ECE 201 Lecture 29 Borja Peleato Phasors 1 Phasors A sinusoidal signal voltage or current can always be expressed as x t Ax cos wt or equivalently x t Re X ejwt where X Ax ej X is a complex number known as the PHASOR corresponding to x t The magnitude of X is the same as the amplitude maximum value of x t The phase of X is the same as that of x t i e the real part of X is equal to x 0 Phasors do not depend on time We can go back and forth between phasors and time signals if x t Ax cos wt then X Ax ej x t Re X ejwt 2 Circuit elements We have studied three main circuit elements op amps have to be dealt with separately R L and C Each characterized in terms of an equation relating its voltage drop with its current R gave Ohm s law V t R I t L gave C gave We can always use these to analyze the circuit but when all the inputs are sinusoidal these equations can be significantly simplified 3 Impedance For sinusoidal v t Av cos wt Re V ejwt i t Ai cos wt Re I ejwt Resistor v t R i t R Ai cos wt or in phasor notation V R I Inductor v t L i t L w Ai sin wt or in phasor notation V jwL I Capacitor i t C v t C w Ai sin wt or in phasor notation I jwC V or V jwC 1 I So all three elements follow Ohm s law in phasor notation but with complex numbers Instead of resistance the proportionality factor is called IMPEDANCE and denoted by Z jw Instead of conductance the inverse impedance is called ADMITANCE and denoted by Y jw The impedance or admittance of inductors and capacitors changes with the frequency w For resistors it stays constant 4 Summary of impedances Circuit Element Impedance Admittance Circuit analysis We can use phasors to simplify the analysis of circuits in sinusoidal steady state If all the inputs indep sources oscillate with the same frequency all the currents and voltages in the circuit will also oscillate with that same frequency We can temporarily ignore the time varying oscillation and solve the circuit using phasors as if everything was constant in time but complex If there are multiple inputs with different frequencies we will have to use superposition and find the output as a sum of components with different frequencies General method for analyzing circuits FOR EACH FREQUENCY w 1 2 3 4 5 Convert the w sinusoidal inputs to phasors deactivate the others Convert all R L C elements into impedances USING THE CURRENT w Analyze the circuit assuming everything is constant but complex Ohm s law KCL and KVL is all you need Use nodal or loop analysis Once you have the phasor that you wanted convert it back to time domain USING THE CURRENT w After you have found all the different TIME DOMAIN components add them up by superposition DO NOT APPLY SUPERPOSITION TO THE PHASORS 6 Example 7 8 9
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