* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Ch 33 - A.C. Circuits
Survey
Document related concepts
Power MOSFET wikipedia , lookup
Audio power wikipedia , lookup
Crystal radio wikipedia , lookup
Surge protector wikipedia , lookup
Opto-isolator wikipedia , lookup
Power electronics wikipedia , lookup
Integrated circuit wikipedia , lookup
Radio transmitter design wikipedia , lookup
Regenerative circuit wikipedia , lookup
Standing wave ratio wikipedia , lookup
Valve RF amplifier wikipedia , lookup
Zobel network wikipedia , lookup
Rectiverter wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Transcript
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. Resistor Capacitor Inductor Vp Ip Vp Ip wt Ip wt wt Vp Reactance - Phasor Diagrams Resistor Capacitor Inductor Vp Ip Vp Ip wt Ip wt wt Vp “Impedance” of an AC Circuit R L ~ C The impedance, Z, of a circuit relates peak current to peak voltage: Ip Vp Z (Units: OHMS) “Impedance” of an AC Circuit R L ~ C The impedance, Z, of a circuit relates peak current to peak voltage: Ip Vp Z (Units: OHMS) (This is the AC equivalent of Ohm’s law.) Impedance of an RLC Circuit R E ~ L C As in DC circuits, we can use the loop method: E - V R - VC - VL = 0 I is same through all components. Impedance of an RLC Circuit R E ~ L C As in DC circuits, we can use the loop method: E - V R - VC - VL = 0 I is same through all components. BUT: Voltages have different PHASES they add as PHASORS. Phasors for a Series RLC Circuit Ip VLp VRp f (VCp- VLp) VP VCp Phasors for a Series RLC Circuit Ip VLp VRp f (VCp- VLp) VP VCp By Pythagoras’ theorem: (VP )2 = [ (VRp )2 + (VCp - VLp)2 ] Phasors for a Series RLC Circuit Ip VLp VRp f (VCp- VLp) VP VCp By Pythagoras’ theorem: (VP )2 = [ (VRp )2 + (VCp - VLp)2 ] = Ip2 R2 + (Ip XC - Ip XL) 2 Impedance of an RLC Circuit R Solve for the current: Ip Vp Vp Z R2 (X c X L )2 ~ L C Impedance of an RLC Circuit R Solve for the current: Ip ~ L C Vp Z R2 (X c X L )2 Impedance: Vp Z 1 R wL wC 2 2 Impedance of an RLC Circuit Vp Ip Z 1 R wL wC Z 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: 2 2 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 We can also find the phase: VRp (VCp- VLp) f VP tan f = (VCp - VLp)/ VRp or; or VCp tan f = (XC-XL)/R. tan f = (1/wC - wL) / R Phase in an RLC Circuit Ip VLp We can also find the phase: VRp (VCp- VLp) f VP tan f = (VCp - VLp)/ VRp or; or VCp tan f = (XC-XL)/R. tan f = (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 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 an AC Circuit The power 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 T P 0 P( t )dt T (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 ) I PVP sin( w t )sin( w t f ) I PVP 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 ) I PVP sin( w t )sin( w t f ) I PVP 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 P 1 2 I PV P cos f Power in an AC Circuit Now: sin( wt f ) sin( wt )cos f cos(wt )sin f P( t ) I PVP sin( w t )sin( w t f ) I PVP 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 P 1 2 I PV P cos f 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. Power in an AC Circuit P IrmsVrms cos f What if f is not zero? I P V Here I and V are 900 out of phase. (f 900) wt (It is purely reactive) The time average of P is zero. Transformers Transformers use mutual inductance to change voltages: N2 V2 V1 N1 N1 turns Iron Core V1 Primary Power is conserved, though: (if 100% efficient.) N2 turns V2 Secondary I1V1 I 2V2 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)