Lesson+17

Lesson+17 - In your own words explain Kirchoffs two laws 1...

• Notes
• 38

This preview shows pages 1–12. Sign up to view the full content.

1 Lesson 17 In your own words explain Kirchoff’s two laws.

This preview has intentionally blurred sections. Sign up to view the full version.

Challenge 16 Steady State Sinusoidal Response Chapter 13.7 Challenge 17 Lesson 17 2 Lesson 17
Challenge 16 Using time-domain methods, convolve x(t) = e -t u(t) and h(t) = e -2t u(t). That is, determine y(t) = x(t) h(t). 0 ; ) ( ) ( ) ( ); ( ) ( 2 0 2 2 0 2 t e e d e e d e e t y t u e t x t u e t h t t t t t t t t 3 Lesson 17

This preview has intentionally blurred sections. Sign up to view the full version.

Using the convolution theorem, convolve x(t) = e -t u(t) and h(t) = e -2t u(t). That is, determine y(t) = x(t) h(t). 0 ; ) ( ) ) 2 /( 1 ) 1 /( 1 ( )) ( ( ) ( 2 1 1 t e e t y s s L s Y L t y t t (same as before) 4 Lesson 17 )) 1 /( 1 ) 2 /( 1 ( )) 2 /( 1 ))( 1 /( 1 ( ) ( ) ( ) ( s s s s s H s X s Y
0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 ; ) ( ); ( ) ( ); ( ) ( 2 2 t e e t y t u e t x t u e t h t t t t » t=0:0.01:10; x=exp(-t);h=exp(-2*t); » y=conv(x,h); yy=y(1:1001); » plot(t,x,t,h,t,yy/100) x(t) h(t ) y(t ) Machine production using discrete-time simulation. 5 Lesson 17

This preview has intentionally blurred sections. Sign up to view the full version.

Revisit the importance of convolution We make a bold claim that the physical world behaves as a natural lowpass filter. Can this explain why normal noise (a.k.a., Gaussian) called normal? 1 0 -1 PDF 1 0 -1 h(t) (channel) Random signal x(t) (binary valued heads/tail) y 1 (t)= x(t)*h(t) h(t) x(t) y 1 (t) PDF 6 Lesson 17
h(t) (channel) y 1 (t) (previous output) y 2 (t)= y 1 (t)*h(t) 1 0 -1 PDF 1 0 -1 h(t)] y 1 (t) y 2 (t) PDF 7 Lesson 17

This preview has intentionally blurred sections. Sign up to view the full version.

Steady State Frequency Response Author’s tale. Assume that the input to an LTI having h(t) H(s) is sinusoidal (consult trig, table): x(t) = A cos( t+ ) = A cos( ) cos( t) - A sin( ) sin( t) It therefore follows that (consult Laplace table): 2 2 2 2 2 2 ) sin( ) cos( ) sin( ) cos( ) ( s s A s A s s A s X 8 Lesson 17
Author’s tale. According to past studies, the solution should consist of a steady-state component define by the input and Eigenvalues, plus a natural response defined by the Eigenvalue of H(s). If all the Eigenvalues (poles) of H(s) have negative real parts, the natural solution will eventually converge to zero leaving only the only the steady- state solution. At steady-state solution would have the (Heaviside) form: 2 2 ) sin( ) cos( ) ( ) ( ) ( ) ( s s A s H s X s H s Y j s K j s K s Y 2 1 ) ( ; t for 9 Lesson 17

This preview has intentionally blurred sections. Sign up to view the full version.

Author’s tale . 1 2 2 1 K K j s K j s K s Y ; ) ( ω ω j e j H j H Since K 2 =K 1 *, the filter function is: j j s j s e A j H j s s A j H s Y j s K 2 1 sin cos ) ( ) ( 1 10 Lesson 17
Author’s tale. Examine: where: * 1 2 1 2 K K e j H A K j Interpreting

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern