Tutorial 6 - Power in AC Circuits

Learning Objective

• To work out the power in a resistive AC circuit.

• To work out power in circuits with reactive elements;

• To understand the power triangle

• To derive and prove the equations.

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 Circuits

In 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:

• The power is always positive.  Negative voltage × negative current = positive power.

• The peak power occurs exactly at the same time as the peak current and peak voltages, both positive and negative.

• The minimum power is 0.

• The power varies at twice the frequency of the current and voltage. 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: Question 1 When does the minimum power occur? Question 2 An alternating voltage has a peak voltage of 16 V and a peak current of 2.0 A.  Calculate the average power. Question 3 The alternating voltage in Question 2 has a frequency of 25 Hz.  Calculate the instantaneous power after 25 ms from t = 0.  Is the power increasing or decreasing? ### Power in a Capacitor Circuit

Let’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:

• There are four power peaks, while there are two voltage and current peaks.

• The power peak is NOT the product of the peak voltage and the peak current.

• The area under the graph is energy (power × time).  The areas are equal above and below the x-axis.  This means that the average power = 0.

 What is the power at the instances I is a maximum, and V is a maximum?  Explain your answer by referring to the variation of the equation for the voltage and current. Question 5 A series CR circuit has a resistor of resistance 33 ohms and a capacitor of capacitance 150 mF connected to a 12 VRMS supply of frequency 200 Hz.    Calculate: a.       The reactance of the capacitor; b.      The impedance of the circuit; c.       The current; d.      The phase angle; e.       The power factor; f.    The power dissipated across the resistor. ### Power in an inductive circuit

Let’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:

• There are four power peaks, while there are two voltage and current peaks.

• The power peak is NOT the product of the peak voltage and the peak current.

• The area under the graph is energy (power × time).  The areas are equal above and below the x-axis.  This means that the average power = 0.

In both reactive circuits, the energy is being fed to the component on the forward half-cycle, and back to the source on the reverse half-cycle. 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.

 Why is there a minus sign in the current equation? Question 7 What is the power at the instances I is a maximum, and V is a maximum?  Explain your answer by referring to the variation equation for the voltage and current. Question 8 Estimate the energy in each half-cycle using the graph above. ### Power in RC Circuits

Consider 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, XC, 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 x-axis.

2.    A small fraction is below the x-axis.

To get the average power, draw a line half way between the crests and troughs of the wave.   The area above the x-axis 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: Question 9 How do we know that the reactance is lagging?  How much does it lag by? ### Power in RL Circuits

Consider 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, XC, 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 x-axis.

• 2.    A small fraction is below the x-axis.

To get the average power, draw a line half way between the crests and troughs of the wave.   The area above the x-axis 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:   Estimate the energy in each half-cycle of voltage using the graph Question 11 Calculate the power factor for the example above.