Electricity. Coil in the AC circuit /11. Electrodynamics. What you need:

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1 Electrodynamics Electricity Coil in the AC circuit -01/11 What you can learn about Inductance Kirchhoff s laws Maxwell s equations AC impedance Phase displacement Principle: 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. Set-up of experiment P with FG-Module What you need: Experiment P with FG-Module Experiment P with oscilloscope Function generator Oscilloscope, 30 MHz, 2 channels Difference amplifier Digital counter, 4 decades Screened cable, BNC, l = 750 mm Connecting cord, l = 100 mm, red Coil, 300 turns Coil, 600 turns Resistor in plug-in box 50 Ω Resistor in plug-in box 100 Ω Resistor in plug-in box 200 Ω Connection box Connecting cord, l = 500 mm, red Connecting cord, l = 500 mm, blue Cobra3 Basic Unit Power supply, 12 V RS232 data cable Cobra3 Universal writer software Measuring module function generator PC, Windows 95 or higher Complete Equipment Set, Manual on CD-ROM included Coil in the AC circuit P /11 Tangent of the current-voltage phase displacement as a function of the frequency used for calculation of the total inductance of coils connected in parallel and in series. 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 impedance of coils connected in parallel and in series. PHYWE Systeme GmbH & Co. KG D Göttingen Laboratory Experiments Physics 183

2 Coil in the AC circuit LEP -01 Related topics Inductance, Kirchhoff s laws, Maxwell s equations, a. c. impedance, phase displacement. Principle 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 Coil, 300 turns Coil, 600 turns Resistor in plug-in box 50 Ohms Resistor in plug-in box 100 Ohms Resistor in plug-in box 200 Ohms Connection box Difference amplifier Function generator Digital counter, 4 decades Oscilloscope, 20 MHz, 2 channels Screened cable, BNC, l = 750 mm Connecting cord, l = 100 mm, red Connecting cord, l = 500 mm, red Connecting cord, l = 500 mm, blue 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 impedance of coils connected in parallel and in series. Set-up and procedure The experimental set up is as shown in Fig. 1. Since normal voltmeters and ammeters generally measure only rms (root mean square) values and take no account of phase relationships, it is prefereable to use an oscilloscope. The experiment will be carried out with sinusoidal voltages, so that to obtain rms values, the peak-to-peak values measured on the oscilloscope (U - ) are to be divided by 222. In accordance with I = U/R, the current can be deduced by measurement of the voltage across the resistor. The circuit shown in Fig. 2 permits the simultaneous display of the total current and the coil voltage. If, by means of the time-base switch of the oscilloscope, one half-wave of the current (180 ) is brought to the full screen width (10 cm) possibly with variable sweep rate the phase displacement of the voltage can be read off directly in cm (18 /cm). The Y-positions of the two base-lines (GND) are made to coincide. After switching to other gain settings, the base-lines are readjusted. In order to achieve high reading accuracy, high gain settings are selected. The inputs to the difference amplifier are non-grounded. Fig.1: Experimental set up for investigating the a. c. impedance of the coil. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

3 LEP -01 Coil in the AC circuit Fig. 2: Circuit for display of current and voltage with the oscilloscope. Fig. 3: Impedance of various coils as a function of the frequency. 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. The resistance and impedance values are then equal: R Ω = vl = X L (1) with the phase displacement f given by vl tan f = (3) R and U 0 I 0 = (4) 2R 2 1vL2 2 The phase displacement between the terminal voltage and the total current can be measured using a similar circuit to Fig. 2, but with channel B measuring the total voltage and not the voltage across the coil. 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. It is customary to treat complex impedances as operators Coil Rˆ L ivl, Ohmic resistance Rˆ R. With parallel connection, 1 ai Rˆ i 1 Rˆ i : 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 IR L di dt 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 = 2 f and the waveform U = U 0 cos vt, then the solution of (2) is I = I 0 cos (vt f), (2) The real impedance of a circuit is the absolute value of and the phase relationship, analogous to (2), is the ratio of the imaginary part to the real part of. From the regression line to the measured value of Fig. 3 and the exponential statement Y = A X B there follows the exponent B 1 = 1.02 ± 0.01 (see (1)) B 2 = 1.01 ± 0.01 With the regression line to the measured values of Fig. 3 and the linear statement Y = A + B X PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

4 Coil in the AC circuit LEP -01 Fig. 4: Phase displacement (tan f) between total current and total voltage as a function of frequency tan f Fig. 5: Phase displacement (f) between total current and total voltage as a function of frequency. f the slope B 1 = ± (see (1)) B 2 = ± The frequency at which the total impedance of the coils was equal to the reference of 200 Ω was determined with coils connected in parallel and in series. is obtained. From this, with R = L the inductances L 1 = 2.38 mh L 2 = 10.4 mh Table: Total inductance of coils L i connected in parallel (line 1) and in series (line 2). Coil f (200 Ω) L tot L 1 L khz 1.93 mh L 1 + L khz mh are obtained. From the regression line to the measured values of Fig. 4 and the exponential statement Y = A X B the exponent B = 0.97 ± 0.01 follows (see (2)) PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

5 LEP -01 Coil in the AC circuit PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

6 Coil in the AC circuit with Cobra3 LEP -11 Related topics Inductance, Kirchhoff s laws, Maxwell s equations, a. c. impedance, phase displacement. Principle 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 Cobra3 Basic Unit Power supply, 12 V RS 232 data cable Cobra3 Universal writer 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, l = 250 mm, red Connecting cord, l = 250 mm, blue Connecting cord, l = 500 mm, red Connecting cord, l = 500 mm, blue 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. 1: Experimental set up for the measurement of the coil impedance. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

7 LEP -11 Coil in the AC circuit with Cobra3 Fig. 2a: Circuit for measurement of the coil impedance. then the solution of (2) is I = I 0 cos (vt f) with the phase displacement f given by and vl tan f = (2) R Fig. 2b: Circuit for measurement of total current and total voltage. U 0 I 0 = (3) 2R 2 1vL2 2 It is customary to treat complex impedances as operators Coil Rˆ L ivl, Ohmic resistance Rˆ R. With parallel connection, Rˆ i : 1 ai Rˆ i 1 The real impedance of a circuit is the absolute value of and the phase relationship, analogous to (2), is the ratio of the imaginary part to the real part of. 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 IR L di dt 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 = 2 f and the waveform U = U 0 cos vt, Fig. 3: Measuring parameters., (1) 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 = vl = 2pf L (4) The masured 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 + (1/2pL) R We receive for the inductance: L = 1/2pb 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 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

8 Coil in the AC circuit with Cobra3 LEP -11 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)) Both values are very close to theoretical values of the used inductances L(300) = 2 mh, L(600) = 9 mh. y = a + b x tan(phi) = a + (2pL/R) f We receive for the inductance: L = br/2p 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 Fig. 6: Phase displacement between total current and total voltage for 600 turns coil and 47 ohm resistor as a function of the frequency. Fig. 5: Measurement of current and voltage amplitudes and of phase displacements with the "Survey Function". Fig. 7: The tangent of phase displacement as a function of frequency for a 600 turns coil. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

9 LEP -11 Coil in the AC circuit with Cobra3 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. Fig. 8: Calculation of the total inductance of coils connected in parallel and in series PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen