University of Pittsburgh

University of Pittsburgh Experiment #11 Lab Report Inductance/Transformers Submission Date: 12/04/2017 Instructors: Dr. Minhee Yun John Erickson Yanhao Du Submitted By: Nick Haver & Alex Williams Station #16 ECE 1201: Electronic Measurements and Circuits Laboratory

Introduction The purpose of this experiment was to construct and analyze an inductor and transformer. In many applications, including earlier labs in this course, inductors are implemented in the form of discrete 2-terminal components. While the inductance of a given inductor is influenced by various parameters related to the inductor s construction, these parameters are often ignored, as they can be easily observed or modified when using typical discrete inductor components. In this lab, an inductor will be constructed using epoxy-coated 24 AWG wire and a toroid-shaped ferrite core. When the wire is wound around the core and current is passed through the inductor, magnetic flux is produced inside the core, creating inductance. Physical parameters including the size and cross-sectional area of the core and the number of windings influence the inductance of the device. Later in the lab, an additional set of windings will be added to the opposite side of the toroid-shaped core to create a transformer. The primary purpose of a transformer is to increase or decrease the voltage of an AC signal. For a given number of turns of wire on the input side of the core (referred to as the primary), a given amount of electric flux is created inside the core. At the output side of the core (referred to as the secondary), output voltage is determined by the number of windings on the secondary relative to the number of windings on the primary. Procedure A: Construction of a 0.01 H Inductor As mentioned in the introduction, the inductor was constructed using epoxy-coated wire and a ferrite core. To construct the inductor, an F-material core and 24 AWG epoxy-coated wire were chosen. Based on a fixed toroid-shaped core size, inductance was manipulated by changing the number of windings around the core. To calculate inductance, the following equation was employed: L = µ oµ r N 2 a h (1) In this equation, µ o represents the permeability of free space, while µ r represents the relative permeability of the core. N represents the number of turns, a represents the cross-sectional area of the core, and h represents the height for a straight core. Since Eq. 1 is meant for a straight core, the equation was adapted for a toroid-shaped core by replacing the height (h) with the circumference of the toroid. The mean circumference of the toroid was calculated given the total area (A) of the toroid as follows: C = 2πr = 2π A π = 2π 37 mm2 = 21.5628 mm (2) π The rectangular cross-sectional area of the toroid (a) was calculated by multiplying the difference between outer and inner radius by the height: a = (29 mm 19 mm) (7.43 mm) = 74.3 mm 2 (3) Now, for a given inductance of 10 mh, the required number of turns can be calculated as follows applying Eq. 1: L = 10mH = µ oµ r N 2 a h = (4π 10 7 H m )(3000)N2 (74.3 mm 2 ) (21.5628 mm) (4) Lastly, solving for N, we calculate that N = 27.746 turns. For purposes of constructing our inductor, 28 turns were used. In electromagnetics, it is generally defined that magnetic field intensity (H) and magnetic flux intensity (B) are linearly related by permeability. However, this relationship only holds true until magnetic flux intensity (B) reaches approximately

half of its saturation level (B m). From the core s catalog/datasheet, it was noted that B m for our core is approximately 490 mt, so it was concluded that B and H are linear until B reaches 245 mt. We then looked to calculate the necessary current needed to obtain a magnetic flux intensity of 245 mt inside the inductor. The equation for the magnetic field inside a toroid was employed as follows: B = 245mT = µ oµ r NI 2πr = (4π 10 7 H m ) (3000)(28) I 2π (2.9316 mm) (5) Solving for I, we find that to achieve B = 245 mt, a current of I = 42.7525 ma is needed. Before energizing the inductor, the epoxy coating on the ends of the wires was scraped off to allow a conducting path to and from the inductor. Using the function generator, a 400 Hz AC signal was applied to the inductor. The multimeter was used to monitor inductor current until current reaches the 42.7525 ma calculated above. This current was achieved with the 400 Hz signal with a peak-to-peak voltage (V pp) of 6.37 V. To create a 10 mh inductor, a ferrite core wound with wire was used. Our inductor had the following parameters: Table 1: 10 mh Inductor Physical Parameters Parameter Variable Value Number of Wire Turns N 28 turns Toroid Circumference C 21.5628 mm Toroid Cross-Sectional Area a 74.3 mm 2 Relative Permeability µ r 3000 Procedure B: Experimental Verification of Inductor In Procedure B, we were looking to experimentally confirm the value of the inductance of the inductor constructed I Procedure A. To do so, an RL series circuit was constructed using our inductor and a 1 kω resistor (measured to be 993.6 Ω), as shown in Fig. 1. Figure 1: RL Circuit used to Measure Inductance of 10 mh Inductor Eq. 6 below defines the frequency (f) of the AC source signal in Fig. 1 at which the voltages across the resistor and inductor are equal. This equation can be easily derived from phasor analysis of the circuit in Fig. 1. L = 3 R 2π f (6) Using the multimeter, function generator, and some trial and error, it was found that the voltages across the inductor and resistor are equal when the AC signal was set to a frequency of 51 Hz. Applying Eq. 6 with this frequency and accounting for the 50 Ω internal resistance of the function generator, inductance was calculated as follows:

