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Download Ch 33 - A.C. Circuits
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Alternating Current Circuits Chapter 33 (continued) Phasor Diagrams • A phasor is an arrow whose length represents the amplitude of an AC voltage or current. • The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. • Phasor diagrams are useful in solving complex AC circuits. • The “y component” is the actual current or voltage. Resistor Capacitor Inductor VRp Ip VLp Ip wt Ip wt wt VCp Impedance in AC Circuits R V ~ C L The impedance Z of a circuit or circuit element relates peak current to peak voltage: Vp Ip Z (Units: Ohms) (This is the AC equivalent of Ohm’s law.) Phasor Diagrams Circuit element Impedance Amplitude Phase Resistor R VR= IP R I, V in phase Capacitor Inductor Xc=1/wC XL=wL VC=IP Xc VL=IP Xc I leads V by 90° I lags V by 90° Resistor Capacitor Inductor VRp Ip VLp Ip wt Ip wt wt VCp RLC Circuit R V ~ C L Use the loop method: V - V R - VC - VL = 0 I is same through all components. BUT: Voltages have different PHASES they add as PHASORS. RLC Circuit Ip VLp VRp f (VCp- VLp) VP VCp RLC Circuit Ip VLp VRp f (VCp- VLp) VP VCp By Pythagoras’s theorem: (VP )2 = [ (VRp )2 + (VCp - VLp)2 ] = Ip2 R2 + (Ip XC - Ip XL) 2 RLC Circuit R Solve for the current: Ip Vp Vp Z R 2 (X c X L ) 2 V ~ C L RLC Circuit R Solve for the current: Ip V Vp Z R 2 (X c X L ) 2 Impedance: Vp ~ C L 1 Z R wL wC 2 2 RLC Circuit Ip Vp The current’s magnitude depends on the driving frequency. When Z is a minimum, the current is a maximum. This happens at a resonance frequency: Z 2 1 2 Z R wL wC The circuit hits resonance when 1/wC-wL=0: w r=1/ LC When this happens the capacitor and inductor cancel each other and the circuit behaves purely resistively: IP=VP/R. IP R =10W L=1mH C=10mF R = 1 0 0 W 0 1 0 wr 2 1 0 3 1 0 4 1 0 5 w The current dies away at both low and high frequencies. Phase in an RLC Circuit Ip VLp VRp f VP We can also find the phase: tan f = (VCp - VLp)/ VRp = (XC-XL)/R (VCp- VLp) VCp = (1/wC - wL) / R Phase in an RLC Circuit Ip VLp We can also find the phase: VRp f VP tan f = (VCp - VLp)/ VRp = (XC-XL)/R (VCp- VLp) VCp = (1/wC - wL) / R More generally, in terms of impedance: cos f R/Z At resonance the phase goes to zero (when the circuit becomes purely resistive, the current and voltage are in phase). Power in an AC Circuit The power dissipated in an AC circuit is P=IV. Since both I and V vary in time, so does the power: P is a function of time. Use V = VP sin (wt) and I = IP sin (w t+f ) : P(t) = IpVpsin(wt) sin (w t+f ) This wiggles in time, usually very fast. What we usually care about is the time average of this: 1 P T T 0 P(t)dt (T=1/f ) Power in an AC Circuit Now: sin( wt f) sin( wt)cos f cos(wt)sin f Power in an AC Circuit Now: sin( wt f) sin( wt)cos f cos(wt)sin f P(t) IPVP sin( w t)sin( w t f ) IPVP sin 2 (w t)cos f sin( w t)cos(w t)sin f Power in an AC Circuit Now: sin( wt f) sin( wt)cos f cos(wt)sin f P(t) IPVP sin( w t)sin( w t f ) IPVP sin 2 (w t)cos f sin( w t)cos(w t)sin f Use: and: So sin (w t) 2 1 2 sin( w t)cos(w t) 0 1 P IPVP cos f 2 Power in an AC Circuit Now: sin( wt f) sin( wt)cos f cos(wt)sin f P(t) IPVP sin( w t)sin( w t f ) IPVP sin 2 (w t)cos f sin( w t)cos(w t)sin f Use: and: So sin (w t) 2 1 2 sin( w t)cos(w t) 0 1 P IPVP cos f 2 which we usually write as P IrmsVrms cos f Power in an AC Circuit P IrmsVrms cos f (f goes from -900 to 900, so the average power is positive) cos(f)is called the power factor. For a purely resistive circuit the power factor is 1. When R=0, cos(f)=0 (energy is traded but not dissipated). Usually the power factor depends on frequency, and usually 0<cos(f)<1. Power in a purely resistive circuit V f= 0 p I 2p wt V(t) = VP sin (wt) I(t) = IP sin (wt) (This is for a purely resistive circuit.) P P(t) = IV = IP VP sin 2(wt) Note this oscillates twice as fast. p 2p wt Power in a purely reactive circuit P IrmsVrms cos f The opposite limit is a purely reactive circuit, with R=0. I V P This happens with an LC circuit. 0 Then f 90 wt The time average of P is zero. Transformers Transformers use mutual inductance to change voltages: Iron Core Np turns Ns turns Vs Vp Primary (applied voltage) Secondary (produced voltage) Faraday’s law on the left: If the flux per turn is f then Vp=Np(df/dt). Faraday’s law on the right: The flux per turn is also f, so Vs=Ns(df/dt). Ns Vp Vs Np Transformers Transformers use mutual inductance to change voltages: Np turns Iron Core Ns turns Vs Vp Primary (applied voltage) Secondary (produced voltage) Ns Vs Vp Np In the ideal case, no power is dissipated in itself. the transformer Then IpVp=IsVs Np Is Ip Ns Transformers & Power Transmission Transformers can be used to “step up” and “step down” voltages for power transmission. 110 turns Power =I1 V1 V1=110V 20,000 turns V2=20kV Power =I2 V2 We use high voltage (e.g. 365 kV) to transmit electrical power over long distances. Why do we want to do this? Transformers & Power Transmission Transformers can be used to “step up” and “step down” voltages, for power transmission and other applications. 110 turns Power =I1 V1 V1=110V 20,000 turns V2=20kV Power =I2 V2 We use high voltage (e.g. 365 kV) to transmit electrical power over long distances. Why do we want to do this? P = I2R (P = power dissipation in the line - I is smaller at high voltages)