EE 3324 Electromagnetics Laboratory

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Lorena Dickerson
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1 EE 3324 Electromagnetics Laboratory Experiment #3 Inductors and Inductance 1. Objective The objective of Experiment #3 is to investigate the concepts of inductors and inductance. Several inductor geometries will be investigated and precision devices will be used to measure the magnetic flux distributions surrounding these devices. The magnetic flux measurements will be compared to analytical magnetic flux density expressions. 2. Introduction The basic geometry of the ideal inductor (long solenoid) is shown in Figure 1. The length of the cylindrical inductor core l is assumed to be much larger that the radius of the core r o. The cylindrical core is wound with N turns of wire which carry a current I. The vector magnetic field H (A/m) and the vector magnetic flux density B (Wb/m 2 or T) are both assumed to be uniform throughout the inductor core which is characterized by a permeability : (H/m). The uniform field approximation within the inductor becomes more accurate as the ratio l/r o grows larger. The assumption of a uniform magnetic field within the long solenoid is accurate everywhere except close to the ends of the solenoid. The vector magnetic field is related to the vector magnetic flux according to: (1) The direction of the magnetic field within the long solenoid is related to the current direction by the right hand rule. The magnetic field on the solenoid axis (z-axis) is found by approximating the current flowing around the inductor as a uniform cylindrical surface current defined by (2) Figure 1. Ideal inductor. and using the Biot-Savart law which gives (3)

2 Equation (3) can be applied to the long solenoid (l o r o ) to determine the magnitude of the magnetic field at the center of the inductor ( z = l/2) which yields Using the same approximations in Equation (3) to determine the magnetic field at the ends of the long solenoid yields a value which is exactly one-half of Equation (4). The inductance L in henries of the ideal inductor (long solenoid) of Figure 1 is found according to the definition of inductance: where 7 defines the flux linkage (Wb). Each magnetic field line for the ideal inductor in Figure 1 emanates from the top of the inductor core, bends downward outside the inductor, and re-enters the inductor core at the bottom, forming a closed loop which links the inductor current N times. Thus, the flux linkage for the ideal inductor is defined by where Q m is the total magnetic flux over the cross-section of the inductor. Since B is assumed to be uniform everywhere within the ideal inductor, the total magnetic flux over the inductor cross-section is simply the product of the magnetic flux density and the cross-sectional area. Combining Equations (1), (4) and (6) yields the inductance of the ideal inductor. (4) (5) (6) The inductance of a seemingly less complex geometry actually yields a more complicated computation. Consider the inductance of a single wire loop (loop radius = r o, wire radius = a) as shown in Figure 2(a.). The determination of the flux linkage in this case involves a magnetic flux density which is not uniform over the cross-section of the loop. Evaluation of the inductance of the single wire loop yields (7) (8) where E(m) and K(m) are the complete elliptic integrals of the first and second kinds defined by (9) (10)

3 and where (11) For a large loop (a/r o is small), Equation (9) reduces to (12) Although Equation (12) was derived for a single wire loop, this equation can be modified to account for loops of similar geometry with multiple turns as shown in Figure 2(b.). The radius a in Figure 2(b.) represents the equivalent radius of the bundle of N conductors while the loop radius r o represents the mean radius to the center of the conductor bundle. Equation (12) must be multiplied by N 2 to account for the multiple turns of this loop which yields: (13) Figure 2. (a.) Single wire loop (b.) multiple wire loop. Note that Equation (13) is an approximation that is only valid if the loop radius r o is large in comparison to the equivalent radius of the wire bundle. The axial magnetic field along the axis of the single wire loop (the loop is assumed to be centered at the origin lying in the x-y plane) is given by (14) Equation (14) may also be used for the multiple wire loop (r o o a) by simply replacing the current I with NI. 3. Equipment List Gauss/Tesla Meter, axial probe, transverse probe Probe mount

4 LCR Meter High Current Power Supply 20 gauge wire (1 m length) 3 inductors: Multiple turn wire loop (N = 320, r o = 13.6cm, a = 9mm, I max = 1.5A) Fixed length coil (N = 120, r o = 45 mm, l = 42 cm, I max = 10A) Adjustable length coil (N = 30, r o = 40 mm, l max = 40 cm, I max = 10A) 4. Procedure Measurement of Magnetic Flux density The Model 5080 Gauss/Tesla Meter utilizes a Hall effect probe to measure magnetic flux density in units of Gauss, Tesla or ampere-meter. The Hall effect probe contains a device known as a Hall generator as shown in Figure 3. A constant current (I h ) is forced through the Hall generator (a thin slice of semiconductor material) and the Hall voltage (V h ) is measured between the edges of the Hall generator which are transverse to the current direction. With no magnetic flux normal to the Hall generator, the Hall voltage is zero. When the magnetic flux density through the Hall generator is non-zero, the current bends closer to one edge and produces a Hall voltage which is linearly proportional to the flux density. The polarity of the Hall voltage allows the user to determine the direction of the flux density, also. The Model 5080 comes with two probes for measuring magnetic flux density. The transverse and axial probes are shown in Figure 4. Note the difference in the probe shapes and the orientation of the Hall generator. Also note that the Hall generator is located very close to the tip of the Figure 3. Hall generator. respective probe. Warning. Handle the Hall probe with care. Do not bend the stem or apply pressure to the probe tip as damage may result. Do not allow the probe to come in contact with any voltage source greater than 30 V-rms or 60 VDC. When inserting the probe into the gaussmeter, carefully pull the base of the probe away from the meter so that the probe can be inserted completely. Make sure that the probe is connected to the Model 5080 prior to turning the device on since a self calibration test is performed on power up. To make a magnetic flux density measurement, first rotate the function selector to MODE and press the SELECT button to select AC or DC measurement. All of the measurements in this laboratory experiment are static (DC) measurements. Turn the function selector to UNITS and press the SELECT button to the proper units. All of the magnetic flux density measurements should be performed in units of Teslas. Rotate the function selector to RANGE to select the range of magnetic flux density to be measured. The Model 5080

5 Figure 4. Transverse and Axial Hall effect probes. allows for an automatic mode (AUTO) which allows the device to select the proper range. Select the AUTO mode. The probe must be zeroed before making any measurement since the Hall effect probe may produce a small DC voltage even when no field is present. This error voltage may be caused by variations in materials in construction, or by other sources of magnetic flux in the vicinity (motors, magnets, the magnetic field of the earth, etc.). The Model 5080 allows for an automatic zeroing of the probe. Rotate the function selector to ZERO and press the AUTO pushbutton. The zeroing process normally takes from 5 to 15 seconds. 1. Take the leads of the LCR meter and connect them together. Record the inductance measured. Connect the 1-m long 20 gauge wire between the terminals of the LCR meter and form the wire in the shape of a circle as shown in Figure 5(a.). Measure the inductance of the circular wire loop. Compare the measured inductance with the analytical formula given in Equation (12). Grasp the wire on the both sides of the circle and pull the wire outward so that the conductors become approximately parallel as shown in Figure 5(b.). Measure the inductance of this wire configuration. Discuss your results as they relate to flux linkage. 2. Measure the inductance and resistance of the variable length coil as the coil length is varied from 20 cm to 40 cm in 2 cm steps. The variable length coil is characterized by N = 30 and r o = 40 mm. Plot the measured inductance verses inductor length. On the same plot, show the corresponding theoretical inductance.

6 3. Measure the inductance and resistance of the fixed length coil (N = 120, r o = 45 mm and l = 42 cm). Compare your measured inductance with that of the ideal inductor (long solenoid) equation. Place the first metal rod into the inductor and record the inductance. Repeat for the second metal rod. Discuss your results at they relate to the characteristics of the two metal rods. 4. Measure the inductance and resistance of the multiple-turn wire loop (N = 320, r o = 13.6cm, a = 9mm). Connect the multiple-turn inductor in series with the constant length inductor and measure the resulting inductance. Note that the cross-section of the multiple-turn wire loop is rectangular rather than cylindrical as assumed in the derivation of Equation (13). The given equivalent radius (a = 9mm) represents one-half of the axial dimension of the conductor bundle. Equation (13) should yield sufficiently accurate results if r o is large in comparison to a. Connect the multiple-turn inductor in parallel with the constant length inductor and measure the inductance. Using the measured inductances, compare your results with the expected values. 5. Align the axial probe of the Gauss/Teslameter to measure the magnetic flux density along the axis of the coil. Take care to carefully align the probe along the coil axis. With the probe mount fully extended, position the probe tip at the midpoint of the coil (there is a small hole in the plastic core of the fixed length coil that marks the midpoint of the coil). Turn on the power to the Gauss/Teslameter and setup the device for DC measurements in the AUTO range mode. Be sure to zero the probe before making any measurements. Connect the high current power supply to the fixed length coil. With the current limit and voltage limit dials turned to the minimum values, turn on the DC source. Carefully increase the voltage and current dials slightly and monitor the current produced. You should be able to determine what voltage will produce a current of 2 A across the fixed length coil resistance measured in part (3). Adjust the current and voltage until a current of 2 A is obtained. Using the end of the coil as your reference (z = 0), measure the magnetic flux density along the axis of the coil in 1 cm steps from the coil center (z = 21 cm) to 5 cm beyond the end of the coil (z =!5 cm). This data will be included in your report on a plot with the analytical expression for the axial field (see the additional questions). Measure the axial magnetic flux density on the outside of the solenoid at its midpoint by placing the axial probe against the outer surface of the solenoid. Record your result. 6. Turn the Gauss/Teslameter off, remove the axial probe and replace it with the transverse probe. Align the transverse probe of the Gauss/Teslameter to measure the magnetic flux density through the multiple-turn wire loop from the center of the loop to the inside edge of the loop. Note that the transverse probe should be positioned parallel to the plane of the loop. Position the probe tip at the midpoint of the loop. Connect the high current power supply to the multiple-turn wire loop. Turn on the power to the Gauss/Teslameter and setup the device for DC measurements in the AUTO range mode making sure to zero the probe before making any measurements. Carefully adjust the current and voltage dials to obtain a current of 500 ma across the multiple-turn wire loop resistance measured in part (4). Using the center of the loop as your reference (r = 0), measure the magnetic flux density along a radial line in 1 cm steps from the loop center outward to a distance of (r = 26 cm). Plot the measured magnetic flux density verses radius in your report. Compare your measured value of flux density at the loop center to that obtained by the analytical equation in Equation (14) (modified to account for the multiple turns).

7 5. Additional Questions 1. Use MATLAB to determine the inductance of the multiple turn wire loop according to Equation (8). Record the values of m, E(m) and K(m) in your report. Compare your answer to your measured value and the analytical value obtained using Equation (12) modified to account for the multiple turns. 2. Use MATLAB to plot the analytical expression for the magnetic field along the axis of the long solenoid [Equation (3)] over a range of z =!l/2 to z = 3l/2 for the fixed length coil. Plot the data points obtained in part (5) the laboratory procedure (convert the measured B to H). Discuss your results. 3. Compute how much energy is stored in the magnetic field of the air-core solenoid and the magnetic field of the steel-core (: r =150) solenoid assuming each inductor carries a DC current of 200mA. Use the value of air-core inductance measured for the fixed length coil in part (3) of the laboratory procedure.

Experiment No: EM 4 Experiment Name: Biot-Savart Law Objectives:

Experiment No: EM 4 Experiment Name: Biot-Savart Law Objectives: Measuring the magnetic field of a current passing through long straight and conductor wire as a function of the current. Measuring the magnetic