01400_acpower.pdf

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14: Power in AC Circuits


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem

Power Factor Correction

Ideal Transformer

Transformer Applications

Summary

E1.1 Analysis of Circuits (2014-4121) AC Power: 14 1 / 11

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)

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3/103

Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)

Intantaneous Powerdissipated in R: p(t) = v2(t)R

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)50 100 150 200 250

0

0.5

1

t (s)

2


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5/103

Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)50 100 150 200 250

0

0.5

1

t (s)

2


Average Powerdissipated inR:

P = 1

T T

0 p(t)dt

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)50 100 150 200 250

0

0.5

1

t (s)

2



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)50 100 150 200 250

0

0.5

1

t (s)

2



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt =v2

R

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)50 100 150 200 250

0

0.5

1

t (s)

2 v2(t)

mean(v2)



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt =v2

Rv2

is the value ofv2(t)averaged over time

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)

v(t)

VRMS

50 100 150 200 2500

0.5

1

t (s)

2 v2(t)

mean(v2)



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt =v2

Rv2


We define theRMS Voltage(Root Mean Square): Vrms =v2

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)

v(t)

VRMS

50 100 150 200 2500

0.5

1

t (s)

2 v2(t)

mean(v2)



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt =v2

Rv2



The average power dissipated inR isP =v2

R = (Vrms)

2

R

Vrms is the DC voltage that would causeR to dissipate the same power.

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Average Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


50 100 150 200 250-1

0

1

t (s)

v(t)

VRMS

50 100 150 200 2500

0.5

1

t (s)

2 v2(t)

mean(v2)



P = 1

T T

0 p(t)dt = 1

R 1

T T

0 v2

(t)dt =v2

Rv2



The average power dissipated inR isP =v2

R = (Vrms)

2

R

Vrms is the DC voltage that would causeR to dissipate the same power.

We usesmall lettersfor time-varying voltages andcapital lettersfor

time-invariant values.

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)

Cosine Wave: v(t) = 5 cos t. Amplitude isV = 5 V.

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)0 2 4 6 8 10

0

10

20

t (ms)

2


Squared Voltage: v2(t) =V2 cos2 t=V212 +

12cos 2t

Mean Square Voltage: v2= V22 sincecos 2t averages to zero.

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)0 2 4 6 8 10

0

10

20

t (ms)

2

2



12cos 2t

Mean Square Voltage: v2= V22 sincecos 2t averages to zero.

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)0 2 4 6 8 10

0

10

20

t (ms)

2

2



12cos 2t

Mean Square Voltage: v2= V22 sincecos 2t averages to zero.RMS Voltage: Vrms =

v2 = 12

V = 3.54 V

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)0 2 4 6 8 10

0

10

20

t (ms)

2

2



12cos 2t


v2 = 12

V = 3.54 V

Note:Power engineersalwaysuse RMS voltages and currents exclusively

and omit the rms subscript.

For example UK Mains voltage = 230 V rms = 325 V peak.

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Cosine Wave RMS


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-5

0

5

t (ms)0 2 4 6 8 10

0

10

20

t (ms)

2

2



12cos 2t


v2 = 12

V = 3.54 V =VNote:Power engineersalwaysuse RMS voltages and currents exclusively

and omit the rms subscript.

For example UK Mains voltage = 230 V rms = 325 V peak.

In this lecture course only, a ~ overbar means

2: thus

V = 1

2V.

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


Suppose voltage and current phasors are:

V = |V| ejV v(t) = |V| cos(t + V)I= |I| ejI i(t) = |I| cos(t + I)

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)



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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)



Power dissipated in loadZ isp(t) =v(t)i(t) = |V| |I| cos(t + V) cos (t + I)

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)= 12

|V

| |I

|cos(V

I) +

12

|V

| |I

|cos (2t + V + I)

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)

0 2 4 6 8 10

0

0.5

1

1.5

t (ms)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)= 12

|V

| |I

|cos(V

I) +

12

|V

| |I

|cos (2t + V + I)

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)

0 2 4 6 8 10

0

0.5

1

1.5

t (ms)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)= 12

|V

| |I

|cos(V

I) +

12

|V

| |I

|cos (2t + V + I)

Average power: P = 12 |V| |I| cos() where =V I

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)

0 2 4 6 8 10

0

0.5

1

1.5

t (ms)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)= 12

|V

| |I

|cos(V

I) +

12

|V

| |I

|cos (2t + V + I)

Average power: P = 12 |V| |I| cos() where =V I=V I cos() cos is thepower factor

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Power Factor


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


0 2 4 6 8 10-2

0

2

t (ms)

v(t)i(t)

0 2 4 6 8 10

0

0.5

1

1.5

t (ms)




= |V| |I| 12cos(2t + V + I) + 12cos (V I)= 12

|V

| |I

|cos(V

I) +

12

|V

| |I

|cos (2t + V + I)

Average power: P = 12 |V| |I| cos() where =V I=V I cos() cos is thepower factor

>0 alagging power factor(normal case: Current lags Voltage)

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


If V = 12|V| ejV and I= 1

2|I| ejI

Thecomplex powerabsorbed byZ isS=VIwhere * means complex conjugate.

