Problem 1. (25 pts)  For this problem, consider a right-hand…

Problem 1. (25 pts)  For this problem, consider a right-hand coordinate system wherein the x-axis is directed to the right on your paper, the y-axis is directed toward the top of your paper, and the z-axis is pointing out of your paper. A winding with 4 turns is wound around the rectangle with corners (x,y,z) = (0,2,1), (0,2,3), (2,2,3), and (2,2,1).  The winding is wound such that the direction for positive flux is in the same direction as the positive y-axis. Now suppose that within the rectangle B=(1+2yxz)aysin(100t) T{“version”:”1.1″,”math”:”B=(1+2yxz)aysin(100t) T”} where ay{“version”:”1.1″,”math”:”ay”} is the unit vector is the y direction, and t represents time. The winding (coil) is open circuited.  Using the definition of flux, flux linkage, and Faraday’s law determine an expression for the voltage (as a function of time) that would be measured across the coil, assuming the usual passive sign convention we used in class. The answer should be of the form v=57sin(100t+π/4){“version”:”1.1″,”math”:”v=57sin(100t+π/4)”} (which is incorrect, but you get the idea).  Problem 2. (25 pts)  Consider the EI core device below. Derive an expression for the absolute value of electromagnetic force on the I-core in terms of N, i, ds, ws, w=wb=we=wi=wc/2, g, μr{“version”:”1.1″,”math”:”μr”}, and the permeability of free space μ0{“version”:”1.1″,”math”:”μ0″}. Problem 3. (25 pts)  Consider the transformer T-equivalent circuit (referred to the primary). The primary to secondary turns ratio is 3. Given the primary voltage and that the radian frequency is ωe{“version”:”1.1″,”math”:”ωe”}, we want to find the actual secondary voltage. (a) 6 pts. If the physical resistance of 4 Ohms is placed on the secondary, what value of resistance do I use when performing calculations on the equivalent circuit? (b) 6 pts. Write an expression for Zb as we used it in class. Use symbols when you don’t have numerical data. Something like Zb=3r1+jωeLl1+jωeLm4jωeLm+4{“version”:”1.1″,”math”:”Zb=3r1+jωeLl1+jωeLm4jωeLm+4″} (which is incorrect). (c) 7 pts. Suppose the phasor representing the magnetizing voltage is denoted v~m{“version”:”1.1″,”math”:”v~m”} and the radian frequency is ωe{“version”:”1.1″,”math”:”ωe”}.  Derive an expression for the phasor representing the referred secondary voltage in terms of v~m{“version”:”1.1″,”math”:”v~m”}, ωe{“version”:”1.1″,”math”:”ωe”}, and the equivalent circuit parameters (using symbols where you don’t have numerical values as in part (b)). (d) 6 pts. Suppose v~2’=30∠20°{“version”:”1.1″,”math”:”v~2’=30∠20°”}. Write a corresponding expression for the secondary voltage in the time domain as we would measure it with an oscilloscope.  Here your answer should look like v2=257cos(ωet+5π180){“version”:”1.1″,”math”:”v2=257cos(ωet+5π180)”}, which is wrong but again you get the idea. Problem 4. (25 pts)  Consider the following flux linkage equations. Determine the following:  (a) {5 pts} if the system is magnetically linear (b) {15 pts} the field-energy (c) {5 pts} electromagnetic force in terms of x{“version”:”1.1″,”math”:”x”}, λ1{“version”:”1.1″,”math”:”λ1″}, and λ2{“version”:”1.1″,”math”:”λ2″}. Also note that the system is conservative, so you won’t need to verify this.  i1=λ1+10(λ1+2λ2)3(5+x)i2=2λ2+20(λ1+2λ2)3(5+x){“version”:”1.1″,”math”:” i1=λ1+10(λ1+2λ2)3(5+x)i2=2λ2+20(λ1+2λ2)3(5+x)”} Congratulations, you are almost done with Exam 1.  DO NOT end the Honorlock session until you have submitted your work to Brightspace.  When you have answered all questions:  Use your smartphone or scanner to scan your answer sheet, and save the scan as a PDF. Make sure your scan is clear and legible.  Submit your PDF  as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.).  Submit your exam to this assignment: Exam 1 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam.  End the Honorlock session.   
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