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Unformatted text preview: teady tate AC Analysis I Steady-State AC Analysis I © Fred Terry all, 2008 Fall, 2008 1 Material Order • Alexander & Sadiku Chapters 9, 10, 14 • ecture: Somewhat Different Order than Lecture: Somewhat Different Order than Text Chapters oday: Big Picture (Part of Chp 14) • Today: Big Picture (Part of Chp 14) – Finding AC Steady State Response – Relationship to Transient Response/Differential Equations 2 AC Circuit Analysis • Objective: Find the Steady-State Response of a inear Circuit to a Fixed AC Signal Linear Circuit to a Fixed AC Signal – V in (t)=V cos( ω t) he Steady tate Response is the orced Solution • The Steady-State Response is the Forced Solution of the Differential Equation for the Circuit hy Do This? • Why Do This? – Through Fourier Series, this gives us the response to ANY periodic signal – Directly Useful for All Analog Circuit Design, Understanding Clock Skew on Digital Lines, etc. 3 Example: General RLC Problem C L 2 2 1 2 1 2 1 1 c c c General Differential Equation V V i R R R V i L t LC C t LC t V i ∂ ∂ ∂ + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∂ ∂ ⎞ ⎞ i o (t)=I cos( ω t)u(t) ( ) ( ) 2 2 1 1 2 1 2 ( ) 2 c c c V V i R V i f t t C t L C t R R L α ω α ∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞ + + = + = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ R 2 R 1 1/2 1 cos( ) LC Now i I t ω ω ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ = ater we will use the following 1 ( ) f t C ⎛ = − ⎜ ⎝ 1 sin( ) cos( ) ( ), ( ), ( ) ( ) cos ( ) sin ( ) R I t I t LC Trial Forced Solutionof formof f t f t f t v t A t B t ω ω ω ω ω ⎛ ⎞ ⎞ + ⎟ ⎜ ⎟ ⎠ ⎝ ⎠ ′ ′ ′ = + Later, we will use the following...
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