L = 3 R 3 (993.6 Ω + 50 Ω) = = 5.641 mh (7) 2π f 2π (51000) While this inductor value is slightly lower than expected, as shown in Table 2, there are a variety of factors that may have played a role in this discrepancy. In particular, gaps between the windings of our inductor allows for leakage flux, resulting in a decrease in inductance. Additionally, the internal resistance of the inductor itself is not accounted for in Eq. 7. Including this internal resistance would slightly increase the value of R. Ignoring physical defects such as leakage flux and assuming an inductance of exactly 10 mh, the internal resistance of our inductor could be approximated as follows: L = 10 mh = 3 R 2π f = 3 (993.6 Ω + 50 Ω + R int) 2π (51000) (8) Solving for internal resistance (R int) gives an internal resistance value of R int = 806 Ω. Table 2: Theoretical and Experimental Inductance Values Theoretical Inductance 10.0 mh Experimental Inductance 5.641 mh Procedure C: Excitation Current in the Inductor In Procedure C, we looked to observe the current in the inductor when an AC signal was applied. While the oscilloscope cannot be used to measure current, the relationship between voltage can be used to gain insight into the inductor current: v(t) = L d i(t) dt (9) For a sinusoidal voltage signal, it can be concluded that inductor current will also be a sinusoid. The current sinusoid will be shifted 90 with respect to the voltage sinusoid. Its amplitude will also be increased by a factor of 1/L. Figure 2: Inductor Voltage with 400 Hz AC Signal

Procedure D: Construction of a Transformer To create the transformer, another ten turns of winding were added, separate to the original 28 turns of wire. The transformer voltage gain can be predicted using Eq. 10, yielding 0.357 V across the ten turn secondary winding when 1 V is applied across the 28 turn primary winding. V 2 = N 2, V V 1 N pp2 = N 2 V 1 N p1 = 10 (1.0 V) = 0.357 V (10) 1 28 The actual voltage across the secondary coil was then measured. Next, conditions under which B > B m/2 were observed. It was predicted that the linear relationship between V and I on the secondary coil would diminish because more magnetic flux would be lost, so output current would begin to remain constant under increasing voltage. To test this prediction, input voltage amplitude and frequency were varied independently, and the resulting voltage gain was observed. First, input voltages with approximate amplitudes of 0.5 V, 2.0 V, and 3.5 V, all at 400 Hz, were applied to the primary. Primary and secondary voltages for each of these three cases are shown in Fig. 3, Fig. 4, and Fig. 5. Across these three cases, a constant voltage gain of approximately 0.39 V/V was observed, shown in Table 3. Next, input voltages with frequencies of 60 Hz, 200 Hz, and 800 Hz were applied to the primary. This time, however, voltage gain appeared to increase with frequency, as shown in Table 4. Table 3: Analysis of Primary and Secondary with 400 Hz Input Signal of Varying Amplitudes Vi (V) Vo (V) Vo/Vi (V/V) 0.492 0.192 0.3902 1.96 0.78 0.3980 3.38 1.3 0.3846 Table 4: Analysis of Primary and Secondary with Varying Frequencies and Steady V Across Primary f (Hz) Vi (V) Vo (V) Vo/Vi (V/V) 60 0.548 0.212 0.3869 200 0.552 0.216 0.3913 800 0.556 0.228 0.4101 Figure 3: Primary and Secondary Voltages for Vi = 0.492 V at 400 Hz Figure 4: Primary and Secondary Voltages for Vi = 1.96 V at 400 Hz

Figure 5: Primary and Secondary Voltages for Vi = 3.38 V at 400 Hz Procedure E: Loading the Transformer To apply a load to the secondary of the transformer, a 10 Ω resistor was placed across the secondary terminals. A 0.5 V pp AC voltage at 400 Hz was applied to the primary. The multimeter was used to measure currents and voltages at both the primary and secondary, shown in Table 5. The oscilloscope was then used to observe V 1 and V 2 simultaneously. I 2 was then calculated using Ohm s Law given a 10 Ω load and the measured value of V 2. This calculated value was then compared to the measured value of I 2 obtained earlier. Using the measured values of V and I at the primary and secondary, it was calculated that the transformer produced a power loss of 91.96%. The loaded transformer circuit was then simulated in PSPICE using an AC voltage source, a linear transformer (XFRM_LINEAR) element, a transformer internal resistance, and a 10 Ω resistor, as shown in Fig. 6. Using PSPICE, primary and secondary voltage (bottom), current (top), and power (middle) were plotted with time, as shown in Fig. 7. In the top plot, it was observed that current passing through the primary was inducing a current through the secondary. Interestingly, while a sinusoidal voltage was seen at the primary (V 1), no voltage was seen at the secondary (R 1). This can be explained, however, by looking at the power plot. Oscillating R 2 power at the primary can be seen in red. The primary appears to have a peak power of approximately 240 µw. Power at point V 2 (shown in purple), however, indicates that significantly less power is seen at the secondary, only approximately 8 µw. While troubling, the results of this simulation align with the significant power loss seen with our transformer. The transformer was then measured a second time with a 5 V pp 5.0 khz signal. Based on the positive correlation between voltage gain and frequency observed in Part D, it was predicted that the increase in frequency would decrease the power loss of our transformer. As was done before, the multimeter was used to measure currents and voltages at both the primary and secondary, shown in Table 6. I 2 was again calculated using Ohm s Law and the calculated value was then compared to the measured value of I 2. Using the measured values of V and I at the primary and secondary, it was calculated that transformer power loss had decreased significantly to just 34.18%. Table 5: Primary and Secondary Coil Voltage, Current, and Power with Vpp = 0.5 V at 400 Hz Primary Secondary Vpp (mv) 110 52 I (measured) (ma) 7.08 1.20 I (calculated) (ma) 1.84 Power (uw) 275.15 22.14 Vrms (mv) 38.89 18.38

Figure 6: The Schematic Used for the PSPICE Simulation Figure 7: PSPICE Simulation Comparing the Primary and Secondary Voltage, Current, and Power Table 6: Primary and Secondary Coil Voltage, Current, and Power, with Vpp = 5.0 V at 5000 Hz Primary Secondary Vpp (mv) 2620 1020 I (measured) (ma) 19.08 32.25 I (calculated) (ma) 36.06 Power (mw) 17.67 11.63 Vrms (mv) 926.30 360.60 Conclusion Our parameter estimates and inductor construction a measured 5.641 mh inductor. While this is a bit less than the intended 10 mh inductor, there were many factors that may have played a role in the discrepancy. The calculations were done assuming ideal configurations, which were not the circumstances in many ways. Given that the coil was not tightly wound, flux leakage occurred through the air, and the coils were also perfectly circular and were flawed loops. Overall, there were many factors contributing to losses endured by the inductor, and so that low inductance is not a particularly surprising result. The measured waveform from Fig. 2 forms a sinusoidal output as expected. The input is a sinusoidal wave and the output current is going to be a wave that is shifted 90 degrees from the voltage across the inductor. The transformer constructed in Procedure D was created with a ten loop secondary, and according to Eq. 10, a 0.357 V/V gain was to be expected across the secondary coil. Performing measurements at 400 Hz, it can be seen in Table 3 that the gain is near 0.390 V/V. The gain is somewhat different, but may be because the current is not increasing as much as it should due to internal resistance. Fig. 3-5 illustrate the oscilloscope measurements across the primary and secondary when higher voltages are applied. It was clear that the higher input voltages were outside of the cutoff range of the transformer, resulting in distortion.

Table 3 exhibits a similar phenomenon, where increasing the frequency lead to a higher gain. This is most likely because increasing the frequency causes an increase in the resistance of the transformer s current to change from the magnetic flux, and so the current is less able to increase under higher frequencies, meaning the voltage does not scale proportionally as much as it should. The transformer current did not scale as expected when checking the effectiveness of the transformer. Table 5 shows a power loss of about 92% across the transformer. Performing an analysis in PSPICE shows something similar, as Fig. 7 also indicated a considerable power loss. Our hypothesis was that V pp = 0.5 V is too low, and so the internal resistance and flux leakage is so great that the majority of the power is lost. To test this hypothesis, we performed another set of measurements with greater V pp and higher frequency. It appeared that a higher voltage was less affected by the internal resistance and flux leakage. Table 6 shows that the resulting power loss was considerably less than in our first experiment, only about 34%. Even this improved power loss, however, is considerably inefficient. For practical applications, a transformer of much higher performance would be necessary. References ECE 1201 Website: http://engrclasses.pitt.edu/electrical/faculty-staff/gli/1201/ Magnetics Materials Catalog: http://www.engrclasses.pitt.edu/electrical/faculty-staff/gli/1201/ferrite_b_h.pdf Magnetics Ferrite Cores Catalog: http://www.engrclasses.pitt.edu/electrical/faculty-staff/gli/1201/magnetics.pdf