Related Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage

Transcription

1 Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is produced in a ong coi. The votages induced across thin cois which are pushed into the ong coi are measured depending on the frequency of the current in the fied coi and the strength of its magnetic fied as we as the number of turns and the diameter of the induction coi. Materia 1 Fied coi, 750 mm, 485 turns/m Induction coi, 300 turns, d = 40 mm Induction coi, 300 turns, d = 32 mm Induction coi, 300 turns, d = 25 mm Induction coi, 200 turns, d = 40 mm Induction coi, 100 turns, d = 40 mm Induction coi, 150 turns, d = 25 mm Induction coi, 75 turns, d = 25 mm Digita function generator Muti-range meter, anaogue Connecting cord, = 750 mm, red Connecting cord, = 750 mm, bue Fig. 1: Experimenta set-up with one induction coi sid into the fied coi. P PHYWE Systeme GmbH & Co. KG A rights reserved 1

2 TEP Magnetic induction Tasks Measure the induction votage as a function of 1. the current in the fied coi at a constant frequency, 2. the frequency of the magnetic fied at a constant current, 3. the number of turns of the induction coi at constant frequency and current and 4. the cross-sectiona area of the induction coi at constant frequency and current. Set-up The experimenta set-up is as shown in fig. 1. One digita muti-range meter is set up in series connection to the fied coi and the digita frequency generator in order to measure the current in the fied coi. The second muti-range meter is connected to the induction coi to measure the induced votage. Procedure Set both muti-range meter to AC (aternating current) mode. For the measurements choose ranges up to 300 ma and up to 3 V respectivey. Be carefu to read the correct scaes for the measurements. For a detaied description of the operation of the digita function generator pease refer to the manua. Task 1: Tune the current in the fied coi by turning up the ampitude of the sinus signa of the digita frequency generator. Start with an ampitude of 0.5 V and increase to a maximum of 10 V in steps of 0.5 V. Task 2: Choose the current in the fied coi between 20 and 40 ma. The effect of frequency shoud be studied between 1 khz and 12 khz, since beow 0.5 khz the coi practicay represents a short circuit and above 12 khz the accuracy cannot be guaranteed. Increase the frequency in steps of 0.5 khz. In order to maintain a constant current in the fied coi for various frequencies, you have to adjust the ampitude of the sinus signa very accuratey for each frequency. Task 3 and 4: Choose frequency and signa-ampitude and maintain these settings throughout the measurements. Note down the induced votage, number of turns and diameter for each induction coi. Theory To understand the fundamentas of this experiment, two cases must be considered. First we treat the tempora variation of the magnetic fux through an area which induces a votage in a conductor. In this experiment, this votage wi be measured. Second the tempora variation of a current in a conductor, which induces a magnetic fied, wi be regarded whereby the current is the second measurand. The tempora variation of the magnetic fux eads to Faraday's aw of induction. The magnetic fux φ through an area A is obtained by integrating the magnetic fux density B over this area (1). φ = B da A (1) After the aw of induction the tempora variation of the fux φ induces the votage (2). Considering the fux φ through A which is encosed by a conductor oop, the induced votage is the integra of the eectric fied E in the conductor oop over the area s boundary C. 2 PHYWE Systeme GmbH & Co. KG A rights reserved P

3 Magnetic induction TEP = φ = t E ds C (2) This reationship for one conductor oop is the second of Maxwe s equations. For n parae conductor oops equation (3) is vaid, if φ is the same for a oops. = n φ t (3) Now we have a reation between the induced votage and the number of cois. We need to find a way to cacuate the right side of equation (3). Therefore we consider the magnetic fux density in a ong coi, which is constant, so that equation (1) simpifies to the foowing reation. φ = B da = B A (4) A Utiizing Maxwe s first equation, we can find an expression for B which depends ony on measurands and fundamenta constants. Maxwe s first equation (5) states that in a conductor a current generates a magnetic fied, of which the cosed fied ines circe around the currents. ȷ A da = 1 μ B ds C (5) There μ is the magnetic conductivity (a materia s constant), C is the inductor coi which is penetrated by the fied coi s magnetic fied density B and encoses the area A. In the fied coi fows a current I which is given by integrating the inductor coi s area A over the current density ȷ, so I = ȷ da. For a ong coi with n turns the absoute vaue of B can be approximated by the foowing equation: A B = n μ I (6) There, is the ength of the coi which must be significant higher than the diameter. In air, μ can be approximated by the magnetic constant μ 0 = 4 π 10 7 V s A 1 m 1. (7) If an aternating current I(t) = I 0 sin(ωt) with the frequency ν = ω 2π fows through the fied coi, then from (6) the fied density in the fied coi is a function of time and aternates in phase with the current: B(t) = n μ 0 I 0 sin(2π ν t) (8) The induced votage can be cacuated by appying equation (6) to equations (3) and (4). Execution of the time derivation gives the foowing reation for the induced votage P PHYWE Systeme GmbH & Co. KG A rights reserved 3

