explanation of voltage?

Question

If 230 volts is the r.m.s value, then 325.27 volts is the peak value and 207.07 volts is the average value for half the sine wave. So if 207.07 volts is the average value, how come 230 volts is the value that is equivalent to a d.c. voltage. I understand the maths when using the r.m.s voltage with r.m.s current to find the power, but I can't understand why the r.m.s voltage (or current) alone is equivalent to d.c and not the average of half a sine wave.

Answer 1

Rms voltage and current are derived from average power flow in a resistive circuit. The DC power flow to the circuit is V(dc)I(dc), and is constant over time , so the average flow for any time period is V(dc)I(dc). The instantaneous power flow in an AC circuit is V(pk)cosωtI(pk)cosωt, or V(pk)I(pk)cos^2ωt, the average of which over a complete cycle is (1/2)V(pk)I(pk). V(rms) of a sine wave is (1/√2)V(pk) and I(rms) = (1/√)I(pk). Multiplying V(rms) by I(rms) yields (1/2)V(pk)I(pk). Therefore it is the rms value, not the half-wave average value of a sine wave that is equivalent to a constant DC value.

Answer 2

An rms voltage of an AC is that DC voltage which will produce the same power in a component (typically a resistor).

power = V^2/R
so if we apply 10V (AC) to a 50 ohm resistor we should get a heating effect of:
10^2/50 = 2 W,
If we apply 10V (DC) we should get the same result.

If we have a varying voltage (ANY varying voltage and not just a sinusoidal voltage) we have:

average P = average (V^2/R)

assuming that R is constant we get:

average P = {average (V^2)} /R

But for the equivalent DC:
P = V^2 /R

The P's are equal, the R's are equal so
the V^2 in DC is equivalent to
{average (V^2)} in AC.

Taking square roots:
V in DC is equivalent to
sqrt{average (V^2)} in AC,
that is, the square root of the average of the squares of the values, or RMS, root-mean-square. (As an aside, the word "mean" is far more precise and unambiguous than "average".)

This applies to all varying voltages.
An equivalent argument can be made for currents. One would start with P = I^2 *R

In the case of a sinusoidal voltage it can be shown mathematically that:
rms value = peak value/ sqrt2
and the 325.27 and 230 quoted in the question are in the ratio of sqrt2.

Answer 3

For a sinusoidal wave form(which for the mains in the uk is near dead on) rms voltage is close to 1.11 times average voltage.RMS voltage is used because it is the one which is the effective voltage,mathematically it comes from the time average of the area under the graph of sine(t)for the half wave.Thus it is the the equivalent of the dc voltage where the load is resistive.

Answer 4

Because P = IV for both ac and dc hence the rms voltage and rms current would be the equivalent in a dc circuit.

Answer 5

RMS the AREA under the curve, not simple the quantity.