ece_350_lecture_3_v7_sp10

ece_350_lecture_3_v7_sp10 - ECE 350 Lecture 3 Lecture...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 350 D. van Alphen 1 ECE 350 – Lecture 3 Lecture Overview • Pseudo-quiz • Review: Zero-Input Response, Cases 1 and 2 • Zero-Input Response, Case 3 (Complex Roots) • Zero-State Response – Impulse Response, h(t) – Convolution • Interconnected Systems
Background image of page 1

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

View Full DocumentRight Arrow Icon
ECE 350 D. van Alphen 2 Pseudo-Quiz Questions 1. A system is linear if and only if it obeys what 2 properties? ___________________ and ______________________ 2. The differential equation for a time-invariant system has ______________ coefficients. 3. Consider the system described by differential equation: Is this system linear or non-linear? … time-invariant or time-varying? 4. = _______ ) t ( x 4 y 7 dt dy t 3 dt y d 2 2 = + τ τ δ τ d ) t ( ) ( x
Background image of page 2
ECE 350 D. van Alphen 3 Review: Zero-Input Response of Linear Systems, Cases 1 and 2 To find the characteristic equation: Set Q(D) = 0 Q( λ ) = 0; Solve for characteristic equation roots = system eigenvalues, λ i – Case 1: (Real) Distinct eigenvalues • Characteristic modes: e λ i t – Case 2: Real, repeated eigenvalues • Characteristic modes: e λ i t , te λ i t , t 2 e λ i t , … – Case 3: Complex roots (still to come) … • Zero Input Response, y 0 (t): weighted sum of characteristic modes
Background image of page 3

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

View Full DocumentRight Arrow Icon
ECE 350 D. van Alphen 4 Zero-Input Response, Case 3: Characteristic Equation has Complex Roots Say we get λ 1 = α + j β , λ 2 = α -j β (complex conjugate pairs); Characteristic modes: e λ 1 t , e λ 2 t (as before) y 0 (t) = c 1 e ( α + j β )t +c 2 e ( α β )t (zero-input response, as before) or, equivalently (as we will show later), y 0 (t) = c e α t cos( β t+ θ ) Example : Find the zero-input response (or homogeneous solution) for the system with differential equation: (D 2 + 4D + 40) y(t) = x(t) – Characteristic Equation: λ 2 + 4 λ + 40 = 0 – Roots of Quadratic Equation: j 6 2 2 ) 40 )( 1 ( 4 4 4 2 ± = ± = λ α β
Background image of page 4
ECE 350 D. van Alphen 5 Zero-Input Response, Case 3 - Example, continued - Exponential Approach: y 0 (t) = c 1 e (-2+6j)t + c 2 e (-2-6j)t = c 1 e -2t e 6jt + c 2 e -2t e -6jt Note: real solution c 1 = c 2 *; let c 1 = re j θ , c 2 = re -j θ ; y 0 (t) = r e j θ e -2t e 6jt + r e -j θ e -2t e -6jt = r e -2t (e j(6t+ θ ) + e -j(6t+ θ ) ) = 2 r e -2t cos(6t + θ ) = c e -2t cos(6t + θ ) Alternative Approach (by inspection!): y 0 (t) = c e -2t cos(6t + θ ) αβ Later: Constants c and θ will be determined from the I.C.’s
Background image of page 5

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

View Full DocumentRight Arrow Icon
ECE 350 D. van Alphen 6 Zero-Input Response – Case 1 Example: Using I.C.’s to Find the Unknown Constants Example: Find the zero-input response, y 0 (t), of an LTIC system with differential equation: (D 2 + 3D + 2) y(t) = D x(t) if y 0 (0) = 0, and Characteristic Equation: λ 2 + 3 λ + 2 = 0 ( λ + 1) ( λ + 2) = 0 ⇒λ = -1, -2 (Case 1) Characteristic Modes: e -1t , e -2t Zero-Input Response: y 0 (t) = c 1 e -1t + c 2 e -2t . 5 ) 0 ( y = & t 2 2 t 1 0 e c 2 e c dt ) t ( dy = d/dt, to be able to use the 2 nd I.C.
Background image of page 6
ECE 350 D. van Alphen 7 Zero-Input Response, Case 1 Example, continued Repeating the 2 equations, with the I.C.’s: y 0 (t) = c 1 e -1t + c 2 e -2t c 1 + c 2 = 0 -c 1 –2c 2 = -5 Solution: y 0 (t) = -5 e -1t + 5 e -2t t 2 2 t 1 0 e c 2 e c dt ) t ( dy = t = 0 c 1 = -5, c 2 = 5
Background image of page 7

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

View Full DocumentRight Arrow Icon
ECE 350 D. van Alphen
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 02/27/2010.

Page1 / 34

ece_350_lecture_3_v7_sp10 - ECE 350 Lecture 3 Lecture...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online