Lecture 21. AC Circuits, Reactance.

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1 Lecture 1. AC Circuits, Reactance. Outline: Power in AC circuits, Amplitude and RMS values. Phasors / Complex numbers. Resistors, Capacitors, and Inductors in the AC circuits. Reactance and Impedance. Conflict final exam: December 7, 500 PM: Last day and time to request conflict exam for Final Exam (you can request conflict exam if you have another exam at same time OR have 3 exams in 4 hour period). You must Professor Cizewski with details of why you are requesting conflict exam. November 5 (Rutgers Thursday): Required E&M Post-test during lecture times. You can attend any lecture. Makeup post tests will be in early December. NO Recitations the week of November 4. 1

2 Oscillations in L-C Circuits L-C circuits: the circuits with TWO elements that can store energy (ideally, without dissipation). The energy flow back and forth between L and C results in harmonic oscillations of q t and i t. electric field energy in the capacitor U E = q C U B = LL magnetic field energy in the inductor Let s say, at t = 0 the capacitor is fully charged, i=0. q 0 C = q t C + L i t

3 Oscillations in L-C Circuits (cont d) ω = 1 LL L t T 1 f π ω q t C = 0, t = π LL U E = q C = q 0 C ccc ωω = q 0 4C U B = LL = Lω q 0 d q t dt, Solution: q t = q 0 ccc ωω + φ ccc ωω sss ωω = Lω q 0 4 amplitude i t = 1 + sss ωω dq t d q t dt + 1 q t = 0 LL angl. frequency phase at t=0 = ωq 0 sss ωω + φ 0 T/ T 3

4 L-C-R Circuits L t q t C ii = 0, d q t dt + R L dq t + 1 q t = 0 LL Solution for weak damping 1 LL R 4L : q t = q 0 eee RR L ccc ω t + φ 0 ω = 1 LL R 4L weak damping ( small R) strong damping ( large R) 6

5 Impedance of AC circuits So far we have considered transient processes in R-C and R-L circuits: the approach to the stationary (time-independent) state after some perturbation (switch on / off). Today we ll discuss how these circuits behave being connected to the alternating current (AC) power supply: the circuits driven by a steady external drive, e.g. the AC voltage source. We disregard all transient processes and instead consider the steady-state AC currents: currents and voltages vary with time as ccc ωω + φ 0, but their amplitudes are t-independent. We ll describe the response of an L-C-R circuit to a harmonic drive using the notion of impedance. 13

6 V t Amplitudes, rms Values, and Power in AC Circuits I t RLC Instantaneous values: Instantaneous power: V t = V 0 cos ωω I t = I 0 cos ωω + φ P t = V t I t Currents and voltages are NOT necessarily in phase, φ is the phase shift between V and I (the phase angle ). P t = V 0 cos ωω I 0 cos ωω + φ = 1 V 0I 0 cos ωω + φ + cos φ P aa P t = 1 V 0I 0 cos φ Root mean square (rms): the square root of the average of the square of the quantity: a rrr = a t V rrr = V 0 Power, being expressed in the rms values: I rrr = I 0 P aa = V rrr I rrr cos φ cos φ - the power factor 10V wall outlet: f = 60HH, ω = π 60 rrr s = 377rrr/s, T = 1 f 17mm V rrr = 10V, V 0 170V 14

7 Instantaneous value φ 1 φ Phasors / Complex Number Representation phasor Problem: To find current/voltage in R-L-C circuits, we need to solve differential equations. Solution: The use of complex numbers / phasors allows us to replace linear differential equations with algebraic ones, and reduce trigonometry to algebra. We represent voltages and currents in the R-L-C circuits as the phase vectors (phasors) on the D plane. Quantity: A t = A 0 cos ωt. Corresponding phasor: a vector of length A 0 rotating counter-clockwise with the angular frequency ω. Instantaneous value of A t is the projection of the phasor onto the horizontal axis. If all the quantities oscillate with the same ω, we can get rid of the term ωω by using the rotating (merry-goaround) reference frame. We ll consider the steady state of AC circuits, when all amplitudes (the phasor lengths) are t-independent, and the only time dependence remaining is in the single frequency sinusoidal oscillation of voltages and currents. The angle between different phasors represents their relative (t-independent) phase. 15

8 Complex Numbers, Phasors b b φ φ Z a + ii = RR Z + i II Z RR Z = r cccφ II Z = r sssφ a Z = re iφ Z a ii = RR Z i II Z Imaginary unit: i 1 e iφ = cos φ + i sin φ e iπ = i e iπ = i A ccc ωt + φ = A A sss ωt + φ = A i = 1 ei ωω+φ + ei ωω+φ 1 i = i Euler s relationship i ωω+φ e i ωω+φ e i The absolute value (or modulus or magnitude): = RR A ei ωω+φ = II A ei ωω+φ complex conjugate of Z Z Z Z Z = r cccc + i ssss r cccc i ssss = r ccc φ + sss φ = r Phasor: refer to either A e i ωω+φ or just A e i φ. In the latter case, it is understood to be a shorthand notation, encoding the amplitude and phase of an underlying sinusoid. 16

9 Complex Numbers, Phasors (cont d) The use of complex numbers / phasors allows us to replace linear differential equations with algebraic ones, and reduce trigonometry to algebra: Aition: a + ii + c + ii = a + c + i b + d Multiplication: Differentiation: Ae iω 1t Be iω t = AAe i ω 1+ω t d AAiii = iωaa iii 17

10 I t Resistor V t AC current through a resistor and AC voltage across the resistor are always in phase. Power dissipated in a resistor: P t = V t I(t) I V P P aa = 1 V 0I 0 cccc = 1 V 0I 0 = V rrr I rrr π π phase ωt 18

11 V t I t V t Q t C = 0 V t = V 0 e iii i ωω+φ I t = I 0 e Capacitor I t = dv t dq t t = 1 C dq t = 1 I t C = ii V 0 e iii iicv 0 = I 0 e ii i = e iπ φ = π/ For a capacitor, voltage LAGS current by P t = RR V t I(t) I t V t = RR V 0 I 0 e i ωω π = V 0 I 0 RR cos ωω + i sin ωω cos π + i sin π π I V P π phase ωt 0-1 = V 0 I 0 sin ωω Current (reference phasor) Voltage P aa = 1 V 0I 0 cccc = φ = π = 0 The power IS NOT dissipated in a capacitor: it is stored in the capacitor for half a period, and returned to the circuit for another half. 19

12 V t I t V t L t V t = V 0 e iii Inductor = 0 V t = L t V 0 = ii LL 0 e ii = ω LL 0 e i π +φ i ωω+φ I t = I 0 e t i ωω+φ = ii I 0 e φ = π/ In an inductor, current LAGS voltage by 90 0 I t P t = RR V t I(t) V t = V 0 I 0 RR e i ωω+π = V 0 I 0 RR cos ωω + i sin ωω cos π + i sin π = V 0 I 0 sin ωω Current (reference phasor) Voltage π I V P π time t phase ωt P aa = 1 V 0I 0 cccc = φ = π = 0 The power IS NOT dissipated in an inductor: it is stored in the inductor for half a period, and returned to the circuit for another half. 0

13 Reactance Resistor V = IR = IX R X R = R Capacitor t = 1 C I t iiv 0e iii = I C I 0e i ωω+π V 0 = I 0 ωω X C = 1 ωω V rrr = I rrr ωω V rrr = I rrr X C V t = I t i X C Inductor V t = L t V 0 e iii = L iωi 0 e i ωω π V 0 = ωl I 0 V rrr = ωω I rrr V rrr = I rrr X L X L = ωl V t = I t ix L Wiki: reactance is the opposition of a circuit element to a change of electric current due to that element's inductance or capacitance. The reactance is measured in Ohms. Major differences between reactance and resistance: the reactance for L and C changes with frequency, it reflects (being combined with ±i) the phase shift between V and I, and it dissipates no energy. 1

14 Impedance Impedance Z is a measure of the overall opposition of a circuit to current, in other words: how much the circuit impedes the flow of current. Both the reactance and resistance are components of the impedance. The magnitudes of V and I are the rms voltage and current respectively, and the various reactances behave mathematically just like resistances, except that they are complex. V t = I t Z I For R, C, and L in series: Z = Z Z = R + ωω 1 ωω V s = I R + i ωω 1 ωω Z = R + ix L ix c = R + i ωω 1 ωω V s = V s V s = I R + ωω 1 ωω II L II C IR V s I V = I Z For V rrr = 10V, 60HH, R = 300Ω, C = μμ, L = 3H: ωω 1 ωω = = = 195Ω V I rrr = Ω = 0.335A 5

15 Next time: Lecture 3. Resonance in AC circuits, Transformers

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