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ContentsInitial Problem Statement 2 Narrative 3-9 Notes 10 Appendices 11-14

AC ElectricityHow can you characterise AC electricity for production supply to the distribution grid?

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4AC ElectricityInitial Problem StatementThere are two distinct forms of electrical current

in use. Direct current, or DC, is a current of

constant magnitude that flows in a single

direction. The most familiar source of DC

electricity is provided by batteries. The other

form is alternating current, or AC. AC electricity

differs from DC in that both the magnitude

and the direction of current flow is constantly

changing. This makes it very useful for high

efficiency transmission as this characteristic

allows voltages to be changed using a simple

transformer. Transforming to a high voltage

supply reduces power losses as the electricity is

transmitted.

The frequency with which the change in

magnitude and direction occur is an important

parameter and power stations must ensure that

they produce electricity of the same frequency

as the electricity already on the grid. They must

therefore ensure that their generator turbines

spin at an appropriate speed as this determines

the frequency. In addition the AC electricity

also has to be synchronised to exactly the

same phase as that already on the grid. This

means that the peak voltage from the generated

electricity must occur at the same time as the

peak voltage on the grid.

How can you characterise AC electricity for production supply to the distribution grid?

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NarrativeIntroductionThe instantaneous voltage, V, between two terminals delivering a domestic AC supply varies sinusoidally with time. The sine curve repeats every 0.02 seconds. The peak voltage is called V

0 and

has a value of 325 V.

MultimediaAC Electricity Interactive 1 is available to demonstrate the frequency and amplitude of a sinusoidally varying wave.

Activity 1Sketch and label a graph showing a typical variation of V with time.

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0

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0 0.00 5 0.01 0.01 5 0.02 0.02 5 0.03 0.035 0.04 0.04 5 0.05 0.05 5 0.06Time (s)

Voltage, V (volts)

Figure 1.

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Discussion MultimediaIf the sine wave repeats every 0.02 seconds what is the frequency of the wave, i.e. how many waves would there be in 1 second?

AC Electricity Interactive 2 is available to demonstrate the amplitude, period and frequency of a number of waves.

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2. Finding a mathematical function to describe AC electricity

Activity 2Write a function that might describe the graph we have just sketched. Think about: • Theshapeofthecurve. • Themaximumandminimumvalueofthecurve. • Howtheshapechangeswithtime.

MultimediaAC Electricity Interactive 1 is available to demonstrate the frequency and amplitude of a sinusoidally varying wave.

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3. What happens if you double or halve the angle in the function?In many engineering problems it is better to use radians as an angular measure than degrees. As there are 2π radians in a circle you can see that the graph of sinθ will repeat every 2π where the angle is measured in radians.

Activity 3

Draw a graph of V = 325sinθ, V = 325 12

sin θ and V = 325 2sin θ

for θ large enough to show several cycles. What is the distance between two successive peaks on a graph and how does this change between graphs?

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0.00π0.00π0.00π0.00π 1.00π1.00π1.00π1.00π 2.00π2.00π2.00π2.00π 3.00π3.00π3.00π3.00π 4.00π4.00π4.00π4.00π 5.00π5.00π5.00π5.00πθ (radians)

Voltage, V (volts)

Figure 2.

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4. Introducing the angular frequencyFigure 2 has an x-axis that measures seconds so if θ = t then the period (in seconds) is the same as the distance between the peaks. Looking at the results in Figure 3:

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0.00π0.00π0.00π0.00π 0.50π0.50π0.50π0.50π 1.00π1.00π1.00π1.00π 1.50π1.50π1.50π1.50π 2.00π2.00π2.00π2.00π 2.50π2.50π2.50π2.50π 3.00π3.00π3.00π3.00π 3.50π3.50π3.50π3.50π 4.00π4.00π4.00π4.00π 4.50π4.50π4.50π4.50π 5.00π5.00π5.00π5.00πθ (radians)

Voltage, V (volts) V = 325 sin θ V = 325 sin 2θV = 325 sin ½θ

Figure 3.

