1• Complex forcing functionECE 201: Lecture 28Borja PeleatoMath review• Trigonometry:–– cos(wt - 90) = sin(wt) • Complex exponential:––Euler identityConverselyReal partImaginary partGiven a sinusoid functionIt can be represented asComplex number representationRelation:Polar coordinates:Rectangular coordinates:A complex sinusoidal function can be interpreted as a point rotating over a circle of radius K on the complex plane, with angular velocity w and starting phase θ at t=0.is the projection on the real (horizontal) axisMath Review: Complex OperationsGiven two complex numbers(in rectangular coordinates)(in polar coordinates)MultiplicationDivisionSummation (difference)Sinusoidal Steady State (SSS)5• Until now we have studied circuits with constant or step inputs– Capacitors and inductors react to step inputs (or equivalently, switches), create transient responses– After a long time with a constant input, transient responses die out and the circuit settles down to its DC state (constant V and I everywhere)• If the inputs (independent sources) keep changing and never stay constant, the circuit never settles to a DC state– In general, we must describe the circuit with differential equations and solve them to find the desired outputs– However, if the circuit is linear, a sinusoidal input will create a sinusoidal output WITH THE SAME FREQUENCY (the magnitude and phase can change)• Sinusoidal Steady State refers to the response to a sinusoidal input, once all the transient behaviors have died out– It can be understood as the output after a long time with the same sinusoidal input– Superposition still applies. If we have multiple sinusoidal inputs with different frequencies, the output will have multiple components with different frequencies– The analysis of the circuit can be simplified significantly using phasors (next lecture)Application to circuitsWhy does this work?are exchangeableAlternativelyTherefore, differentiation operation is preserved by the representationApplying Complex Representation in SSS AnalysisComplex forcing functionProcedures to obtain the response for Vs=cos(ωωωω t)1. Replace Vs=cos(ωωωω t) by the complex forcing function Vs=ejωωωω t;2. Compute the (complex) response for the newforcing function;3. Take the real part of the response, this will bethe response for Vs=cos(ωωωω t) equivalentreal part
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