ece3101_Homework2_SOLNS_F19.docx - B Olson 1 a Determine an expression for the RMS value of i A(t as a function of the Fourier Coefficients shown below

ece3101_Homework2_SOLNS_F19.docx - B Olson 1 a Determine an...

This preview shows page 1 - 5 out of 16 pages.

B. Olson 1) a) Determine an expression for the RMS value of i A (t) as a function of the Fourier Coefficients shown below. b) Determine an expression for the average power of i A (t) and v(t) using the Fourier Coefficients shown below. c) Determine an expression for the average power of i B (t) and v(t) using the Fourier Coefficients shown below. a) i A ( RMS ) 2 = 1 T to to + T i A ( t ) i A ( t ) dt = 1 T to to + T ( a ( A ) v + k = 1 a ( A ) k cos ( 0 t ) + b ( A ) k sin ( 0 t ) )( a ( A ) v + n = 1 a ( A ) n cos ( 0 t ) + b ( A ) n sin ( 0 t ) ) dt = a ( A ) v 2 + 1 2 k = 1 a ( A ) k 2 + b ( A ) k 2 i A ( RMS ) = a ( A ) v 2 + 1 2 k = 1 a ( A ) k 2 + b ( A ) k 2 -1 0 1 2 3 t -1 0 1 2 3 t -1 0 1 2 3 t F. Coefficients: aAv, aAk, bAk F. Coefficients: aBv, aBk, bBk F. Coefficients: aCv, aCk, bCk iA (t) v (t) iB (t)
Image of page 1
b) P A ( ave ) = 1 T to to + T i A ( t ) v ( t ) dt = 1 T to to + T ( a ( A ) v + k = 1 a ( A ) k cos ( 0 t ) + b ( A ) k sin ( 0 t ) )( a ( C ) v + n = 1 a ( C ) n cos ( 0 t ) + b ( C ) n sin ( 0 t ) ) dt = a Av a Cv + 1 2 k = 1 a ( A ) k a ( C ) k + b ( A ) k b ( C ) k c) P B ( ave ) = 1 T to to + T i B ( t ) v ( t ) dt Signal i B ( t ) : ^ Τ = 2 ; ^ ω 0 = 2 π ^ Τ = π Signal v ( t ) : ~ Τ = 1 ; ~ ω 0 = 2 π ~ Τ = 2 π = 2 ^ ω 0 Signal i B ( t )⋅ v ( t ) : ^ Τ = 2 ; ^ ω 0 = 2 π ^ Τ = π working out a few terms : i B ( t )= a ( B ) v + k = 1 a ( B ) k cos ( k ^ ω 0 t ) + b ( B ) k sin ( k ^ ω 0 t ) = a ( B ) v + a ( B ) 1 cos ( ^ ω 0 t ) + a ( B ) 2 cos ( 2 ^ ω 0 t ) + a ( B ) 3 cos ( 3 ^ ω 0 t ) + a ( B ) 4 cos ( 4 ^ ω 0 t ) + ... k = 1 b ( B ) k sin ( k ^ ω 0 t ) v ( t )= a ( C ) v + k = 1 a ( C ) k cos ( k ~ ω 0 t ) + b ( C ) k sin ( k ~ ω 0 t ) = a ( C ) v + k = 1 a ( C ) k cos ( k 2 ^ ω 0 t ) + b ( C ) k sin ( k 2 ^ ω 0 t ) ( sin ce : ~ ω 0 = 2 ^ ω 0 ) = a ( C ) v + a ( C ) 1 cos ( ~ ω 0 t ) + a ( C ) 2 cos ( 2 ~ ω 0 t ) + a ( C ) 3 cos ( 3 ~ ω 0 t ) + ... k = 1 b ( C ) k sin ( k ~ ω 0 t ) = a ( C ) v + a ( C ) 1 cos ( 2 ^ ω 0 t ) + a ( C ) 2 cos ( 4 ^ ω 0 t ) + a ( C ) 3 cos ( 6 ^ ω 0 t ) + ... k = 1 b ( C ) k sin ( k ~ ω 0 t ) These basis functions do not have a match with the basis functions used to describe v(t)
Image of page 2
Because the basis functions are orthogonal, when i B (t) and v(t) are multiplied and integrated only the cross terms that contain the same basis functions will remain. Specifically: 1 T a ( B ) v a ( C ) v to to + T 1 dt = a ( B ) v a ( C ) v 1 T a ( B ) 2 k a ( C ) k to to + T cos ( k 2 ^ ω 0 t ) cos ( k 2 ^ ω 0 t ) dt = 1 T a ( B ) 2 k a ( C ) k ( T 2 ) = 1 2 a ( B ) 2 k a ( C ) k 1 T b ( B ) 2 k b ( C ) k to to + T sin ( k 2 ^ ω 0 t ) sin ( k 2 ^ ω 0 t ) dt = 1 T b ( B ) 2 k b ( C ) k ( T 2 ) = 1 2 b ( B ) 2 k b ( C ) k All others are zero This leaves P B ( ave ) = 1 T to to + T i B ( t ) v ( t ) dt = = a ( B ) v a ( C ) v + 1 2 ( a ( B ) 2 a ( C ) 1 + a ( B ) 4 a ( C ) 2 + a ( B ) 6 a ( C ) 3 + ..... ) + 1 2 ( b ( B ) 2 b ( C ) 1 + b ( B ) 4 b ( C ) 2 + b ( B ) 6 b ( C ) 3 + ..... ) 2) The input shown below is applied to the circuit. Determine the first three non-zero terms of the Fourier Series of vo(t). Assume that T = p msec. If you would like you can use the relationship below to determine the Fourier Coefficients. Determining Fourier Coefficients Method 1: using equation provided 60 -60 T/2 T t 2 5 m H 1 0 0 + Vin(t) - + Vo(t) - -T/2 A -A T/2 T t -T/2 ) sin( 1 4 ) ( 0 .. 5 , 3 , 1 t n n A t f k p
Image of page 3
vin ( t )= f ( t + T 4 ) = 240 π π n = 1,3,5.. 1 n sin ( 0 ( t + T 4 ) ) = 240 n = 1,3,5 ..
Image of page 4
Image of page 5

You've reached the end of your free preview.

Want to read all 16 pages?

  • Fall '18
  • James Kang

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes