lecture 3 outline 3-1 Sinusoidal steady-state analysis Phasor analysis is a technique to find the steady-state response when the system input is a sinusoid. That is, phasor analysis is sinusoidal analysis. Phasor analysis is a powerful technique with which to find the steady-state portion of the complete response. Phasor analysis does not find the transient response. Phasor analysis does not find the complete response. Original circuit Add an imaginary sine source to obtain Use Euler's relation to obtainlecture 3 outline 3-2 The differential equation becomes R IL ejωt + L d ( IL ejωt )/dt = Vs ejωt R IL ejωt + jωL IL ejωt = Vs ejωt R IL + jωL IL = Vs The complex currents and voltages in the equations above are called phasors—phasor currents and phasor voltages. Many use the convention of using RMS values when using phasor analysis in electrical circuits. That’s what we’ll do in this course from now on. Notice that, in the equation above, the inductance appears as a "resistance" of jωL. This quantity is referred to as the inductance's impedance. Impedance The algebraic relationship between a phasor voltage and a phasor current is a generalization of resistance and is termed an element's impedance. The unit of impedance is the ohm. Resistance Let's assume that all the voltages are of the form V ejωt and all the currents are of the form I ejωt. Let's look at the resistance's element relation, ohm's law. V ejωt = R I ejωt V = R I The impedance of the resistance Zr is just its resistance. That is, Zr = V / I = Rlecture 3 outline 3-3 Inductance From the inductance's element relation: V ejωt = L d (I ejωt)/dt = jωL I ejωt V = jωL I ZL = jωL Capacitance From the capacitance's element relation: I ejωt = C d (V ejωt)/dt = jωC V ejωt I = jωC V ZC = 1/ jωC = -j /ωC Resistance Inductance Capacitance impedance Zr = R ZL = jωL Zc = -j/ωClecture 3 outline 3-4 Example Find vc(t) and ix(t). 1. Find phasor circuit (give sources in RMS) 2. Write nodal equations 3. Solve for nodal analysissystem 4. Find the phasor voltages and/or currents of interest 5. Use the phasor information to provide the sinusoidal responses. Time-domain circuit a) Find phasor circuit b) Write the nodal equations to find the phasor voltage across the capacitance, Vc (express in polar form with the phase in degrees). c) Find vc(t), the voltage across the capacitance as a function of time.lecture 3 outline 3-5 Perform nodal analysis on this circuit:lecture 3 outline 3-6 Using Maple Phasor analysis example > restart: > alias(I=`I`,j=sqrt(-1)): > eqns:={v1=7.071*exp(j*20*Pi/180), > (v2-v1)/(j*3)+v2/(-j*2)+(v2-v3)/4=0, > (v3-v2)/4+v3/5=0}; > soln:=solve(eqns): > assign(soln): > vc:=evalf(polar(v2),4); eqns := {9/20 v3 - 1/4 v2 = 0, v1 = (10/√2) exp(1/9 j Pi), - 1/3 j (v2 - v1) + 1/2 j v2 + 1/4 v2 - 1/4 v3 = 0} vc := polar(11.77, -2.205) > evalf(-2.205*180/Pi,4); -126.3 check your work: vc(t) = 11.77 cos (3t - 126.3°) V, ix(t) = 1.31 cos (3t - 126.3°) A Discuss Again, compare with the big gun technique. How many variables would have been required? How many equations? Really good advice on phasor analysis Far and away the best tool for phasor analysis is a good engineering calculator. Experience shows that you will avoid pain, work less, and learn more by taking this simple