ac power, rms and 3-phase circuits

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ower, RMS and 3-Phase circuits

www.animations.physics.unsw.edu.au/jw/power.html[1/1/2014 10:47:56 AM]

RMS and power in single and three

phase AC circuits

Power in AC circuits, the use of RMS quantities and 3 phase AC including answers

to these questions:

What are RMS values?How can you work out the power developed in an AC circuit?How can you get 680 V dc from a 240 V ac supply just by rectifying?When do you need three phases and why do you need four wires?

This page provides answers to these questions. This is a resource page from Physclips. It is a subsidiary page to the main AC circuitssite. There areeparate pages on RC filters, integrators and differentiators, LC oscillationsand motors and generators.

ower and RMS values

The power p converted in a resistor (ie the rate of conversion of electrical energy to heat)s

p(t) = iv = v2/R = i2R.

We use lower case p(t) because this is the expression for the instantaneous powerat timeUsually, we are interested in the mean power delivered, which is normally written P. P

s the total energy converted in one cycle, divided by the period T of the cycle, so:

n the last line, we have used a standard trigonometrical identity that cos(2A) = 1 -2 sin2A. Now the sinusoidal term averages to zero over any num

f complete cycles, so the integral is simple and we obtain

his last set of equations are useful because they are exactly those normally used for a resistor in DC electricity. However, one must remember that he average power, and V = Vm/2 and I= Im/2. Looking at the integral above, and dividing by R, we see that I is equal to the square root of the m

alue of i2, so I is called the root-mean-square or RMS value. Similarly, V = Vm/2 ~ 0.71*Vmis the RMS value of the voltage.

When talking of AC, RMS values are so commonly used that, unless otherwise stated, you may assume that RMS values are intended*. For instanceormal domestic AC in Australia is 240 Volts AC with frequency 50 Hz. The RMS voltage is 240 volts, so the peak value Vm= V.2 = 340 volts. S

ctive wire goes from +340 volts to -340 volts and back again 50 times per second. (This is the answer to the teaser question at the top of the page:ectification of the 240 V mains can give both + 340 Vdc and -340 Vdc.)

An exception: manufacturers and sellers of HiFi equipment sometimes use peak values rather than RMS values, which makes the equipment seemmore powerful than it is.

ower in a resistor.In a resistor R, the peak power (achieved instantaneously 100 times per second for 50 Hz AC) is Vm2/R = im

2*R. As discussed

bove, the voltage, current and so the power pass through zero volts 100 times per second, so the average power is less than this. The average is exa2 2
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s shown above: P = Vm /2R = V /R.

ower in inductors and capacitors.In ideal inductors and capacitors, a sinusoidal current produces voltages that are respecively 90 ahead and beh

he phase of the current. So if i = Imsin wt, the voltages across the inductor and capacitor are Vmcos wt and -Vmcos wt respectively. Now the integr

os*sin over a whole number of cycles is zero. Consequently, ideal inductors and capacitors do not, on average, take power from the circuit.

hree phase AC

Single phase AC has the advantage that it only requires 2wires. Its disadvantage is seen in the graph at the top of thispage: twice every cycle V goes to zero. If you connect aphototransistor circuit to an oscilloscope, you will see thatfluorescent lights turn off 100 times per second (or 120, ifyou are on 60 Hz supply). What if you need a more evensupply of electricity? One can store energy in capacitors, ofcourse, but with high power circuits this would require big,expensive capacitors. What to do?

An AC generatormay have more than one coil. If it has threecoils, mounted at relative angles of 120, then it will producethree sinusoidal emfs with relative phases of 120, as shownin the upper figure at right. The power delivered to a resistive

load by each of these is proportional to V2. The sum of the

three V2terms is a constant. We saw above that the average

of V2

is half the peak value, so this constant is 1.5 times thepeak amplitude for any one circuit, as is shown in the lowerfigure at right.

Do you need four wires? In principle, no. The sum of thethree V terms is zero so, provided that the loads on eachphase are identical, the currents drawn from the three linesadd to zero. In practice, the current in the neutral wire isusually not quite zero. Further, it should be the same guageas the other wires because, if one of the loads were to fail andform an open circuit, the neutral would carry a currentsimilar to that in the remaining two loads.

The voltage (top) and square of the voltage (bottom) in the three active lines of 3 phase supply.

Go to the main AC circuitssite,RC filters, integrators and differentiatorsLC oscillations, or toMotors and generators.

oe Wolfe/ [email protected], phone 61- 2-9385 4954 (UT + 10, +11 Oct-Mar).

chool of Physics, University of New South Wales, Sydney, Australia.

Joe's scientific home page

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ac power, rms and 3-phase circuits

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