For the graph V t= 325sin the period is 2π (s).

For the graph V t= 325 12

sin the period is 4π (s).

For the graph V t= 325 2sin the period is π (s).

Activity 4Find a relationship between the number multiplying t, call it ω, and the spacing between peaks (= period of the wave).

Discussion MultimediaWhat would you have to multiply t by to obtain the peak-to-peak spacing shown in Figure 1?

AC Electricity Interactive 1 is available to demonstrate the frequency and amplitude of a sinusoidally varying wave.

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5. How does the angular frequency relate to the frequency?Consider equation (2) and (5). The equations are:

ftT

= (2)

T = 2πω

(5)

Activity 5Look at equations (2) and (5). Find a relationship between the angular frequency, ω and the frequency, f .

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To learn more about why this is used see “Root mean square voltage and current”

on page 10.

6. Relating the numbers to our domestic supply

DiscussionIn the work above we have used V0=325volts.Howdoesthisrelatetoourdomestic supply of 230 volts?

Thevalueof325voltsisthepeakvoltagesupplied.HoweverinthedesignofitemsthatuseAC,engineers need to consider, amongst other things, how the current will heat the equipment. The heating value is given by an average measure called the “root mean square” or RMS voltage and it is this value that is reported as the domestic supply voltage NOT the peak voltage.

DiscussionWhat do you think average means in this context?

The RMS voltage is given by

V V V or VRMS = ≈0

00

20 707

1 414.

. (7)

Activity 6Check that the RMS voltage given using (7) when V0 = 325 gives an RMS of 230 V for a domestic supply.

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NotesRoot mean square voltage and currentThe heating caused by a current is proportional to the power dissipated in the material that is conducting it. For a purely resistive circuit the power, P, is given by

P I R VR

= =22

where R is the resistance of the circuit. It is seen that P ∝ I ² and P ∝ V ². As these quantities are continuously varying we need to find an average value to give the average heating. In fact because the heating depends on the square of the current or voltage we need to find the average of the squares of the values. This gives the so-called mean squarevalue.However,whentalkingaboutthe voltage of a source it is not common to give its mean square value (which would have units of volt²). Instead the square root is given (which would have the familiar unit of volts). This is called the root mean square value and it gives the effective average voltage to use when calculating power dissipation. It is expected that a DC supply with a voltage equal to the root mean square voltage of an AC supply will provide the same amount of power dissipation.

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Appendix 1using the interactive resources

AC Electricity Interactive 1This resource is available to demonstrate the frequency and amplitude of a sinusoidally varying wave.

The display shows the path traced out by a point on a rotating circle.

Figure 5.

The top red slider allows you to change the speed of rotation. Notice how higher speed means higher frequency and more waves are visible on screen. Changing the frequency does not change the amplitude of the wave.

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Figure 6.

The bottom red slider changes the amplitude of the wave; effectively reducing the radius of the circle used to trace out the path. Changing the amplitude does not change the frequency of the wave.

Figure 7.

There is an additional “pause” button at the bottom of the screen. This freezes the display to allow for discussion.

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AC Electricity Interactive 2This resource is available to demonstrate the amplitude, period and frequency of a number of waves.

Figure 8.

The display shows a wave trace and the user is invited to deduce and enter values for the wave amplitude, period and frequency, giving results to the stated number of decimal places.

The “Check” button allows a user to check that the answers are correct. If one or more answer is incorrect this will also be highlighted, see below – the period has been incorrectly entered as 0.05 rather than 0.04.

Figure 9.

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Appendix 2mathematical coverageUse and apply mathematical modelling to solve engineering problems• Theengineeringproblemisquantifiedusingmathematicalexpressions

Use trigonometry and coordinate geometry to solve engineering problems• Knowthegraphofy = sinx• Usedegreesandradiansandconvertbetweenthem• Solveproblemsinangularmotion

Use algebra to solve engineering problems• Formanalgebraicexpressionfromadescriptioninwords