1

Read: Chapter 10 in Electric Circuits, 9th Edition by Nilsson

Chapter 10 - Power Calculations in AC Circuits

Instantaneous Power: p(t) = v(t)i(t) = instantaneous power

T

-T

lim 1P p(t)dt

T 2T

T

0

1P p(t)dt

T

Average Power:

• Instantaneous power is not commonly used.

• Average power, P, is more useful.

• P = PAVG = PDC = average or real power (in watts, W)

• P can be defined for any waveform as:

• For periodic waveforms with period T

(including sinusoids), P can be expressed as:

1Chapter 10 EGR 272 – Circuit Theory II

Average power can be calculated:

1) by inspection (for simple cases)

2) by integration (for more complex cases)

Example: Find the average power absorbed to the resistor below.

10 v(t)

+

_

i(t)

t [s]

i(t) [A]

10

4

2 4 6 8 10 12 0

2Chapter 10 EGR 272 – Circuit Theory II

Example: Find the average power absorbed to the resistor below.

10v(t)

+

_

i(t)

t [s]

v(t) [V]

20

2 4 6 8 10 1200

3Chapter 10 EGR 272 – Circuit Theory II

RMS Voltage and Current

VRMS = root-mean-square voltage (also sometimes called Veff = effective

voltage)

VRMS = the square root of the average value of the function squared

T

2

R M S

0

1V v (t)dt

T

T

2

R M S

0

1I i (t)dt

T

Note how the name RMS essentially gives the definition.

4Chapter 10 EGR 272 – Circuit Theory II

Calculating average power for a resistive load using RMS values

A key reason that RMS values are used commonly in AC circuits is that they are used in power calculations that are very similar to those used in DC circuits.

Recall that power in DC circuits can be calculated using:

2

2RM S

RM S RM S RM S

VP V I I R

R

2

2VP V I I R

R

Similarly, instantaneous power to a resistor can be calculated using

Show that power can be also be calculated using RMS values as follows:

2

2v (t)p(t) v(t) i(t) i (t) R

R

Development:

5Chapter 10 EGR 272 – Circuit Theory II

Example: Find IRMS for the waveform below and use IRMS to calculate the

average power absorbed to the resistor. Recall that P was calculated earlier

using instantaneous power.

10 v(t)

+

_

i(t)

t [s]

i(t) [A]

10

4

2 4 6 8 10 12 0

RMS values can be calculated:

1) by inspection (for simple cases)

2) by integration (for more complex cases)

6Chapter 10 EGR 272 – Circuit Theory II

10v(t)

+

_

i(t)

t [s]

v(t) [V]

20

2 4 6 8 10 1200

Example: Find VRMS for the waveform below and use VRMS to calculate the

average power absorbed to the resistor. Recall that P was calculated earlier

using instantaneous power.

7Chapter 10 EGR 272 – Circuit Theory II

Development: Derive the expression for VRMS above.

RMS value of sinusoidal voltages and currents

Using the definition of RMS voltage and current, it can be shown that for a sinusoidal voltage v(t) = Vpcos(wt) and a sinusoidal current i(t) = Ipcos(wt) that:

p

R M S p

VV 0 .707V

2

p

R M S p

II 0 .707I

2

Example: Find the power absorbed by a 10 resistor with v(t) = 20cos(377t) V.

8Chapter 10 EGR 272 – Circuit Theory II

Superposition of Power

Recall that, in general, superposition applies to voltage and current, but not to

power.

+ _

+ _

R

V 1

V 2

i

Applying superposition to find the current i:

i = i1 + i2 or

i = current due to V1 + current due to V2

Show that this leads to: P P1 + P2

9Chapter 10 EGR 272 – Circuit Theory II

T

0

2

T

0

Rdt iT

1 p(t)dt

T

1 P

Development:

Power to a Complex Load

If v(t) = Vm cos(wt) then i(t) = Im cos(wt - )

Useful identities:

cos(wt - ) = cos()cos(wt) + sin()sin(wt)

cos2 (wt) = ½ + ½ cos(2wt)

sin(wt)cos(wt) = ½ sin(2wt)

Zа v(t)

i(t)

+

_

m

m

V V 0 since I = I

ZZ

Ð Ð

Ð

2

2RM Sm m

RM S RM S RM S

VV IP cos ( ) V I cos ( ) cos ( ) I Z cos ( )

2 Z

Show that:

10Chapter 10 EGR 272 – Circuit Theory II

T

0

T

0

v(t)i(t)dtT

1 p(t)dt

T

1 P

Development: