EE223 Lab6

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Bluegrass Community and Technical College *

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223

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Electrical Engineering

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Jan 9, 2024

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Laboratory Report Course: EE223 Experiment No: 6 Title: Maximum Power Transfer Author: Elnoel Akwa Instructor: Dr. Caicheng Lu

( ) ( ) 𝑠 𝐿 Executive Summary: Using resistor substitution box available in the lab for R L and bench power supply for V S with varying R L between 0 to 10kΩ voltage and current were measured using DMM across the load and VOM in series. Collected data was used to plot the measured power. Part 2 of the experiment a complex impedance was used in a circuit to measure the input current, voltage, and relative phase between these so that complex load impedance Z be measured. Necessary component was determined to match the Z for maximum power transfer to the load. Using these components, a matching circuit in series with complex Z was constructed so that V L and I L can be measured over the range of frequencies from 200 Hz to 20kHz. 1. Objectives The primary objectives of this experiments are to: Measure the power delivered to a variable resistive load and determine the condition for maximum power transfer to the load. Design a matching circuit for the maximum power transfer to a complex load and verify the design by measuring the time average power on the load with respect to varying frequency. 2. Theory The theoretical tools relevant to this experiment are presented in this section. The notation has been adapted from [Alexander, C. K.2020] i The power delivered to the load is P = V L I L ? 𝐿 𝑉 ? 𝑉 ? = ? 1 + ? 𝐿 𝑉 2 ? = ( ? 1 + ? 𝐿 ) 2 ? 1 + ? 𝐿

𝑉 Figure 1: Series resistive circuit with variable resistance load It can be shown that for a fixed R 1 , the maximum power delivered to the load R L occurs when R L = R 1 In this case we are going to ignore the internal source resistance of the power supply. From the equation (i) if we assume x = R L /R 1 then Df(x)/dx = (1-x)/(x+1) 3 = 0 x = 1 = R L /R 1 2 P = 𝑠 × ? 1 R L = R 1 𝑥 ( 𝑥 +1) 2 For the impedance network contains both resistive and reactive (capacitive and inductive) elements. In the block diagram in Figure 2 Figure 2: Impedance network It can be shown that maximum time average power is delivered to the load Z L is when Z 1 = Z L * The time average power on complex load is

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R L and X L vary independently. To achieve maximum power X L + X 1 = 0 X L = - X 1 For maximum power R L = R 1 For the second part of the lab, we configure the circuit in Figure 3 Figure 3: Complex impedance using the encapsulated circuit To measure Z L using ohm’s law To measure I L place a R in [?] box. 𝑣 𝐿 𝑧 𝐿 = 𝐿 | 𝑣 𝐿 | < ∅𝐿 = | 𝐼 𝐿 | < ∅𝐼 To match Z L replace [?] by Z 1 we can get the value of power in the load P L = 1 ?𝑒 ( 𝑉 𝐼 ∗ ) 2 𝐿 𝐿 𝐼

𝑐 As the frequency f, the source is varied we find the specific frequency 2500 Hz value where time average power delivered to the load is maximum. At this specific frequency is called resonance frequency Z L = Z 1 * or when R 1 = R 2 as specified in this lab 1 𝑗𝜔 𝑐 = − 𝑗 𝜔 𝑐 𝐿 3. Results The following instruments were used in this experiment: Tektronix Digital Phosphor Oscilloscope (DPO), Function Generator, substitute resistance box, encapsulated circuit, capacitor, various resistors. 3.1. Part A of the experiment from 1 to 4 In this part of the lab, we used resistor substitution box available in the lab for R L and use the bench power supply for V S = 19.96. We place the VOM in the series to measure the current and DMM across the load to measure voltage. We measure and document both voltage and current at many points. Using the collected data, we plot the measured power dissipated in the load resistor P L vs R L . R L(KOhms) V L V I L A P L W 0 .7 3.1 2.17 2 4.54 3.5 11.35 4 7.4 2 14.8 5 8.47 1.9 16.393 6 9.37 1.85 17.317 6.5 9.77 1.8 17.586 6.8 10 1.77 17.7 7 10.4 1.74 17.637 7.5 10.6 1.7 17.833 8 10.8 1.68 17.836 10 11.9 1.5 17.865 Table 1: Voltage, current and power in varying resistor

Figure 4: P L vs R L plot From the Figure 4 we see that the peak is close to 6000Ω where we calculated value we have shown that the maximum power delivered to the load is 6.8kΩ which is close. There is difference between collected data and theoretical value, but it is not high. 3.2. Part B of the experiment from 1 to 6 For the second part of the lab, we have used R = 220 Ω and our frequency is 2500 Hz. V R = 1.092 V V L = 931 mv = .931 V R = 100 Δ Φ = 28.13 0 Now we can calculate I L = V R / R = 1.092/100 = .01092 A Z L = (R 1 V L )/V R * (e^(j ΔΦ)) = ((100 * .931)/1.092) (e^j28.13 0 )

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= 114.14 * (e^j28.13 0 ) = Re(114.14 * cos(28.13)) + Im (j114.14 * sin(28.13)) = Re (101.115) +Im (j52.95) So Z L * = 101.115 – j52.951 Using the relationship between capacitor and inductor in theory section we can calculate the value of capacitor from Z L * . -j52.951 = (-j)/ ( 𝜔 C) C = 1.2e-6 Now we use measure and collect data of V L and I over the range frequencies from 200 Hz to 20kHz which is shown in Table 2 For calculating power, we used the following formula P av = (|V L | * |V R |)/(2R) * cos (V L – V R ) Q = (|V L | * |V R |)/(2R) * sin (V L – V R ) Here R = sqrt (101.115 2 +52.92 2 ) = 114.26 Ω f Hz VL V Φ VR Pav Q 2.317 .805mv 25.56 670m 0.0098 0.0129 2.5 .941mv 23.35 612.1m 0.0108 0.01595 10.0 1.14 26.55 493.1m 0.6616 0.0142 13.0 1.24 17.56 0.0095 0.023 15.0 1.30 16.46 435m - 0.0037 - 0.1689 18.0 1.32 15.46 399.9 .0057 - .1798 19.98 1.50 13.46 135.4 .003456 .0897 200 .214 77.75 170.9 .5363 - .0657 1k .649 40.71 636m .0646 .03293

Frequency vs Power 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 frequency 8 10 12 1.5 .702 36.50 366.9 .0465 .01977 Table 2: Average power over as varying frequency And a graph of frequency vs power is shown in figure 5. / ” 0.216 0.10 89 Figure 5: Frequency vs Power From the Table 2 and figure we see that at 5000 Hz load absorbs highest power. As we know from the relationship reactance and resonance, as frequency increases the reactance decreases and the total impedance of the load decreases, that is why power decreases. 4. Conclusion There may some error while collecting data. For example, in Table 2 frequency at 20kHz the phase difference is negative. It may human error or machine glitch, needs further investigation. There was also calculation error in part 2 of the lab for measuring reactance, but it was fixed in the report. power