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SERIES AC CIRCUITS Graphs of sinusoidal functions of time Alternating current Resistance in an AC circuit Inductance in an AC circuit Capacitance in an AC circuit Series LRC circuit Power loss Resonance by Dr. John Dayton AC CIRCUITS Every slide contains valuable and need-to-know information that has to be understood and retained before proceeding. Throughout this PowerPoint a color scheme has been employed. Current is depicted in purple, voltage across a resistor in orange, voltage across an inductor in green, voltage across a capacitor in blue, and generator voltage in red. I VL VR VC E GRAPHS OF SINUSOIDAL FUNCTIONS OF TIME. Important Terms: Period, T Amplitude, A Angular frequency, w Frequency, f Phase shift, f The graph is just a small piece of y Asin wt . A is the amplitude, w is the angular frequency in radians/s. Related to w is f, the frequency in Hz = s-1. Adding a phase angle, f, shifts the graph. y Asin wt f shifts the graph to the left if f is positive. RMS – Root Mean Square, the Effective Value (purple graph) The additional graph shown in green is: y A sin wt 2 T 2 RMS – Root Mean Square, the Effective Value T The average value of any function is the area under its graph divided by the width of the graph. Over one period the average value of y Asin wt is 0 since there is as much positive area above the horizontal axis as there is negative area below the horizontal axis. RMS – Root Mean Square, the Effective Value A2 1 2 A2 T Area = 2A2T T Examine the graphs of y 2 A2 sin2 wt shown above. The two shaded areas in the first period lying above the midpoint can be cut off and used to fill in the troughs as shown. This completes a rectangular area equal to 12 A2T , further shaded in pink. Thus the average value of A2 sin2 wt is 12 A2 over one period. We can now define what is called the root mean square of y to be yrms 1 2 A A 2 2 The pulsating dots show the original and modified distributions of area. EXAMPLE: Given: y 5sin 12t , identify the following: the amplitude, A the angular frequency, w the frequency, f the period, T the rms value of y, yrms The basic sinusoidal equation is y Asin wt . Using this it can be seen that A = 5 (which may contain units) w = 12 rad/s Then: w 1 f 1.91Hz T 0.524 s 2 f yrms A 5 3.54 2 2 PHASORS A phasor can be likened to a rotating vector. It has a specific and constant length equal to the maximum value, amplitude, of a sinusoidally varying quantity such as a current or a voltage. The tail end of the phasor is fixed on the origin of an x-y coordinate system and is the pivot of the phasor’s rotation. As the phasor rotates about this pivot, it sweeps out the angle w t + f and its projection on the y-axis corresponds to the value of the quantity represented by the phasor. y y t phasor PHASORS The diagram shows two phasors with the same frequency but out of phase with each other. The blue phasor leads the red phasor by an angular amount equal to their phase difference. You will note that the blue phasor crosses the +y axis before or ahead of the red phasor, indicating it leads. You can also say the red phasor lags the blue phasor. On the graphs you can see that the peak in the blue quantity occurs before the peak in the red quantity. y y t graphic from web Alternating Current Instantaneous current: i I sin(wt ) AC Power Supply Instantaneous emf: E Emax sin wt PHASOR i I I Graph of i wt t The time varying current, i, in this phasor diagram is represented by the projection of I on the y-axis. Resistance in an AC Circuit Maximum Voltage: VR Instantaneous Voltage: vR VR sin wt Peak-to-Peak Voltage: 2VR This phasor diagram shows VR and I rotating at w. vR and i are represented as projections on the y axis. The Current and Voltage are in Phase. Maximum Current: RMS Current: I rms VR I R I 2 Resistance in an AC Circuit I, V Imax Graphs of i and vR p i R I R sin wt 2 Pavg I R I 1 2 VR t 2 I I rms 2 V IR VR , rms I rms R Effective current, Irms Effective voltage, Erms 2 VR , rms VR 2 2 2 rms R Resistance in an AC Circuit EXAMPLE: An AC generator has a maximum emf of 240 volts at a frequency of 90 Hz. Connected to the generator is a 500 W resistor. What are the rms current through the resistor and the average power loss? I rms Erms E 240V 0.339 A R R 2 500W 2 PAVG I 2 rms R 0.33941A (500W) 57.6W 2 REACTANCE VS RESISTANCE As current flows through a resistor it interacts with the material of the resistor and experiences an opposition to its flow known as resistance. This is expressed by Ohm’s Law. V IR As a changing current flows through an inductor or to and from a capacitor there are voltage changes that occur due to the fields in the respective devices. There is a relation between this voltage and the current that involves an opposition to current flow due to these fields called reactance. Reactance is designated by X and Ohm’s Law can be written: V IX Both resistance and reactance are measured in ohms, W, but there is no power loss due to reactance as there is with resistance. Inductance in an AC Circuit Reactance is an effective resistance to the flow of current due to the presence of electric or magnetic fields. Inductance in an AC Circuit i vL L t I sin wt vL L L wI cos wt t vL wLI sin wt 2 VL sin wt 2 VL IX L Inductive Reactance: X L wL XL w Inductive Reactance Inductive reactance increases linearly with frequency. Inductance in an AC Circuit This phasor diagram shows VL and I rotating at w. vL and i are represented as projections on the y axis. I, V Graphs of i and vL I VL t VL leads I by 90o The current through an inductor lags the voltage across it by 90o. The voltage peak across the inductor will be seen T/4 seconds before the current peak. (T = period = 1/f). The flashing yellow dots on the graphs flash as the respective phasor crosses the +y-axis indicating maximum value. Inductance in an AC Circuit EXAMPLE: A 300 peak voltage ac generator operating at 120 Hz is connected to a 15,000 turn solenoid of length 5.0 cm. and radius 1.0 cm. (A) what is the inductance of the solenoid. (B) What is the inductive reactance of the circuit? (C) What is the rms current in the circuit? (D) At t = T/3 what is the stored energy in the inductor? L 0 N A 2 4 107 TmA 15000 .01m 2 2 1.78H l .05m X L w L 2 fL 2 120 Hz 1.7765H 1.34 103 W I rms Erms E 300V 0.158 A 3 X L X L 2 (1.33945 10 W) 2 E T E 2 T E 2 sin w sin sin XL 3 XL T 3 XL 3 300V 2 i sin 0.19397 A 3 1.33945 10 W 3 i I sin wt U 12 Li 2 12 (1.7765H )(0.19397 A)2 0.0334 J Capacitance in an AC Circuit Capacitance in an AC Circuit q vC C q i t q it q q it I sin wt t q I cos wt I sin wt 2 w w I sin wt 2 vC VC sin(wt 2 ) wC Capacitance in an AC Circuit I sin wt 2 vC VC sin(wt 2 ) wC VC IX C XC 1 XC wC Capacitive Reactance w Capacitive reactance decreases with the reciprocal of the frequency. I, V I Capacitance in an AC Circuit Graphs of i and vC t VC The current to/from a capacitor leads the voltage by 90o. The current peak will be seen T/4 seconds before the voltage peak. This phasor diagram shows VC and I rotating at w. vC and i are represented as projections on the y axis. VC lags I by 90o Capacitance in an AC Circuit EXAMPLE: An AC generator operating at 140 V and 75 Hz is connected to a 12.7 F capacitor. (A) What is the capacitive reactance? (B) What is the rms current in the circuit? (C) What is the average power loss? 1 1 1 2 XC 1.67 10 W 6 wC 2 fC 2 75Hz 12.7 10 F I rms Erms E 140V 0.592 A 2 X C X C 2 (1.6709 10 W) 2 Since there is no resistance, R, there is no power loss in the circuit. Series LRC Circuits The current i in the circuit is the same at all points at any instant of time and varies sinusoidally. i I sin wt The instantaneous voltage across the resistor is in phase with the current. The instantaneous voltage across the inductor leads the current by 90o. The instantaneous voltage across the capacitor lags the current by 90o. Series LRC Circuits I, VR, VL, VC, and Emax are