EE 230 Lecture 6 Spring 2010

EE 230 Lecture 6 Spring 2010 - EE 230 Lecture 6 Linear...

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EE 230 Lecture 6 Linear Systems Poles/Zeros/Stability Stability
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Quiz 5 A system has the transfer function T(s) Determine the poles of the system. () 1 4 s Ts 10 7s s + = ++
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And the number is ? 1 3 8 4 6 7 5 2 9
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And the number is ? 1 3 8 4 6 7 5 2 9 ?
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Quiz 5 A system has the transfer function T(s) Determine the poles of the system. () 1 4 s Ts 10 7s s + = ++ Solution: 71 0 2 1 1 4s 4 4 s 10 ss s ⎛⎞ + + ⎜⎟ ⎝⎠ == + + 10 5 2 11 44 s7 s s 2 s + + Poles at s = -2 and s = -5
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Step Response of First-Order Networks IN OUT Many times interested in the step response of a linear system when the system is first-order X OUT (t)=? 0 C t-T t OUT X = F + (I-F)e I is the intital value, F is the final value and t C is the time constant For any first-order linear system, the unit step response is given by Review from Last Time
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Step Response of First-Order Networks X OUT t I F T 0 T 0 + t C 0 C t-T t OUT X = F + (I-F)e Review from Last Time
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Step Response of First-Order Networks 0 C t-T t OUT V = F + (I-F)e -1 C t= - p=RC Example: () K Ts = s-p Obtain the step response of the circuit shown if the step is applied at time T=1msec and prior to V OUT (t)=0 for t<1msec. Assume R=1K, C=0.1uF Solution: 1 1+RCs 1 RC 1 s+ RC 1 p = - RC F=1V I=1V t-.001 RC OUT V = 1 + (-1)e t-.001 RC OUT V= 1 - e This is first order and of the form: Thus, the output can be expressed as: Review from Last Time
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Impedance and Conductance Notation Circuit Analysis with Impedance Notation (Z) and Conductance Notation (G) Ohms Law V=I Z I=V G KCL ( ) X12 3 K 1 12 23 3 k k V G +G +G +. ..+G = V G +V G +V G +. ..+V G 1 kk Xi i i i=1 VG = V G i = ⎛⎞ ⎜⎟ ⎝⎠ ∑∑ KCL is often the fastest way to analyze electronic circuits G 2 3 k Node with conductance notation Conductance notation is often much less cumbersome than impedance notation when analyzing electronic circuits Why? Why? Formally: Review from Last Time
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Poles and Zeros of Linear Networks () m i i i=0 n i i i=0 as Ts bs = For any linear system, T(s) can be expressed as where a i and b i are all real, , , and n m Numerator often termed N(s) Denominator often termed D(s) Definition: The roots of D(s) are the poles of T(s) and the roots of N(s) are the zeros of T(s) The poles of T(s) are often termed the poles of the system m i i i=0 n i i i=0 Ns Ds == Can always make b n =1 n b 0 m a 0 Linear System X IN X OUT Review from Last Time
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Step Response of First-Order Networks ( ) I 0 pt -T OUT X=
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This note was uploaded on 02/01/2012 for the course EE 230 taught by Professor Mina during the Fall '08 term at Iowa State.

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EE 230 Lecture 6 Spring 2010 - EE 230 Lecture 6 Linear...

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