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI


VI= V ejV I ejI

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI


VI= V ejV I ejI = V I ej(VI)

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI


VI= V ejV I ejI = V I ej(VI)=V I ej

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI


VI= V ejV I ejI = V I ej(VI)=V I ej = V I cos +j V I sin

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI



=P+jQ

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI



=P+jQ

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI



=P+jQComplex Power: S=VI =P+jQmeasured inVolt-Amps(VA)

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38/103

Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI



=P+jQComplex Power: S=VI =P+jQmeasured inVolt-Amps(VA)Apparent Power: |S| =

V Imeasured inVolt-Amps(VA)

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI




V Imeasured inVolt-Amps(VA)Average Power: P =

(S)measured inWatts(W)

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



2|I| ejI






Reactive Power: Q= (S)Measured inVolt-Amps Reactive(VAR)

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem

Power Factor Correction Ideal Transformer


Summary



2|I| ejI






Reactive Power: Q= (S)Measured inVolt-Amps Reactive(VAR)Power Factor: cos = cos

V I= P|S|

Machines and transformers have capacity limits and power losses that are

independent ofcos ; their ratings are always given inapparent power.

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Complex Power


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



2|I| ejI






Reactive Power: Q= (S)Measured inVolt-Amps Reactive(VAR)Power Factor: cos = cos

V I= P|S|

Machines and transformers have capacity limits and power losses that are

independent ofcos ; their ratings are always given inapparent power.Power Company:Costs apparent power, Revenue average power.

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary


For any impedance,Z, complex power absorbed: S=VI =P+jQ

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary


For any impedance,Z, complex power absorbed: S=VI =P+jQUsing (a)V =IZ (b)II = I2 we getS= I2 Z=|V|2Z

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



Resistor: S= I2

R=|V

|2

R = 0

Absorbs average power,no VARs(Q= 0)

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



Resistor: S= I2

R=|V

|2

R = 0


Inductor: S=j I2

L =j |V

|2

L = +90

No average power,Absorbs VARs(Q >0)

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



Resistor: S= I2

R=|V

|2

R = 0


Inductor: S=j I2

L =j |V

|2

L = +90


Capacitor:S= j |I

|2

C = j V2

C = 90No average power,Generates VARs(Q

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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



Resistor: S= I2

R=|V

|2

R = 0


Inductor: S=j I2

L =j |V

|2

L = +90


Capacitor:S= j |I

|2

C = j V2


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Power in R, L, C


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



Resistor: S= I2

R=|V

|2

R = 0


Inductor: S=j I2

L =j |V

|2

L = +90


Capacitor:S= j |I

|2

C = j V2


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Tellegens Theorem


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary


Tellegens Theorem:The complex power,S, dissipated in any circuitscomponents sums to zero.

xn=voltage at nodenVb, Ib=voltage/current in branchb

(obeying passive sign convention)

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Tellegens Theorem


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=

1 ifVb starts from noden+1 ifVb ends at noden

0 else

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Tellegens Theorem


Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=


0 elsee.g. branch 4 goes from 2 to 3 a4= [0,1, 1]

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Tellegens Theorem

14: Power in AC Circuits Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=



Branch voltages: Vb =

n abnxn (e.g. V4 =x3 x2)

KCL @ noden: b abnIb = 0 b abnIb = 0

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Tellegens Theorem


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=





KCL @ noden: b abnIb = 0 b abnIb = 0Tellegen:

b VbI

b =

b

n abnxnI

b

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Tellegens Theorem


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=






b VbI

b =

b

n abnxnI

b

=

n

b abnI

b xn

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Tellegens Theorem


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary





abn=






b VbI

b =

b

n abnxnI

b

=

n

b abnI

b xn=

n xn

b abnI

b =

n xn 0

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary


V= 230. Motor modelled as5||7j .I= VR

+ VZL

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL

= 46j32.9 A

P F C i

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL

= 46j32.9 A = 56.5 36

P F C i

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL

= 46j32.9 A = 56.5 36S=VI = 10.6 + j7.6 kVA

P F t C ti

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL

= 46j32.9 A = 56.5 36S=VI = 10.6 + j7.6 kVA = 1336kVA

P F t C ti

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL

= 46j32.9 A = 56.5 36S=VI = 10.6 + j7.6 kVA = 1336kVAcos = P

|S|= cos 36 = 0.81

P F t C ti

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL


|S|= cos 36 = 0.81

Add parallel capacitor of300 F:

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL


|S|= cos 36 = 0.81

Add parallel capacitor of300 F:ZC=

1jC

= 10.6j IC= 21.7j A


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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem



Summary



+ VZL


|S|= cos 36 = 0.81


1jC

= 10.6j IC= 21.7j AI= 46

j11.2 A= 47

14A

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



+ VZL


|S|= cos 36 = 0.81


1jC

= 10.6j IC= 21.7j AI= 46

j11.2 A= 47

14A

SC=VIC= j5 kVAS=VI = 10.6 + j2.6 kVA

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



+ VZL


|S|= cos 36 = 0.81


1jC

= 10.6j IC= 21.7j AI= 46

j11.2 A= 47

14A

SC=VIC= j5 kVAS=VI = 10.6 + j2.6 kVA = 10.914kVAcos = P

|S

|= cos 14 = 0.97

Average power to motor,P, is10.6 kW in both cases.

