Coil in the AC circuit with Cobra3

Transcription

1 Coil in the AC circuit with Cobra3 TEP Related topics Inductance, Kirchhoff s laws, Maxwell s equations, a.c. impedance, phase displacement. Principle and task The coil is connected in a circuit with a voltage source of variable frequency. The impedance and phase displacements are determined as functions of frequency. Parallel and series impedances are measured. Equipment 1 Cobra3 Basic Unit Power supply, 12 V RS232 data cable Cobra3 Universal Recorder software Cobra3 Function generator module Coil, 300 turns Coil, 600 turns Resistor in plug-in box 47 Ohms Resistor in plug-in box 100 Ohms Resistor in plug-in box 220 Ohms Connection box Connecting cord, 250 mm, red Connecting cord, 250 mm, blue Connecting cord, 500 mm, red Connecting cord, 500 mm, blue Fig. 1: Experimental set up for the measurement of the coil impedance P PHYWE Systeme GmbH & Co. KG All rights reserved 1

2 TEP Coil in the AC circuit with Cobra3 Tasks 1. Determination of the impedance of a coil as a function of frequency. 2. Determination of the inductance of the coil. 3. Determination of the phase displacement between the terminal voltage and total current, as a function of the frequency in the circuit. 4. Determination of the total inductance of coils connected in parallel and in series. Set-up and procedure The experimental set up is as shown in Figs. 1, 2a and 2b. Connect the Cobra3 Basic Unit to the computer port COM1, COM2 or to USB port (for USB computer port use USB to RS232 Converter ). Start the measure program and select Cobra3 Universal Writer Gauge. Begin the measurement using the parameters given in Fig. 3. Fig. 2a: Circuit for measurement of the coil impedance. Fig. 2b: Circuit for measurement of total current and total voltage. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P

3 Coil in the AC circuit with Cobra3 TEP Theory and evaluation If a coil of inductance L and a resistor of resistance R are connected in a circuit (see Fig. 2), the sum of the voltage drops on the individual elements is equal to the terminal voltage U: U = I R + L di dt, (1) where I is the current. The resistors R are selected so that the d. c. resistance of the coil, with a value of 0.2 Ω, can be disregarded. If the alternating voltage U has the frequency ω = 2πf and the waveform U = U 0 cos ωt, Fig. 3: Measuring parameters. then the solution of (2) is I = I 0 cos (ωt φ) with the phase displacement φ given by tan φ = ωl R (2) and I 0 = U 0 R 2 + (ωl) 2 (3) P PHYWE Systeme GmbH & Co. KG All rights reserved 3

4 TEP Coil in the AC circuit with Cobra3 It is customary to threat complex impedances as operators R i : Coil R L = iωl, Ohmic resistance R = R. With parallel connection, R 1 1 tot = R ι ι The real impedance of a circuit is the absolute value of R tot and the phase relationship, analogous to (2), is the ratio of the imaginary part to the real part of R tot. Task 1 To determine the impedance of a coil as a function of the frequency, the coil is connected in series with resistors of known value. The frequency is varied until there is the same voltage drop across the coil as across the resistor (see Fig. 2a). The resistance and impedance values are then equal: R = ωl = 2πf L (4) The measured frequencies for 300 turns and 600 turns coils and for different resistors with the same voltage drops across the coil as across the resistor are shown in Fig. 4. Task 2 With the regression line to the measured values of Fig. 4 and the linear statement (see eq. (4)) y = a + b x f = a + (1/2πL) R We receive for the inductance: L = 1/2πb and with the slopes for 300 turns and 600 turns coils (see Fig. 4): L(300) = (1.98 ± 0.09) mh L(600) = (9.1 ± 0.4) mh Both values are very close to theoretical values of the used inductances L(300) = 2 mh, L(600) = 9 mh. 4 PHYWE Systeme GmbH & Co. KG All rights reserved P

5 Coil in the AC circuit with Cobra3 TEP Fig. 4: Measured frequencies for 300 turns and 600 turns coils and for different resistors when the same voltage drops across the coil as across the resistor. Task 3 The phase displacement between the total voltage and the total current can be measured using a circuit shown in Fig. 2b. Use the "Survey Function" of the Measure Software as it is shown in Fig. 5 for the measurement of phase displacements. Plot the phase displacement (see Fig. 6) and the tangent of phase displacement as a function of the Cobra3 function generator frequency (see Fig. 7). From the regression line to the measured values of Fig. 7 and the linear statement (see eq. (2)) y = a + b x tan(ϕ) = a + (2πL/R) f We receive for the inductance: L = br/2π and with the slopes for 300 turns and 600 turns coils (see Fig. 7): L(300) = (2.0 ± 0.1) mh L(600) = (8.6 ± 0.5) mh P PHYWE Systeme GmbH & Co. KG All rights reserved 5

6 TEP Coil in the AC circuit with Cobra3 Fig. 5: Measurement of current and voltage amplitudes and of phase displacements with the "Survey Function". Both values are very close to theoretical values of the used inductances L(300) = 2 mh, L(600) = 9 mh. Fig. 6: Phase displacement between total current and total voltage for 600 turns coil and 47 ohm resistor as a function of the frequency. 6 PHYWE Systeme GmbH & Co. KG All rights reserved P

7 Coil in the AC circuit with Cobra3 TEP Fig. 7: The tangent of phase displacement as a function of frequency for a 600 turns coil. Task 4 When coils are connected in parallel or in series, care should be taken to ensure that they are sufficiently far apart, since their magnetic fields influence one another. As in Task 3, use the "Survey Function" for the measurement of phase displacements and plot the tangent of phase displacement as a function of the frequency (see Fig. 8). From the slopes of the straight lines for coils connected in parallel in series (see Fig. 8) we receive: L( ) = (2.1 ± 0.1) mh L( ) = (11.8 ± 0.6) mh Both values are close to theoretical values of the used inductances: L( ) = 1.6 mh L( ) = 11 mh. P PHYWE Systeme GmbH & Co. KG All rights reserved 7

8 TEP Coil in the AC circuit with Cobra3 Fig. 8: Calculation of the total inductance of coils connected in parallel and in series. 8 PHYWE Systeme GmbH & Co. KG All rights reserved P