Coil in the AC circuit LEP Related topics Inductance, Kirchhoff s laws, parallel connection, series connection, a. c. impedance, phase displacement, vector diagram Principle The impedance and phase displacement of a coil in an A.C. circuit are observed as functions of frequency and inductance. Parallel and series circuits are studied. Material 1 Coil, 300 turns 06513 1 Coil, 600 turns 06514 1 Resistor plug-in box 50 ohm 06056-50 1 Resistor plug-in box 100 ohm 06057-10 1 Resistor plug-in box 200 ohm 06057-20 1 Connection box 06030-23 1 Difference amplifier 11444-93 1 Digital function generator 13654-99 1 Oscilloscope, 30 MHz, 2 channels 11459-95 2 Screened cable, BNC, l = 750 mm 07542-11 2 Connecting cord, l = 100 mm, red 07359 1 Connecting cord, l = 100 mm, blue 07359-04 4 Connecting cord, l = 500 mm, red 07361 3 Connecting cord, l = 500 mm, blue 07361-04 Fig. 1: Experimental set-up with coil and resistor in series. www.phywe.com P2440401 PHYWE Systeme GmbH & Co. KG All rights reserved 1
TEP Coil in the AC circuit Tasks 1. Measure the impedance of one of the coils as a function of frequency. 2. Determine the inductance of the coils. 3. Measure the phase displacement between terminal voltage and total current as a function of the frequency in the circuit. 4. Determine the total impedance of coils connected in parallel and in series. Set-up The experimental set-up is 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 preferable to use an oscilloscope to study the dynamics of current and voltage in such circuits. The experiment will be carried out with sinusoidal voltages. The rms values are obtained when the peak-to-peak values U p p measured on the oscilloscope are divided by 2 2. In accordance with relation (1), the total current I in the circuit can be deduced by measuring the voltage U across the resistor with resistance R. Fig. 2: Circuit for simultaneous display of voltage across the coil and total current. I = U R (1) The circuit shown in Fig. 2 permits the simultaneous display of the total current and the coil voltage. The phase displacement between the terminal voltage and the total current can be measured using a similar circuit, but with channel B measuring the terminal voltage instead of the voltage across the coil (see Fig. 3). When connecting coils in parallel or series the coils need to be sufficiently far apart, since their magnetic fields influence each other. Procedure In order to achieve high reading accuracy on the oscilloscope, high gain settings should be selected. After selecting the gain setting the Y-position of the two base-lines (GND) have to be adjusted until they Fig. 3: Circuit for simultaneous display of terminal coincide. The peak-to-peak amplitude of the frequency generator s signal should not be higher than voltage and total current. 5 V and should have no offset. The digital frequency generator s antenna output has to be connected with the ground socket of the difference amplifier. For detailed descriptions of the operation of the oscilloscope and the digital function generator please refer to the manuals. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P2440401
Coil in the AC circuit LEP Task 1 and 2: To determine the impedance of a coil as a function of the frequency the coil is connected in series with various resistors of known value (see Fig. 2). For each resistor the frequency is varied until there is the same potential difference across the coil as across the resistor. The resistance and impedance values are then equal and equation (2) is valid. R Ω = ωl = X L (2) From this relation the inductance of a coil can be deduced, if the impedance at one frequency is known. Task 3: Set up the circuit as shown in Fig. 3 to display both terminal voltage and total current of the circuit. There are two major ways to measure the frequency-dependent phase shift between total current and terminal voltage. If, by means of the time-base control of the oscilloscope, one half-wave of the current 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). Another way to determine the phase shift is to read the interval between the terminal voltage and the voltage across the resistor (corresponding to the total current) directly from the oscilloscope and calculate the time gap as well as the resulting phase angle. Attention: If the second procedure is chosen, the variable sweep rate must not be used as it distorts the timeframe up to a factor of 2.5 which makes an accurate calculation of the time gap impossible. Task 4: In order to determine the total