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



+ VZL


|S|= cos 36 = 0.81


1jC

= 10.6j IC= 21.7j AI= 46

j11.2 A= 47

14A

SC=VIC= j5 kVAS=VI = 10.6 + j2.6 kVA = 10.914kVAcos = P

|S

|= cos 14 = 0.97

Average power to motor,P, is10.6 kW in both cases.I

, reduced from56.5 47 A (16%) lower losses.Effect ofC: VARs= 7.6 2.6 kVAR , power factor= 0.81 0.97.

Ideal Transformer

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Ideal Transformer


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


A transformer has 2windings on the same magneticcore.

Ideal Transformer

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Ideal Transformer


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.

Ideal Transformer

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Ideal Transformer


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.

N1 :N2+ N3 shows the turns ratio between the windings.

The indicates the voltage polarity of each winding.

Ideal Transformer

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Ideal Transformer


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.


The indicates the voltage polarity of each winding.Since is the same for all windings, V1

N1= V2

N2= V3

N3.

Ideal Transformer

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dea a s o e


Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.



N1= V2

N2= V3

N3.

Assume N1I1+ N2I2+ N3I3 = 0

Ideal Transformer

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Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.



N1= V2

N2= V3

N3.

Assume N1I1+ N2I2+ N3I3 = 0These two equations allow you to solve circuits and also

imply that

Si= 0.

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Ideal Transformer

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.



N1= V2

N2= V3

N3.


imply that

Si= 0.

Special Case:

For a 2-winding transformer this simplifies toV2 =

N2N1

V1 andIL = I2 = N1N2 I1

Ideal Transformer

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



Ampres law:

NrIr = lA

;Faradays law: VrNr

= ddt

.



N1= V2

N2= V3

N3.


imply that

Si= 0.

Special Case:

For a 2-winding transformer this simplifies toV2 =

N2N1

V1 andIL = I2 = N1N2 I1

Hence V1I1

=N1N2

2V2IL

=N1N2

2Z

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


Power Transmission

Suppose a power transmission cable has1 resistance.100 kVA@1 kV =100 A I2R= 10 kW losses.100 kVA@100 kV =1 A

I2R= 1 W losses.


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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


Power Transmission


I2R= 1 W losses.

Voltage Conversion

Electronic equipment requires 20 V but mains voltage is240 V .

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


Power Transmission


I2R= 1 W losses.

Voltage Conversion

Electronic equipment requires 20 V but mains voltage is240 V .Interference protection

Microphone on long cable is susceptible to interference from nearby mains

cables. AnN : 1transformer reduces the microphone voltage byN butreduces interference byN2.

Isolation

There is no electrical connection between the windings of a transformer so

circuitry (or people) on one side will not be endangered by a failure that

results in high voltages on the other side.

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Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


Complex Power: S=VI =P+jQwhereV =Vrms = 12V. For an impedanceZ:S=

I2 Z=|V|2Z

Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



I2 Z=|V|2Z

Apparent Power:

|S

|=VI

used for machine ratings.

Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



I2 Z=|V|2Z

Apparent Power:

|S

|=VI


Average Power: P = (S) = V I cos (inWatts) Reactive Power: Q= (S) =

V I sin (inVARs)

Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



I2 Z=|V|2Z

Apparent Power:

|S

|=VI



V I sin (inVARs)

Power engineers alwaysuseV andIand omit the ~.

Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



I2 Z=|V|2Z

Apparent Power:

|S

|=VI



V I sin (inVARs)

Power engineers alwaysuseV andIand omit the ~. Tellegen:In any circuit b Sb= 0 b Pb =b Qb = 0

Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary



I2 Z=|V|2Z

Apparent Power:

|S

|=VI



V I sin (inVARs)

Power engineers alwaysuse

V and

Iand omit the ~.

Tellegen:In any circuit b Sb= 0 b Pb =b Qb = 0 Power Factor Correction: add parallelCto generate extra VARs

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Summary

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Average Power

Cosine Wave RMS

Power Factor

Complex Power

Power in R, L, C

Tellegens Theorem


Ideal Transformer


Summary


I2 Z=|V|2Z

Apparent Power:

|S

|=VI



V I sin (inVARs)

Power engineers alwaysuse

V and

Iand omit the ~.

Tellegen:In any circuit b Sb= 0 b Pb =b Qb = 0 Power Factor Correction: add parallelCto generate extra VARs

Ideal Transformer:Vi Ni and NiIi = 0(implies Si = 0)See Irwin & Nelms Chapter 9.

01400_acpower.pdf

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