4 TEP Magnetic induction (t) = n A B(t) t = n A 2π ν n μ 0 I 0 cos(2π ν t), (9) where n is the induction coi s number of turns and A its cross-sectiona area. The induced votage aternates with the same frequency as the current but is phase-shifted by π 2. Evauation and resuts In the foowing the evauation of the obtained vaues is described with the hep of exampe vaues. Your resuts may vary from those presented here. Task 1: Measure the induction votage as a function of the current in the fied coi at a frequency of 10.7 khz and cacuate the magnetic constant μ 0. In order to vary the magnetic fied the current in the fied coi has to be atered. With reation (6) the magnetic constant can be cacuated if the current and the magnetic fied are known. In this experiment the magnetic fied is not measured so we need to find another way. Therefore we consider reation (9). As is easiy shown (see equation 10) the magnetic fied constant is incuded in the sope s of the induced votage s inear dependence of the current. (t) = n A 2π ν n μ 0 cos(2π ν t) I 0 = s(t) I 0 (10) The time-dependence can be disregarded, if we aways measure current and votage in intervas of one period T = 1 ν. Insertion in reation (10) gives = n A 2π ν n μ 0 I 0 (11) as the cosine-function simpy reduces to unity. With equation (11) a other contributions to s are known from the specifics of the used cois and we can cacuate µ 0. Fitting the measurements to a inear function (see Fig. 2) gives equation (12) with the correation coefficient R = mv = I 0 /ma (12) As can be seen in fig. 2 for very sma fieds, the measured vaues differ from the expected vaues and the induced votage tends towards zero. For any currents greater than 3 ma the dependence is indeed inear and the experimenta resuts are we described by equation (12). For the sope s we obtain from equations (11) and (12): s = n A 2π ν n μ 0 = From the experiment we cacuate µ 0 = the iterature with μ it 6 Vs 0 = Am 6 Vs Am, which is of the same order as the vaue given in 4 PHYWE Systeme GmbH & Co. KG A rights reserved P

5 Magnetic induction TEP Fig. 3: The graph shows the induced votage for different frequency at a fied strength of 18 µt in the fied coi (bue). The dependence foows reation (13). The red ine indicates the theoreticay expected vaues. Task 2: Measure the induction votage as a function of the frequency of the magnetic fied at constant current. As can be seen from equation (11) the induced votage depends ineary on the frequency of the magnetic fied and the current in the fied coi respectivey. Measurements were done at a current of 30 ma which corresponds to a fied strength of approximatey 18.3 µt in the fied coi. Fitting the measurements to a inear function (see Fig. 3) gives equation (13) with the correation coefficient R = and a sope s fit = 40 ± 8. mv = 40 ν khz + 9 (13) If we cacuate the sope with the known vaues for the fied coi, induction coi and current, we obtain the theoretica vaue for the sope with s cac = 43.2, which is we within error imits. Fig. 4: induced votage for different cois with a diameter of 26 mm foowing reation (14). P PHYWE Systeme GmbH & Co. KG A rights reserved 5

6 TEP Magnetic induction Task 3: Measure the induction votage as a function of the number of turns of the induction coi at constant frequency and current. Measurements were done at 10.7 khz and 30 ma. A induction cois with diameters of 26 mm and 41 mm were studied. Fitting the measured vaues for the cois with a diameter d = 26 mm to a inear function gives reation (14) corresponding to Fig. 4. mv = n 46 (14) with the correation R = With a greater diameter we can reach significanty higher induction votages as Fig. 5 easiy shows. Fitting the measured vaues for the cois with a diameter d = 41 mm to a inear function gives reation (15) Fig. 5: induced votage for different cois with a diameter of 41 mm foowing reation (15). mv = 2.78 n 100 (15) with a sighty better correation of R = Task 4: Measure the induction votage as a function of the cross-sectiona area of the induction coi at constant frequency and current. The cross-sectiona area is the circuar area encosed by the coi. With reation (16) we can cacuate the cross-sectiona area with the known diameter. A = π 4 d2 Measurements were done at 10.7 khz and 30 ma. A induction cois with 300 turns were used in this experiment. Figure 6 shows the obtained vaues which were fitted to a inear function as in equation (17). (16) mv = A mm² 100 (17) The correation is quite high with R = Fig. 6: Induced votage for various cois with 300 turns. The dependence of the cross-sectiona area foows reation (17). 6 PHYWE Systeme GmbH & Co. KG A rights reserved P