1• PhasorsECE 201: Lecture 29Borja PeleatoPhasors• A sinusoidal signal (voltage or current) can always be expressed as x(t)=Ax*cos(wt+θ) or, equivalently x(t)=Re{X*ejwt} where X=Ax*ejθ• 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+θ) then X=Ax*ejθ– x(t) = Re{X*ejwt} 2Circuit 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 simplified3Impedance• For sinusoidal v(t) = Av * cos(wt+θ) = Re{V * ejwt}, i(t) = Ai * cos(wt+ ψ) = Re{ I *ejwt}– Resistor: v(t)=R*i(t) = R*Ai*cos(wt+ ψ) or, in phasor notation V = R * I– Inductor: v(t)=L*i’(t) = -L*w*Ai*sin(wt+ ψ) or, in phasornotation V = jwL* I– Capacitor: i(t) = C*v’(t) = -C*w*Ai*sin(wt+θ) or, in phasornotation 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 constant4Summary of impedancesCircuit Element Impedance AdmittanceCircuit 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 w1. Convert the w-sinusoidal inputs to phasors (deactivate the others)2. Convert all R, L, C elements into impedances USING THE CURRENT w3. Analyze the circuit assuming everything is constant (but complex). Ohm’s law, KCL and KVL is all you need. Use nodal or loop analysis.4. Once you have the phasor that you wanted, convert it back to time domain USING THE CURRENT w5. After you have found all the different TIME DOMAIN components, add them up by superposition. DO NOT APPLY SUPERPOSITION TO THE
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