1

Chapter 31

2

An alternator

tVtV dsin)( max

3

The Great Divide: 60 Hz vs 50 Hz

is an angular frequency. =2f where f is the frequency in Hertz (Hz)

In the US and the rest of the Western Hemisphere, the frequency is 60 Hz=2(60)=377 s-1 or rad/s

In Europe and Asia, 50 Hz or 314 s-1

4

3 Simple Circuits-- Resistor

VR

tIiR

VILet

itR

ViRtViRVbecause

tVVandVV

dR

R

ddR

dR

sin

sinsin,

sin0

max

maxmax

max

This is sometimes called a “resistive load” on the circuit

5

Phasor Diagrams

Phasor Diagrams A method to analyze the

relative phase between voltage and current

Of great concern since power maximizes when voltage and phase are in phase

A convention is that the driving frequency, d , is measured from the +x axis and rotates counterclockwise around the origin

In the case to the right, the phase angle between VR and IR is zero.

6

3 Simple Circuits– Capacitive Load

0

0maxmax

90

sin

90sincos

1

tIiif

tX

Vt

X

Vi

CX

dCc

dC

dC

c

dC

VC

For symmetry reasons, we introduce“capacitive reactance”, XC

tCVdt

dqiLet

qtCVC

qtV

C

qVbecause

tVVandVV

ddc

ddC

dC

cos

sinsin,

sin0

max

maxmax

max

Vmax=ICXC

Capacitive reactance has units of ohms

7

ICE—For capacitors, current leads EMF

If you monitor iC, the current will reach its maximum before the voltage across the capacitor, VC

8

3 Simple Circuits– Inductive Load

VL

For symmetry reasons, we introduce “inductive reactance”, XL

LLd

LL

dddL

dd

L

ddL

dL

X

VIandt

X

Vi

ttLXLet

tL

ViLet

dt

dit

L

V

dt

diLtV

dt

diLVbecause

tVVandVV

max0max

00

max

maxmax

max

90sin

)90(90sincos

cos

sinsin,

sin0

Vmax=ILXL

Inductive reactance has units of ohms

9

ELI—For inductors, the EMF leads current

If you monitor iL, the voltage reaches its maximum value before the current in the circuit.

10

ELI the ICEman

For resistors, current and EMF are in phase

For inductors, the EMF leads the current (ELI)

For capacitors, the current leads the EMF (ICE)

ELI the ICEman will always help you from getting killed…

11

RLC Circuit

VR

VC

VLVR

VL-VCdt

12

RLC Phasor

VR

VL-VC

From Pythagorean theorem,

R

XX

V

VV

RXXZ

RXX

VI

RXXIV

VVVV

CL

R

CL

CL

CL

CL

RCL

tan

22

22

2222

222V

13

Impedance, Z

The impedance of the circuit is the effective resistance Has units of ohms Consists of

Total resistance, R Total capacitance, C Total inductance, L Frequency of the electric field, d

V=IZ, amplitude of voltage across AC circuit

14

Something missing?

If there is no resistor, set R=0 and solve for Z and

If there is no inductance, set L=0 and solve for Z and

If there is no capacitor, set C=∞, and solve for Z and

If DC then Z=R

15

Resonance

Natural frequency of swing is=1/(LC)

Driving Frequency, d

When the driving frequency is equal to the natural frequency, then energy transfer is maximized. d= or XC= XL

16

Power in AC circuits P=i2R where i=I sin(dt-)

P=I2R* sin2(dt-) Since P=i2R, then average power is

P=i2R/2 and if R is constant, Irms=I/sqrt(2) Or Vrms= V/sqrt(2) RMS means root mean square value

Typical AC voltage values measured by voltmeter

Pav=IrmsVrms cos For pure resistive load, For pure inductive load,

For pure capacitive load,

For RLC load, cosR/Z

17

Transformers

A method of stepping up or stepping down the voltage or current in an AC circuit

A wire is wrapped around an iron core, the current in the wire induces a magnetic flux in the iron core

A second wire is wrapped around the core. The flux in the core is then induces EM in the second wire.

Symbol

18

Transformers cont’d

21

221

1

1

21

2

11

2

2

112

1

2

2

1

1

2

2211

21

1

212

1

2

1

2

N

N

R

VI

N

N

R

V

N

NI

R

V

N

NII

N

N

I

I

V

V

IVIV

PP

N

NVV

N

N

V

V

1 is called “Primary” and 2 is called “Secondary”

The current, I1 is the current drawn by a load, R, placed on transformer which has a turns ratio of N2/N1 with primary voltage V1