We can find voltages and currents in simple circuits containing resistors and voltage or current sources.
We should examine whether these circuits variables obey the Conservation of Power principle:
since a circuit is a closed system, it should not dissipate or create energy.
For the moment, our approach is to investigate first a resistor circuit's power consumption/creation.
Later, we will prove that because of KVL and KCL all circuits conserve power.
As defined on (Reference), the instantaneous power consumed/created by every circuit element equals the product of its voltage and current.
The total power consumed/created by a circuit equals the sum of each element's power.
P=∑kvkik
P
k
k
vk
ik
Recall that each element's current and voltage must obey the convention that positive current is defined to enter the positive-voltage terminal.
With this convention, a positive value of
vkik
vk
ik
corresponds to consumed power, a negative value to created power.
Because the total power in a circuit must be zero
(
P=0
P
0
), some circuit elements must create power while others consume it.
Consider the simple series circuit should in (Reference).
In performing our calculations, we defined the current
iout
iout
to flow through the positive-voltage terminals of both resistors
and found it to equal
iout=vinR1+R2
iout
vin
R1
R2
.
The voltage across the resistor
R2
R2 is the output voltage and we found it to equal
vout=R2R1+R2vin
vout
R2
R1
R2
vin
.
Consequently, calculating the power for this resistor yields
P2=R2R1+R22vin2
P2
R2
R1
R2
2
vin
2
Consequently, this resistor dissipates power because
P2
P2 is positive.
This result should not be surprising since we showed that the power consumed by any resistor equals either of the following.
v2R
or
i2R
v
2
R
or
i
2
R
(1)
Since resistors are positive-valued,
resistors always dissipate power.
But where does a resistor's power go?
By Conversation of Power, the dissipated power must be absorbed somewhere.
The answer is not directly predicted by circuit theory, but is by physics.
Current flowing through a resistor makes it hot;
its power is dissipated by heat.
A physical wire has a resistance and hence dissipates power (it gets warm just like a resistor in a circuit).
In fact, the resistance of a wire of length
L
L
and cross-sectional area
A
A
is given by
R=ρLA
R
ρ
L
A
The quantity
ρ
ρ
is known as the resistivity and presents the resistance of a unit-length of material constituting the wire.
Most materials have a positive value for
ρ
ρ,
which means the longer the wire, the greater the resistance and thus the power dissipated.
The thicker the wire, the smaller the resistance.
Superconductors have no resistance and hence do not dissipate power.
If a room-temperature superconductor could be found, electric power could be sent through power lines without loss!
Calculate the power consumed/created by the resistor
R1
R1 in our simple circuit example.
The power consumed by the resistor
R1
R1
can be expressed as
(vin−vout)iout=R1R1+R22vin2
vin
vout
iout
R1
R1
R2
2
vin
2
We conclude that both resistors in our example circuit consume power, which points to the voltage source as the producer of power.
The current flowing into the source's positive terminal is
−iout
iout
.
Consequently, the power calculation for the source yields
−(viniout)=−(1R1+R2vin2)
vin
iout
1
R1
R2
vin
2
We conclude that the source provides the power consumed by the resistors, no more, no less.
Confirm that the source produces exactly the total power consumed by both resistors.
1R1+R2vin2=R1R1+R22vin2+R2R1+R22vin2
1
R1
R2
vin
2
R1
R1
R2
2
vin
2
R2
R1
R2
2
vin
2
This result is quite general:
sources produce power and the circuit elements, especially resistors, consume it.
But where do sources get their power?
Again, circuit theory does not model how sources are constructed, but the theory decrees that all sources must be provided energy to work.
