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Power & Energy










To understand how energy is converted consider a simple circuit with a 'load' * resistor and a D.C. source.
* this could be a heater element, a motor or indeed any component with resistance



DC circuit PE drop across a resistor



Charge loses potential energy(PE) as it moves through the resistor. This electrical PE is transformed mostly into heat energy dissipated in the resistor.

The PE is defined as*:


work = charge x potential difference


W = QV


(Joules) = (Coulombs) x (Volts)



*for more information on this equation see the definition of the volt





charge = (current) x (time current flows)


Q = It


(Coulombs) = (Amps) x (seconds)



therefore, substituting for Q in the work equation,


W = (It)V




W = VIt



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By definition, 'power' is the rate of working and is equal to the work done divided by the time taken.


equation for power



substituting for W

the power equation #2


cancelling the 't'


power equation #3



(Watts) = (Volts) x (Amps)



note: 1 Watt is a rate of working of 1 Joule per second.




The equation for power can be modified if we make substitutions using Ohm's Law.



modified power equation



substituting in the power equation for V ,


modified power equation #2


substituting in the power equation for I,


modified power equation #3



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The Kilowatt-hour (kWh)


A kilowatt-hour is a unit of energy.


By definition, a kilo-watt hour is the amount of energy consumed when a 'rate of working' (power) of 1 kilowatt is used for 1 hour.


conversion of 1 kWh to Joules:


1 kWh = 1 kW x 1 h = 1000 W x 3600 s = 3600000 J


1 kWh = 3.6 x 106 J



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Energy for direct current(D.C) & alternating current (A.C.)


Direct current does not vary with time and it is always in one direction.


On a plot of power against time, D.C. is a horizontal line.


The area under the plot gives the total work done/energy used.

This is simply the product of the constant power( Pconst.)and the time interval that the power is used for( t' ) .



DC power over time



DC power equation



However, for A.C. the situation is more complex.


Here not only does the current value vary, but its direction varies too.


The power through the resistor is given by:


modified power equation #4


But we must take the average of this power over time 't' to calculate the energy/work.


So the energy/work done is given by,


AC power equation #1



The Root Mean Square ( RMS) current is defined as:


definition of RMS current


IRMS is the square root of the average of the current squared.


Therefore energy/work done is given by,


RMS current equation #3


IRMS is the equivalent D.C. current having the same effect on a resistor as the A.C.

Here is a graph of an A.C. sinusoidal waveform:



A.C. sinusoidal waveform




Io is the maximum current

ω is the angular frequency, ω = 2πf ( π pi , f frequency)




AC sinusoidal waveform



RMS current in terms of peak current






AC current squared against time




Recalling the A.C. energy/work done equation,


RMS current equation #3


and substituting for IRMS


AC power equation



AC poer time graph



Therefore at time t' the energy W dissipated in resistor R is given by:


energy at time t  in terms of peak power



note: to avoid confusion between W in equations and W on the graph



P (W) on the graph means power P in watts.

W in equations is the energy/work done




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