Electrical circuits lecture 3a

Dr inż. Agnieszka Wardzińska

Room: 105 Polanka

agnieszka.wardzinska@put.poznan.pl

cygnus.et.put.poznan.pl/~award

Adding voltage sources - 1

Adding voltage sources - 2

Adding voltage sources - 3



e(t)=0,



E=0,



U

=0,

Adding voltage sources example

Adding voltage sources example

Adding voltage sources example

Adding sources - 2

Adding sources - 3

Adding current sources - example

Adding current sources - example

Adding current sources - example

Adding current sources - example

Adding current sources - example

Adding current sources - example

j

j

Z

E

I

0

.

2

0

.

1

)

2

(

2

1

2











Power – DC circuit

]

[

]

[

]

[

W

V

A

I

U

P









R

U

I

R

I

U

P

_









power is is additive for any configuration of circuit: series, parallel, series/parallel, or otherwise.

Maximum Power Transfer Theorem



Maximum Power Transfer Theorem states that the

maximum amount of power will be dissipated by a

load if its total resistance R

is equal to the source total

resistance R

of the network supplying power.

For maximum power:

The Maximum Power Transfer Theorem does not assume maximum or even high efficiency, what is more important for AC power distribution.

Example



Calculate the total power of the load. Check the

additivity rule. Calculate R

to get the maximum power

transfer.

Power in AC circuits

Instantaneous electric power

The time varying value of the amplitude of the sinusoidally oscillating magnitude S and doubling the frequency around the mean value P. It is measured in voltampere (VA).



















Power in AC circuits







 



























Power in AC circuits







 











 



 









































Power in AC circuits







 











 



 























Constant in time Varying in time



















Power in AC circuits



Active power or Real power

where is an phase shift between current and voltage.

The average value of power (for the period) actually consumed by the device, able to be processed into another form (eg. mechanical, thermal), this power is always non-negative. It is measured in watt (W).

 



cos

2

1

I

U

P





Power in AC circuits



Active power or Real power for phasors

Z I U ) (   





e

I

I

e

U

U

Power in AC circuits



Active power or Real power for phasors

Z I U ) (   





e

I

I

e

U

U

 

e

I

U

UI

I

U

e

I

U

e

I

e

U

UI

note









:

Power in AC circuits



Active power or Real power for phasors

Z I U ) (   





e

I

I

e

U

U

 

UI

 

U

I

P

Re

2

1

Re

2

1





Power in AC circuits



Active power or Real power for phasors

Z I U ) (   





e

I

I

e

U

U

 

UI

 

U

I

P

Re

2

1

Re

2

1





R

U

R

I

P

|

|

2

1

|

|

2

1





Active Power for arbitrary signal

 

UI

k

 

U

I

k

P



Re



Re



Reactive power

The value a purely contractual linked to periodic changes in the energy

stored in the reactive components (coil, capacitor), this power can be

positive (induction, where ' > 0) or negative (capacitive, when ' < 0). It

is measured in volt-ampere reactive (var).



_{Complex power}

It is proportional to the RMS values of current and voltage, and marked with the letter S. Complex power is formally defined as a complex number in the form of a complex product of the RMS voltage U and coupled current I. It is measured in volt-ampere (VA). The complex power is a complex sum of real and reactive power: *

2

1

I

U

S



jQ

P

S





 

sin



2

1

Im

2

1

I

U

I

U

Q







Apparent power



The power resulting from the amplitude of voltage and current,

including both the active power and reactive power. The apparent

power can be also calculated as the magnitude of complex power S. It is

measured in volt-ampere (VA). We can easy calculate the apparent

power: reactive (var).

or RMS RMS m m

I

U

I

U

I

U

S







2

1

2

1

_Q

P

S





power triangle



We can define the power triangle the trigonometric

form showing the relation appearant power to true

power and reactive power. It is presented below:



The angle between the real and complex power ' is a phase

of voltage relative to current. It mean the angle of

difference (in degrees) between current and voltage. The

ratio between real power and apparent power in a circuit is

called the power factor. It’s a measure of the efficiency of a

power distribution. The power factor is the cosine of the

phase angle ' between the current and voltage cos':

The power factor is by definition a dimensionless and its value is between -1 and 1. When power factor is equal to 0, the energy flow is entirely reactive. When the power factor is 1, all the energy supplied by the source is consumed by the load.

Maximum Power Transfer Theorem for

DC



Maximum Power Transfer Theorem

states that the maximum amount of

power will be dissipated by a load if its

total resistance Rl is equal to the

source total resistance Rs of the

network supplying power.



For maximum power:



The Maximum Power Transfer

Theorem does not assume maximum

or even high efficiency, what is more

important for AC power distribution.

Maximum Power Transfer Theorem for AC



maximum amount of

active power

will be dissipated by a load with

conditions:

R

X

X

R









Z

Z



X

R

Z

X

R

Z







