Lesson 11Lesson 11Lesson 11

AC Circuits

AC CiruitsPowerMaximum and Instantaneous voltage drops and currentPhasor DiagramsPhase angle

RLC Circuits Resonance frequency

High and Low pass filtersStep up and Step down Transformers

AC GeneratorAC Generator

AC emf source

t N ddt

N ddt

B A

Nd

dtBACos NBA d

dtsin t

if rotational speed is constant: t t t NBA sin t

maxsin t

max

NBA

rms current

Effective (Integrated ) values of I and V

i t Imax sin t

2 f 2 T

; T is the period of oscillation

Instantaneous power i t v t

Heat dissipated = Power used in load i t 2 R Im2Rsin

2 t Average power over one cycle

Pave 1

TIm

2Rsin

2 t 0

T

dt Im2R

1

Tsin

2 t 0

T

dt Im

2R

2

Define

Pave Irms2R Irms

Im2

Alternating Current Circuits

t)

Veff and v(t)

Ieff and i(t)

ac-R circuit

Phasor diagram for R iR(t)

t

iR(t)IRmsin(t)=

Phasor Diagram

Current through Load Resistance

Phasor diagram for R cont.vR(t)

t

vR(t)VRmsin(t)=

Phasor Diagram

PD across Load Resistance

iR t 1

RvR t

t R

Instantaneous current and voltage

ac-C circuit

t)

vC(t)

iC(t)

vC(t)

t

iC(t)

Current in Circuit and PD across

Capacitor

ac-L circuit

vL(t)

t

iL(t)

t)

vL(t)

iL(t)

Current in Circuit and PD across

Inductor

The phase angle between the current and the voltage:

In the resistor is 0 radIn the capacitor is - rad

( Current Ahead)In the inductor is + rad

(Current Behind)

Summary

Series RLC circuit Series RLC circuit series ac-RLC circuit

Instantaneous currentCurrent through all elements is the same

i t iV t iC t iL t Thus the instantaneous PD’s

must be out of phase

Picture

Total Potential

Drop across R, L & C.

4 4

1

L CR

L C

Rm Lm Cmm

L C

L

C

T Tv t v t

v ti t

R X X

V V VI

R X X

X L

XC

Phasor Diagram for RLC circuit IvL(t)

vR(t)

vC(t)

Instantaneous Potential Change Across R, L & C

v t vR t vL t vC t t Thus circuit phasor is

v RLC t v R(t) v L (t) v C(t)

Instantaneous PD

Phasor Diagram for RLC circuit II

vL(t)

vR(t)vC(t)

vRLC(t)

Instantaneous PD as projection onto y-axis

vRLC(t)

v(t1)v(t2)

Phase Angle

Phase Angle

vRLC(t)

tan VLm VCmVRm

Im XL Im XCIm R

XL XCR

tan1 XL XC

R

series ac-RLC graph

ImpedanceThe magnitude of the Total Potential Phasor is

Vm VR2 VLm VCm 2

Im2R

2 ImXL ImXC 2

Im R2 XL XC 2

ImZ

Impedance : Z R2 XL XC 2

Table of definitions

Impedance and reactance

Im VmZVRm

RV Lm

XLVCmXC

Generalized Ohm's Law.

Impedance Z R2 XL XC 2

1tan L CX X

R

Phase Angle between total PD across circuit

and the current

Power Factor

Power is only used in AC circuit in load resistance

P t i t 2

R

(energy is not used in inductor or capacitor)

I m2sin

2 t - R(current is always in phase with PD across total resistance)

P ave = I rms2R I m

2

2R

rms

Z

I rms R rms I rms

RZ rms I rms cos

cos Power Factor

Angular frequency dependence

Power and current depend on angular

frequency of circuit

Power and current depend on angular

frequency of circuit

Max I ; Min ZIm Vm

Z

mZ

Z R2 XL XC 2

R2 L 1

C

2

R2

2LC 1C

2

Z is a minimum when 2 LC 1 0

which occurs when

0 1LC

XL XC

Pave 1

2Im

2 R 1

2

Vm2R

Z 2

1

2

Vm2R

R2 L 1

C

2 1

2

Vm2R

R2 L

2

2 2 0

2 2

1

2

Vm2R

2

R2 2 L

2 2 0

2 2

Power as a function of

ResonanceCircuit uses most power / current

when it is in RESONANCE with applied frequency

Imax and Pave versus

Im Pave

Width of Power curve is a measure of the QUALITY

of the circuitSmall width - High QualitySharpness of response to

external frequency

Quality of circuit

RC Filters IRC Filters

VVoutout

VVinin

Low Pass Filter

VoutVin

ImXCImZ

XCZ

1C

R2 1

C

2

1

RC 2 2 1 then Vout Vin

Vout (as Vin const.)

Low pass Filter

Low Pass Filter

RC Filters

VVoutout

VVinin

RC Filters IIHigh Pass Filter

VoutVin

ImRImZ

RZ R

R2 1

C

2

then Vout Vin Vout (as Vin const.)

High pass Filter

High Pass Filter

Transformers IStep up and Step

down Transformers

Transformers IIV 1 N 1

d Bdt

V 2 N 2

d Bdt

Fluxes are the same

V 2 N 2

N 1

V 1