all maximum values. i, vR, vL, vC , E are all time-varying values. Color distinction has been used throughout for convenience. Graphs of i, v I VR VC Emax VL t Series LRC Circuits Phasors shown at t = 0. Click to see phasor rotation. View time-varying voltages and current as projections on the yaxis. Phasors Emax f E vR vC vL Emax sin(wt f ) Series LRC Circuits Series LRC Circuits Emax Phase: f VL-VC VR VL VC IX L IX C X L X c tan(f ) VR IR R XL Xc tan(f ) R 1 X L X c f tan R Series LRC Circuits Impedance, Z: VL-VC f IZ Z f f I(XL-XC) R IR VR XL-XC Emax V (VL VC ) I R IX L IX C 2 R 2 2 2 2 Emax I R ( X L X C ) IZ 2 2 Z R (X L XC ) 2 2 Emax I Z Impedance, Z, is the effective resistance due to a combination of resistance and reactance and is measured in ohms, W. Series LRC Circuits In an inductive circuit the generator voltage leads the current. f VL-VC f is positive, XL > XC VR In a capacitive circuit the generator voltage lags the current. VR f VL-VC f is negative, XC > XL EXAMPLE: Series LRC Circuits A series RLC circuit with R=300 W, L = .75 H, C = 4.6F, f = 60 Hz and Emax = 120V. w 2f 376.99 s 1 X L wL 282.74W 1 XC 576.65W wC Emax I 0.286 A Z Z R 2 ( X L X C ) 2 419.98W X L XC f tan R Since the phase angle f is negative the circuit is capacitive and the generator emf peak lags the current peak by 44.4o. VR IR 85.8V VL IX L 80.9V VC IX C 164.9V o 44.4 1 VL VR f Emax R VC f Z XL-XC Series LRC Circuits POWER LOSS No power loss occurs at a pure inductor or a pure capacitor. Power loss occurs at a resistor where energy is lost as thermal energy. P I R I rmsVR, rms I rmsErms cos f 2 rms P I rmsErms cos f cos(f is called the power factor. Series LRC Circuits RESONANCE Generally: I rms At resonance: Erms Z X L XC 1 wo LC A maximum value of the current and power loss occurs when XL = XC, w = wo. Erms R 2 ( X L X C )2 1 wo L woC resonance angular frequency Z R, f 0 Erms I rms R o Series LRC Circuits Resonance is related to the natural frequency of oscillation between a fully charged capacitor and an inductor connected in parallel. Initially the capacitor discharges through the inductor transferring all its stored energy from its electric field to an induced magnetic field inside the inductor. As the magnetic field collapses, its energy flows back to the capacitor by recharging the capacitor and reestablishing the electric field. This cycle repeats with each cycle reversing the polarity of the fields. the natural frequency is the same as the resonance frequency. w0 1 LC image from Wikipedia Series LRC Circuits RESONANCE A resonance: w wo 1 LC X L XC ZR f 0 If there is no resistance, then Z = 0 0 Emax I peak R Erms I peak , rms R Series LRC Circuits C = 12.0 F R = 100 W L = 4.0 H Emax = 200 V Current, I, and power, P, vary with w and reach peak values at resonance when w = wo. wo 1 = 144.338 rads LC Power Current wo w wo w IMPORTANT GRAPHS R At resonance XL = XC, Z = R, f = 0, I and P peak. Series LRC Circuits EXAMPLE: Given the RLC series circuit with R = 150 W, L = 1.25 H, C = 8.4 F and Emax = 150 V. (A) Calculate the resonance frequency, wo; and the peak current and power loss at resonance. (B) Calculate the impedance, current, power loss, and phase angle at w = .5wo. 1 LC 1 A wo I Emax Emax 150V 1.00 A Z R 150W 1.25H 8.4 106 F 308.607 rads I 2 R 1.00 A (150W) 2 P I rms R 75W 2 2 2 Continued on next slide. B w 12 wo 12 (308.607 rads ) 154.303 rads 1 Z R wL wC 2 2 1 Z (150W) 2 154.303 rads (1.25H ) 6 rad (154.303 )(8.4 10 F ) s 2 Z 597.764W E 150V I max 0.251A Z 597.764W I 2 R 0.25094 A (150W) P 4.72W 2 2 1 w L 1 w C f tan R 2 1 rad 154.303 (1.25 H ) s 6 rad 154.303 8.4 10 F s 75.5o f tan 1 150W FILTER CIRCUITS VIN VIN VOUT VOUT A low-pass filter: VOUT IX C XC 1 2 2 2 VIN IZ R XC 1 w RC A high-pass filter: VOUT IR R 2 2 VIN IZ R XC 1 1 1 w RC 2 FILTER CIRCUITS VIN VOUT VOUT IX C XC 1 1 2 VIN IZ X L X C X L w LC 1 X C 1 1 2 wo LC 1 At resonance: w wo LC VOUT VIN w 2 wo2 Exercise: show that Z L w END OF PRESENTATION