Alternating Current Circuits Chapter 33 (continued)

Phasor Diagrams VpVp IpIp  t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor 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. 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.

Reactance - Phasor Diagrams VpVp IpIp  t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor

“Impedance” of an AC Circuit R L C ~ The impedance, Z, of a circuit relates peak current to peak voltage: (Units: OHMS)

“Impedance” of an AC Circuit R L C ~ The impedance, Z, of a circuit relates peak current to peak voltage: (Units: OHMS) (This is the AC equivalent of Ohm’s law.)

Impedance of an RLC Circuit R L C ~ E As in DC circuits, we can use the loop method: E - V R - V C - V L = 0 I is same through all components.

Impedance of an RLC Circuit R L C ~ E As in DC circuits, we can use the loop method: E - V R - V C - V L = 0 I is same through all components. BUT: Voltages have different PHASES  they add as PHASORS.

Phasors for a Series RLC Circuit IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Phasors for a Series RLC Circuit By Pythagoras’ theorem: (V P ) 2 = [ (V Rp ) 2 + (V Cp - V Lp ) 2 ] IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Phasors for a Series RLC Circuit By Pythagoras’ theorem: (V P ) 2 = [ (V Rp ) 2 + (V Cp - V Lp ) 2 ] = I p 2 R 2 + (I p X C - I p X L ) 2 IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Impedance of an RLC Circuit Solve for the current: R L C ~

Impedance of an RLC Circuit Solve for the current: Impedance: R L C ~

The circuit hits resonance when 1/  C-  L=0:  r =1/ When this happens the capacitor and inductor cancel each other and the circuit behaves purely resistively: I P =V P /R. Impedance of an RLC Circuit 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:  The current dies away at both low and high frequencies. rr L=1mH C=10  F

Phase in an RLC Circuit IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp We can also find the phase: tan  = (V Cp - V Lp )/ V Rp or; tan  = (X C -X L )/R. or tan  = (1/  C -  L) / R

Phase in an RLC Circuit At resonance the phase goes to zero (when the circuit becomes purely resistive, the current and voltage are in phase). IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp We can also find the phase: tan  = (V Cp - V Lp )/ V Rp or; tan  = (X C -X L )/R. or tan  = (1/  C -  L) / R More generally, in terms of impedance: cos  R/Z

Power in an AC Circuit V(t) = V P sin (  t) I(t) = I P sin (  t) P(t) = IV = I P V P sin 2 (  t) Note this oscillates twice as fast. V tt   I tt  P  = 0 (This is for a purely resistive circuit.)

The power is P=IV. Since both I and V vary in time, so does the power: P is a function of time. Power in an AC Circuit Use, V = V P sin (  t) and I = I P sin (  t+  ) : P(t) = I p V p sin(  t) sin (  t+  ) This wiggles in time, usually very fast. What we usually care about is the time average of this: (T=1/f )

Power in an AC Circuit Now:

Power in an AC Circuit Now:

Power in an AC Circuit Use: and: So Now:

Power in an AC Circuit Use: and: So Now: which we usually write as

Power in an AC Circuit  goes from to 90 0, so the average power is positive) cos(  is called the power factor. For a purely resistive circuit the power factor is 1. When R=0, cos(  )=0 (energy is traded but not dissipated). Usually the power factor depends on frequency.

Power in an AC Circuit What if  is not zero? I V P Here I and V are 90 0 out of phase. (  90 0 ) (It is purely reactive) The time average of P is zero. tt

Transformers Transformers use mutual inductance to change voltages: PrimarySecondary N 2 turns V1V1 V2V2 N 1 turns Iron Core Power is conserved, though: (if 100% efficient.)

Transformers & Power Transmission 20,000 turns V 1 =110V V 2 =20kV 110 turns Transformers can be used to “step up” and “step down” voltages for power transmission. Power =I 1 V 1 Power =I 2 V 2 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 20,000 turns V 1 =110V V 2 =20kV 110 turns Transformers can be used to “step up” and “step down” voltages, for power transmission and other applications. Power =I 1 V 1 Power =I 2 V 2 We use high voltage (e.g. 365 kV) to transmit electrical power over long distances. Why do we want to do this? P = I 2 R (P = power dissipation in the line - I is smaller at high voltages)