Tutorial 6  Power in AC Circuits  
Learning Objective 


Key Questions 
How does power vary in a resistive AC circuit? How do we work out power in a capacitor circuit? How do we work out power in an inductor circuit? How do we work out power in a CR circuit? How do we work out power in an LR circuit? What is the power triangle? 

AC_07– Power in AC CircuitsIn a purely resistive circuit, the power is simply the voltage multiplied by the current. This is because the voltage and the current are in phase. We can show this on the graph below:
There are one or two things to note:
For the maximum power, we can write:
The average power is half the peak power. From this we can write:
We know that:
And that:
From DC electricity we can extend the power formula to give:
and:
This works with AC as well, as long as you use the RMS values.
There is no reason why we cannot work at power at a particular instant:
Where:
And:
So we can write:




Power in a Capacitor CircuitLet’s have a look at what happens in an ideal circuit involving just a capacitor. The source is perfect, the wires are perfect, the meters are perfect, and the capacitor is perfect.
We know that power is given by: P = VI
We know also that the voltage varies as:
And that the current in a capacitor varies as:
So it doesn’t take a genius to see that the power at any instant is given by:
We will soon see how this becomes:
The term P is the average power, while V and I are both RMS values. The term cos f is called the power factor.
We can plot the power graph for a capacitor.
Notice the following from the graph:


Power in an inductive circuitLet’s have a look at what happens in an ideal circuit involving just an inductor. The source is perfect, the wires are perfect, the meters are perfect, and the inductor is perfect.
We know that:
P = VI We know also that the voltage varies as:
And that the current varies as:
So it doesn’t take a genius to see that the power at any instant is given by:
We can plot the power graph for an inductor.
Notice the following from the graph:
In both reactive circuits, the energy is being fed to the component on the forward halfcycle, and back to the source on the reverse halfcycle.
Although the average energy transfer is zero, it does NOT mean that energy is not transferred at all. Large amounts of energy are being transferred forwards on the forward half cycle, and backwards on the reverse half cycle.


Power in RC CircuitsConsider this RC circuit.
We have seen how the circuit gives the following phasor diagram.
In real circuits there are always resistive elements (like resistance of wires) that we need to take into account. We represent these as the resistor R. The phasor diagram for the reactance of the capacitor, X_{C}, the resistance, R, and the impedance, Z is shown below.
Here is the graph:
From this graph we can see that: 1. More of the power graph is above the xaxis. 2. A small fraction is below the xaxis.
To get the average power, draw a line half way between the crests and troughs of the wave. The area above the xaxis is positive while the area below is negative. The energy transferred by the resistive components is the positive area – negative area.
The power being dissipated as heat (or more useful energy) across the resistive components can be worked out as:




Power in RL CircuitsConsider this RL circuit:
We have seen how the circuit gives the following phasor diagram.
In real circuits there are always resistive elements (like resistance of wires) that we need to take into account. We represent these as the resistor R. The phasor diagram for the reactance of the capacitor, X_{C}, the resistance, R, and the impedance, Z is shown below.
Here is the graph:
From this graph we can see that:
To get the average power, draw a line half way between the crests and troughs of the wave. The area above the xaxis is positive while the area below is negative. The energy transferred by the resistive components is the positive area – negative area.
The power being dissipated as heat (or more useful energy) across the resistive components can be worked out as:




