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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