ECE 201 Lecture 28 Borja Peleato Complex forcing function 1 Math review Trigonometry cos wt 90 sin wt Complex exponential Euler identity Conversely Real part Given a sinusoid function It can be represented as Imaginary part Complex number representation Rectangular coordinates Polar coordinates Relation 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 at t 0 is the projection on the real horizontal axis Math Review Complex Operations Given two complex numbers in rectangular coordinates in polar coordinates Summation difference Multiplication Division Sinusoidal Steady State SSS 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 5 The analysis of the circuit can be simplified significantly using phasors next lecture Application to circuits Why does this work Alternatively are exchangeable Therefore differentiation operation is preserved by the representation Applying Complex Representation in SSS Analysis Procedures to obtain the response for Vs cos t 1 Replace Vs cos t by the complex forcing j t function Vs e 2 Compute the complex response for the new forcing function 3 Take the real part of the response this will be the response for Vs cos t equivalent real part of Complex forcing function
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