ECEN340_2024_HW3_Solution

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Apr 3, 2024

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ECEN 340 Electric Energy Conversion – 2024 Spring Homework 3 Assigned on Feb. 9, 2023 || Due on Feb. 16, 2023 at 11:59 PM (In the upper left-hand corner of the first page of your homework solution, please put the number of hours that you spent on this homework assignment. ) 1. Consider the converter below. You may assume that the inductor is large enough that the current through it can be considered constant. 𝑉 𝑖? 𝑉 ? + - 𝐿 ? 𝑅 𝑖 𝐿 𝑖 ? ? q(t) 𝑖 ? 𝑖 ? (a) (10 points) Draw the circuits when the active switch is on and off.

ECEN 340 Electric Energy Conversion – 2024 Spring (b) (5 points) Based on your answer to a, derive an equation in terms of D that relates the input voltage Vin to the output voltage Vout. (c) (5 points) What does this circuit do? 2. We have a PV module running at its maximum power point with 𝑉 ? = 40 𝑉 and 𝐼 ? = 5 𝐴 . We wish to deliver power to a 12-volt battery to charge it. We want to use a DC-DC converter to accomplish this while extracting maximum power from the source. (a) (10 points) Propose a converter topology and specify the reason for that. Since the output voltage is smaller than the input voltage, two topologies are possible: a buck converter and a buck-boost converter

ECEN 340 Electric Energy Conversion – 2024 Spring (b) (10 points) Solve for the duty cycle ( ? ) For a buck converter, 𝑉 ? = ?𝑉 𝑖 ? = 𝑉 ? 𝑉 𝑖 = 12 40 = 0.30 For a buck-boost converter, 𝑉 ? = 𝑉 𝑖 ? 1 − ? ? = 𝑉 ? 𝑉 𝑖 + 𝑉 ? = 0.23 3. (30 points) A solar panel circuit is shown in Figure 3, where the solar panel can be viewed as a DC voltage source. The switch is controlled by a square wave signal 𝑞(𝑡) with the period T and duty cycle D . Assume that the switch and the diode are ideal, and the capacitor is very large. Figure 1 Solar panel circuit for Question 3

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ECEN 340 Electric Energy Conversion – 2024 Spring (a) (10 points) Draw the equivalent circuits when 𝑞(𝑡) = 1 (the switch is on) and when 𝑞(𝑡) = 0 (the switch is off). (b) (10 points) Find the relation between the output voltage 𝑉 ? and the input voltage 𝑉 𝑖? , in terms of the duty cycle D . (Hint: The average voltage of the inductor should be zero)

ECEN 340 Electric Energy Conversion – 2024 Spring (c) (10 points) Suppose that the solar panel voltage 𝑉 𝑖? = 3𝑉 . The output current through the load is 𝐼 ? and follows the relation 𝐼 ? = −𝑉 ? − 15 𝑉 ? ⁄ + 10 A. To achieve maximum power point tracking (MPPT), namely the output power reaches the maximum value, what should the duty cycle D be? 4. A full-wave rectifier is connected to an ideal transformer. The load resistor 𝑅 = 1000 𝛺 . Voltage drop across each diode 𝑣 𝐷 = 0.7 𝑉 . (a) (10 points) Find the peak output voltage ( 𝑣 ?,??𝑎𝑘 ), (b) (10 points) Derive the relationship between the peak output voltage (vo,peak) and the average output voltage (vo,avg). (c) (10 points) Find the average output voltage ( 𝑣 ?,𝑎𝑣𝑔 ) and the average output current ( 𝑖 ?,𝑎𝑣𝑔 ).

ECEN 340 Electric Energy Conversion – 2024 Spring 240 V rms 60 Hz 10:1 + - 𝑣 ? 𝑅 𝑖 𝑅 A) 𝑉 ?1,??𝑎𝑘 = √2 𝑉 ?1,??? = √2 × 240 = 339.4 𝑉 𝑉 ?2,??𝑎𝑘 = 𝑉 ?1,??𝑎𝑘 𝑎 = 33.9 𝑉 𝑉 ?,??𝑎𝑘 = 𝑉 ?2,??𝑎𝑘 − 2 × 𝑉 ? = 32.5 𝑉 b) Derivation of Vavg:- C) 𝑣 ?,𝑎𝑣𝑔,???𝑎? = 2 𝜋 𝑉 ?,??𝑎𝑘 = 20.7 𝑉 𝑖 ?,𝑎𝑣𝑔,???𝑎? = ⟨𝑣 ?,???𝑎? ⟩ 𝑅 = 20.7 ?𝐴

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