Tutorial 6 - Extension

 

Proving Power Equations in AC Circuits

CR Circuits

Consider an alternating voltage across a series CR circuit.  The instantaneous voltage is given by the equation:

 

The current leads the resultant voltage by f rad:

 

We can see this on the graph:

So we can write an expression for the power:

We can rewrite this:

 

 

So we can write this horrid looking expression:

Watch out:

cos(2wt +f) ¹ cos 2wt + cos f

 

This equation consists of two main terms:

1. A sinusoidal term:

    The mean value over 1 cycle is 0.

 

2. A constant term:

So the average power is given by:

Substitute the RMS values where:

 

We get:

It doesn’t take a genius to see the final relationship:

This is true for a capacitative or an inductive circuit, series or parallel.

 

Since the power is only dissipated across the resistive element, we can also say that:

The term cosf is often called the power factor.

 

 

LR Circuits

Consider an alternating voltage across a series CR circuit.  The instantaneous voltage is given by the equation:

The current leads the resultant voltage by f rad:

We can see this on the graph:

 

So we can write an expression for the power:

We can rewrite this:

So we can write this horrid looking expression:

We can tidy it up to give:

So we can now write:

                                                   

Watch out:

cos(2wt +f) ¹ cos 2wt + cos f

 

 

This equation consists of two main terms:

            1 A sinusoidal term:

               

              The mean value over 1 cycle is 0.

 

            2 A constant term:

So the average power is given by:

If we substitute the RMS values where:

we get:

It doesn’t take a genius to see the final relationship:

This is true for a capacitative or an inductive circuit, series or parallel.

 

Since the power is only dissipated across the resistive element, we can also say that:

 

 

The term cosf is often called the power factor.

 

There is not much different between the two derivations except the +f and -f terms.

 

 

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