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Solving Alternating Current Circuits Using Complex Numbers Electromagnetism 1, PHYS1090 Dr Joachim Rose Department of Physics and Astronomy University of Leeds EM1-L20-1 Overview The plan for todays lecture • RLC circuit Calculation of current amplitude and phase using complex numbers • Complex impedances RLC circuits and analogous DC circuits • Summary EM1-L20-2 RLC circuit and complex numbers EM1-L20-3 Differential equation for RLC circuit Kirchhoff’s loop rule L· dI Q + R · I + = Emax · cos (ωt) dt C R Use Q = (dQ/dt) dt = dI 1 L· +R·I + · dt C Z R I dt I dt = Emax · cos (ωt) Note that the right side of the above equation is the real part of the complex function Emax · eiωt = Emax · (cosωt + i sin ωt) RLC circuit and complex numbers EM1-L20-4 Complex differential equation A very similar differential equation but for a complex function z: dz 1 L· +R·z+ · dt C Z z dt = Emax · eiωt The real part of the solution z is the current I I = Re(z) As a solution for the differential equation try z = I0 · eiωt which then implies dz =iωz dt Z 1 z dt = z iω Substituting into the differential equation gives iωL z + R z + 1 Emax z = Emax eiωt = z iωC I0 RLC circuit and complex numbers EM1-L20-5 Complex differential equation (continued) Divide each term by z, write ωL = XL and (1/ωC) = XC , and use XC /i = iXC /i2 = −iXC then iωL z + R z + 1 Emax z = Emax eiωt = z iωC I0 becomes i(XL − XC ) + R = Emax I0 The complex constant I0 is then I0 = Emax i(XL − XC ) + R The denominator expressed in polar form is q i(XL − XC ) + R = (XL − XC )2 + R2 · eiδ = Z · eiδ where Z is the impedance and δ the phase. tan δ = XL − XC R RLC circuit and complex numbers EM1-L20-6 Current in a RCL circuit Using the polar form the constant I0 is Emax Emax −iδ I0 = = e Z eiδ Z and the complex solution z is z = I0eiωt = Emax −iδ iωt Emax i(ωt−δ) e e = e Z Z The current I is thus given by I = Re(z) = Emax cos (ωt − δ) Z RLC circuit and complex numbers EM1-L20-7 Complex Impedances EM1-L20-8 Generalized Ohms law Differential equation for complex function z dz 1 L· +R·z+ · z dt = Emax · eiωt dt C As a solution for the differential equation try Z z = I0 · eiωt This gives iωL z + R z + Emax 1 z= z iωC I0 Multiply by I0/z 1 I0 · iωL + R + iωC = Emax Complex impedances: iωL, R, 1/iωC Complex impedances EM1-L20-9 Complex impedances Resistor: Zeq = R Inductor: Zeq = i ω L = i XL Capacitor: i 1 Zeq = =− = −i XC iωC ωC Complex impedances EM1-L20-10 Complex impedance for RCL circuit Complex impedance: Zeq = iωL + R + 1 iωC = i XL + R − i XC Complex impedances EM1-L20-11 Current, impedance and phase Current I= Emax cos (ωt − δ) Z Phase XL − XC tan δ = R Impedance Z= Complex impedances q (XL − XC )2 + R2 EM1-L20-12 Summary • Resistor Zeq = R • Inductor Zeq = i ω L = i XL • Capacitor 1 i Zeq = =− = −i XC iωC ωC • Complex equivalent impedance: Add up the complex impedances in a circuit with external generator. Similar to finding equivalent resistance for a DC circuit. • Current I= Emax cos (ωt − δ) Z Reading: Tipler, chapter 31, page 980-981 Summary EM1-L20-13