AC Review

Discussion D12.2

Passive Circuit Elements

v iR

1i v

R

div L

dt

1i vdt

L

1v idt

C

dvi C

dt

+

-

Rv

+

-

Lv

+

-

Cv

ii

i+

-

Energy stored in the capacitor

The instantaneous power delivered to the capacitor is

( )t tdv

w p t dt C v dt C vdvdt

( )dv

p t vi Cvdt

The energy stored in the capacitor is thus

21( ) joules

2w Cv t

Energy stored in the capacitor

221 ( )( )

2 2

q tw Cv t

C

q CvAssuming the capacitor was uncharged at t = -, and knowing that

represents the energy stored in the electric field established between the two plates of the capacitor. This energy can be retrieved. And, in fact, the word capacitor is derived from this element’s ability (or capacity) to store energy.

Parallel Capacitors

i1i 2i Ni

1C 2C NCv+

-

ii

eqCv+

-

1 1

dvi C

dt 2 2

dvi C

dt N N

dvi C

dt

1 2 1 2N N eq

dv dvi i i i C C C C

dt dt

1

N

eq kk

C C

Thus, the equivalent capacitance of N capacitors in parallel is the sum of the individual capacitances. Capacitors in parallel act like resistors in series.

Series Capacitors

11

1v idt

C

1 21 2

1 1 1 1N

N eq

v v v v idt idtC C C C

1

1 1N

keq kC C

The equivalent capacitance of N series connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitors. Capacitors in series act like resistors in parallel.

DC

1v 2v Nv

1C 2C NC

vi

+ + + -- -

i

eqCv+

-

DC

22

1v idt

C

1N

N

v idtC

Energy stored in an inductor

The instantaneous power delivered to an inductor is

( ) ( )t t

L

diw t p t dt L i dt L idi

dt

( )di

p t vi Lidt

The energy stored in the magnetic field is thus

21( ) ( ) joules

2Lw t Li t

Series Inductors

1 1

div L

dt 2 2

div L

dt N N

div L

dt

1 2 1 2N N eq

di div v v v L L L L

dt dt

1

N

eq kk

L L

The equivalent inductance of series connected inductors is the sum of the individual inductances. Thus, inductances in series combine in the same way as resistors in series.

DC

1L 2L NL

1v 2v Nv+ + + -- -

v

i

i

eqLv+

-

DC

Parallel Inductors

11

1i vdt

L

1 21 2

1 1 1 1N

N eq

i i i i vdt vdtL L L L

1

1 1N

keq kL L

The equivalent inductance of parallel connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

22

1i vdt

L

1N

N

i vdtL

1i 2i Ni

1L 2L NLv+

-

i

i

v+

-

eqLi

Complex Numbers

real

imag

A

x jy A

jAe A

1tany

x

cos sinje j Euler's equation:

cos sinA jA A

2 2A x y

cosx A

siny A

ComplexPlane

measured positivecounter-clockwise

cos sinj j je e e j

Note:

sin cos2

t t

cos sin2

t t

sin sint t

cos cost t

tt

t = 0

0 0

t = 0

ReRe

Im Im

cos

sin

Phasor projectionon the real axis

cos( )t

sin( )t

sin sin cos cos sint t t

cos cos cos sin sint t t

Relationship between sin and cos

Comparing Sinusoids

sin 45 cos 135t t

sin cos2

t t

cos cost t

sin sint t

Note: positive angles are counter-clockwise

cos sin2

t t

cos sin by 90t t leads

cos - sin by 90t t lags

sin t

Re

Im

cos t

-sin t

-cos t

sin t

Re

Im

cos t

45

45

135

cos 45 cos by 45 and sin by 135t t t leads leads

Phasors

jM MX e X X

XM

Recall that when we substituted j te

cancelled out.

We are therefore left with just the phasors

A phasor is a complex number that represents the amplitude and phase of a sinusoid.

( ) j tMi t I e

in the differential equations, the

Impedance

Impedancephasor voltage

phasor current

VZ

I

2 2 2Z R L

R j L

V

I

zR j L Z V

ZI

1tanz

L

R

Note that impedance is a complex number containing a real, or resistive component, and an imaginary, or reactive, component.

Units = ohms

AC

R

LcosMV t

i(t)

+

-

VR

VL

+ -

-

+

R jX Z

resistanceR

reactanceX

Admittance

Admittance1 phasor current

phasor voltage

IY = =

Z V

2 2 2

RG

R L

R j L

V

I

2 2 2

1 R j LG jB

R j L R L

I

Y =V

conductance

Units = siemens

susceptance2 2 2

LB

R L

AC

R

LcosMV t

i(t)

+

-

VR

VL

+ -

-

+

Re

Im

Re

Im

Re

Im

V

V

V

V

V

V

I

I

I

I

I

I

I in phasewith V

I lags V

I leads V

Z1

DC

Z2

Z3 Z4I1

I2V

I

Zin

3 41 2

3 4in

Z Z

Z Z ZZ Z

We see that if we replace Z by R the impedances add like resistances.

Impedances in series add like resistors in series

Impedances in parallel add like resistors in parallel

Voltage DivisionZ1

DCZ2

+

VI

V1

V2

+

-

-1 2

V

IZ Z

1 1V Z I

2 2V Z I

But

Therefore

11

1 2

Z

V VZ Z

22

1 2

Z

V VZ Z

Instantaneous Power

( ) cosM vv t V t

( ) cosM ii t I t

( ) ( ) ( ) cos cosM M v ip t v t i t V I t t

( ) cos cos 22

M Mv i v i

V Ip t t

Note twice the frequency

Average Power

0 0

0 0

1 1( ) cos cos

t T t T

M M v it tP p t dt V I t t dt

T T

0

0

1cos cos 2

2

t TM M

v i v it

V IP t dt

T

2T

1 cos2 M M v iP V I

0v i 90v i Purely resistive circuit Purely reactive circuit

12 M MP V I 1 cos 90 02 M MP V I

Effective or RMS ValuesWe define the effective or rms value of a periodic current (voltage) source to be the dc current (voltage) that delivers the same average power to a resistor.

0

0

2 21( )

t T

eff tP I R i t Rdt

T

0

0

21( )

t T

eff tI i t dt

T

eff rmsI I root-mean-square

0

0

2 21 ( )t Teff

t

V v tP dt

R T R

0

0

21( )

t T

eff tV v t dt

T

eff rmsV V

Effective or RMS Values

( ) cosM vv t V t

1

22

01 1 cos 2 22 22rms M vV V t dt

2 1 1cos cos 22 2 2T

0

0

21( )

t T

rms tV v t dt

T

11 2 2

22

00

1

2 2 2 2 2M

rms M M

VtV V dt V

Using and

Ideal Transformer - Voltage

1 1( )d

v t Ndt

2 2( )d

v t Ndt

1 1 1

2 2 2

dv N Ndt

dv N Ndt

22 1

1

Nv v

N

11

1( )v t dt

N

The input AC voltage, v1, produces a flux

This changing flux through coil 2 induces a voltage, v2 across coil 2

1v 2v

2i1i

+ +

- -2N1NAC Load

Ideal Transformer - Current

12 1

2

Ni i

N

The total mmf applied to core is

NiF

Magnetomotive force, mmf

1 1 2 2N i N i F R

For ideal transformer, the reluctance R is zero.

1 1 2 2N i N i

1v 2v

2i1i

+ +

- -2N1NAC Load

Ideal Transformer - Impedance

11 2

2

N

NV V

Input impedance

2

2L

VZ

I

21 2

1

N

NI I

1v 2v

2i1i

+ +

- -2N1NAC Load

Load impedance

1

1i

VZ

I

2

1

2i L

N

N

Z Z

2L

i n

ZZ 2

1

Nn

NTurns ratio

Ideal Transformer - Power

12 1

2

Ni i

N

Power delivered to primary

P vi

22 1

1

Nv v

N

1 1 1P v i

1v 2v

2i1i

+ +

- -2N1NAC Load

Power delivered to load

2 2 2P v i

2 2 2 1 1 1P v i v i P

Power delivered to an ideal transformer by the source is transferred to the load.

Force on current in a magnetic field( )q v F BForce on moving charge q -- Lorentz force

Current density, j, is the amount of charge passing per unit area per unit time. N = number of charges, q, per unit volume moving with mean velocity, v.

v t

Sj

j Nqv

( )N V q v F B

( ) V F j B

L

dQ Nq Sv ti j S

dt t

V S L

( ) L F i B

i BForce per unit length on a wire is

Rotating Machine

B

i

i

X

Force in

Force out

+-

brushes

commutator

B

i

iX

Force in

Force out

+-

X

r

B

2 cosw r

flux 2 cosA rl B Β

area 2 cosA lw l r

l

d

dt

emf flux 2 cos d rlt t

E s B

emf 2 sin 2 sinabe rl rlt

B B

a

b

Back emf

2 sinabe rl B

Back emf

abe

B

i

i

X

Force in

Force out

+-

brushes

commutator

Armature with four coil loops

XXX

X

abe

N

S

aE

aR

tV

aI

Motor Circuit

t a a aV E I R

a aE K

Power and Torque

d a a dP E I

/a t a aI V E R

d a aK I

/a t a aI V K R

2at ad

a a

KV K

R R