 in mH, respectively. 4. Verify that the mutual reactances X12 = X21 = XM and that the mutual inductances L12 = L21 = M found above are equal. 5. Compute the coefficient of coupling k of the two-winding transformer from the measured values of L1, L2, L12, and L21 and check that it is less than unity. 6. Using the measurements in Parts (4) and (5), compute the inductances Lp and Ln of the series-aiding and series-opposing coil connections. (Use the measured resistances of R1 and R2 from the lab procedure.) Then compute L12 = L21 = M = Lp − Ln 4 and compare this value to the previously calculated values from questions (2) and (3). 7