inductance of the coils in parallel and series, include the coils appropriately into the circuit and determine the frequency for R = X L = 100 Ω. The inductance can then be calculated analog to task 2. Theory If a coil of inductance L and an ohmic resistor of resistance R are connected in a circuit, the sum of the potential differences across the coil and the resistor is equal to the terminal voltage U t which gives relation (3). U t = I R + L di (3) dt This corresponds to Kirchhoff s second law, which states, that the directed sum of all electrical potential differences in a closed circuit is zero. Care must be taken regarding the direction of the potential differences. The terminal voltage has the reversed direction of the voltage across the coil and the resistor. Relation (4) gives the formula of Kirchhoff s voltage law for the studied circuits, where U Ω = I R is the voltage across the resistor and U L = L di the voltage across the coil. dt U L + U Ω U t = 0 (4) Such resistors R have been selected so that the coils d. c. resistances of 0.8 Ω (n=300) and 2.5 Ω (n=600) can be disregarded as R L (d. c. ) R. If the alternating voltage U t has the frequency ω = 2πν and the waveform U t = U 0 cos ωt, (5) inserting (5) into equation (3) gives the following solution for the current I: www.phywe.com P2440401 PHYWE Systeme GmbH & Co. KG All rights reserved 3
TEP Coil in the AC circuit I = I 0 cos(ωt φ) (6) There φ is the phase angle and the phase displacement is given by tan φ = ωl R (7) with the peak current U I 0 = 0. (8) R 2 +(ωl) 2 As is seen from relation (4) the total current follows the terminal voltage. If there are more than one coil in the circuit, the total inductivity can be calculated with equations (9) for series and (10) for parallel connection, if the coils are magnetically uncoupled. L tot = L i (9) L tot = 1 L i (10) Results and Evaluation In the following the evaluation of the obtained values is described with the help of example values. Your results may vary from those presented here. Task 1: Measure the impedance of one of the coils as a function of frequency. From equation (2) the linear relation between frequency of the signal and impedance of the coil is obvious. Linear regression of the measured values (see Fig. 4) gives relation (11) with the correlation coefficient R = 0.9978. X L (ν) Ω = 13.8 ν 0.9 (11) khz For the coil with n=600 should also be found a linear relation. 4 PHYWE Systeme GmbH & Co. KG All rights reserved P2440401
Coil in the AC circuit LEP Fig. 4: Impedance values for various frequencies. Equation (11) gives the relation of the linear fit. Task 2: Determine the inductance of the coils. Considering relations (2) and (11) for the first coil leads to equation (12) which gives the inductance of the coil with L 1 = 2.19 mh. X L = 2πν L = 13.75 ν L = 13.75 2π = 2.19 mh (12) The analog calculation for the second coil results into L 2 = 10.4 mh. Task 3: Measure the phase displacement between the terminal voltage and total current as a function of the frequency in the circuit. As mentioned above there are two ways to determine the phase angle and the phase displacement respectively. The first way allows to read off the phase angle directly from the oscilloscope s display. Then the phase displacement can easily be calculated. According to equation (7) the phase displacement should show linear dependence with respect to the frequency as is shown in Fig. 5. The alternative method requires some calculation, as only the time shift dt is obtained. Obviously the ratio between time shift and one full period T is the same as between the phase angle and one full circle: dt T = φ 360 The period of the oscillation is nothing more than the reciprocal of the frequency. Thus the phase angle can be calculated with relation (13): www.phywe.com P2440401 PHYWE Systeme GmbH & Co. KG All rights reserved 5
TEP Coil in the AC circuit Fig. 5: Frequency-dependent phase angle (blue) and phase displacement (black) between current and voltage in the circuit. φ = 360 ν dt (13) Note: Special attention should be given to the units of the various measurands. Task 4: Determine the total inductance of coils connected in parallel and in series. For series (s) and parallel (p) connection with R = 100 Ω we find the frequencies of 1.34 khz and 8.62 khz respectively. Calculation analog to equation (12) results into L s = 11.88 mh and L p = 1.85 mh. One can easily verify that the found total inductances follow equations (9) and (10). 6 PHYWE Systeme GmbH & Co. KG All rights reserved P2440401