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(2*wt +f*)
¹ cos 2*wt*
+ 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 cos*f* 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(2*wt +f*)
¹ cos 2*wt*
+ 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 cos*f* is often called
the **power factor**.

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

Back to Tutorial 6