Re: Average, RMS and Power Factor made easy!

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Terry Fritz" <twftesla-at-uswest-dot-net>

Hi Al,

	I have heard of average power and peak power for various things, but the
root mean square of "power" has no real use.

Cheers,

	Terry

At 04:31 AM 1/11/2001 -0500, you wrote:
>Hi members.  But does not R.M.S. power have significance when used as a
>means of rating various electrical components such as loudspeakers and
>power amplifiers?  And would not these same ideas apply to a Tesla  coil?
> After all, a loudspeaker is a coil within a magnetic field that becomes
>excited when a voltage is applied to it, just like a Tesla coil.  And we
>all know how terrible our sound equipment begins to sound once we have
>exceeded the R.M.S. value of the rated components.      AL.
>
>On Mon, 08 Jan 2001 22:53:14 -0700 "Tesla list" <tesla-at-pupman-dot-com>
>writes:
>> Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>" 
>> <free0076-at-flinders.edu.au>
>> 
>> 
>> Many people seem to have trouble with the ideas of RMS current, RMS
>> voltage, average power and Power Factor. I will take some time here 
>> to
>> explain what it is all about, and hopefully any remaining doubts 
>> will be
>> dispelled forever and we can get back to coiling (one of the finer 
>> points
>> of life! =)
>> 
>> 
>> RMS stands for Root Mean Squared, and it can be
>> applied to anything at all. It will only be useful in electronics to 
>> talk
>> about RMS current and voltages however and average power, however. 
>> RMS
>> energy doesn't make much sense but there would be applications where 
>> you
>> might want the average energy per pulse, for example, and you might 
>> choose
>> RMS rather than average. But I very much doubt that anybody on this 
>> list
>> has a use for talking about any RMS measurements that aren't voltage 
>> or
>> current.
>> 
>> The method of finding RMS is as follows. Say I have five numbers, I 
>> can
>> find the mean quite easily. Just add them and divide by five. Now 
>> lets say
>> I want to find the mean squared value. I first square all 5 numbers, 
>> then
>> I find out what the mean of the new list is. So I have the Mean 
>> Squared
>> value of all 5 numbers. Now lets say that I want to square-root the 
>> mean
>> squared value, I end up with the Root Mean Squared value, or RMS 
>> value (of
>> the 5 numbers). If the mean of the values is equal to 0, ie some 
>> numbers
>> are positive, some negative, and they add up to 0, then the RMS 
>> value
>> equals the standard deviation as found in statistics.
>> 
>> Talking about a list of numbers is fine, but doesn't really apply to 
>> a
>> continuous function of time, such as voltage or current. I will get 
>> to
>> that as soon as I explain why we even use RMS.
>> 
>> Lets consider an AC waveform. Assume that it is periodic, like the 
>> mains
>> voltage. If I connect a heater and turn it on, a certain power is
>> dissipated. If I then use a DC supply and find the voltage that 
>> produces
>> exactly the same average heating (and at 50/60Hz you wouldn't notice 
>> that
>> the power is fluctuating) then I can say that the mains waveform has 
>> the
>> same heating effect as that voltage, and that's the basis of why we 
>> use
>> RMS. If we consider only the _average_ power dissipation, we can 
>> quote it
>> as say 2400W.
>> 
>> So how do we find the DC equivalent voltage? Lets say that I have a
>> resistance hooked up to a voltage or a current source (could be AC 
>> or DC).
>> Then the equation for instantaneous power is (i.e. at any instant in 
>> the
>> cycle):
>> 
>> P = i^2 x R   or   P = v^2 / R
>> 
>> Note that the instantaneous power depends on the current or voltage
>> squared. If we took an average of the power, since we want average 
>> power
>> of course, then we would get either the Mean Squared value of 
>> current,
>> multiplied by the resistance, or the Mean Squared value of voltage, 
>> then
>> divide by the resistance. Either way we get the average power. In a 
>> DC
>> system the Mean Squared value is just the DC value squared, since it 
>> never
>> changes. So we have a way of finding the DC equivalent. If instead 
>> of
>> finding the Mean Squared value of voltage or current and comparing 
>> it to
>> the DC value squared, we can just square root the whole thing. So we 
>> see
>> that the DC equivalent of either a voltage or current waveform is 
>> just the
>> RMS value of that waveform.
>> 
>> Now it should be clear why we talk about RMS voltage or current, we 
>> can
>> use it in Ohm's law, we can use it to find power, we can use it 
>> where we
>> like as if it were a DC circuit. But only if we are talking about
>> resistors. The danger is that many people don't know that when you
>> multiply RMS voltage by RMS current you _only_ get the average power 
>> _if_
>> the voltage and current waveforms are proportional to each other at 
>> each
>> instant in time. In other words, the two waveforms must be exactly 
>> in
>> phase, like in the case of a resistor where V = I x R shows that at 
>> every
>> instant in time V and I must be proportional by the constant R.
>> 
>> If you have inductors and capacitors the current and voltage are not
>> necessarily in phase, one signal lags behind the other. Then the 
>> actual
>> average power will be lower than the value you get by multiplying 
>> RMS
>> current by RMS voltage. It turns out that the formula for real power 
>> in an
>> AC circuit is:
>> 
>> P = V I cos(theta)                 <--- theta is often replaced with 
>> phi
>> 
>> The extra cos(theta) bit is called the power factor, or PF. It can 
>> vary
>> between 1 and -1, where 1 is called unity power factor and 
>> corresponds to
>> a purely resistive load. Why -1? If we were talking about a source 
>> of
>> energy like the wall socket, we might like to talk about a negative 
>> power
>> like -2400W in the heater example, because it is delivering power. 
>> The
>> cos(theta) term can handle that too, since a negative answer comes 
>> out if
>> you use an angle between 90 and 270 degrees. By the way, the angle
>> represents the phase difference. Just imagine the whole cycle being 
>> 360
>> degrees, then a quarter of a cycle out of phase (like a capacitor or
>> inductor) would give 90 degrees, or a PF of 0. Note that in that 
>> case NO
>> power flows (if the capacitor or inductor were perfect with no
>> resistance). Even though we have our regular voltage present across 
>> the
>> thing and a measurable current through it, NO POWER FLOWS!!
>> 
>> Note in the above example you might still get charged for the power 
>> you
>> aren't using... The metering systems that I have actually seen in my 
>> own
>> area won't compensate for the power factor, and I doubt that yours 
>> does
>> either, they just measure the current and charge you for it. So if 
>> you
>> draw 10A and your capacitor isn't getting warm you probably are 
>> still
>> paying for it.
>> 
>> That's what PF correction is all about. If you run your coil at a PF
>> that's less than unity, you draw more current than you need to. In 
>> the
>> case of a coil, it's usually an inductive load so the answer is to 
>> place a
>> capacitor in parallel with the circuit and keep changing the 
>> capacitance
>> until the capacitance best cancels the inductance in your coil 
>> (mainly
>> transformer inductance I suspect). The result is less current drawn 
>> from
>> the mains and maybe a little lower bill. It may be that you can 
>> actually
>> get more power into the coil if you were close to blowing a fuse, 
>> since
>> the current drawn has gone down. It's as if the capacitor supplies 
>> the
>> extra current needed by the inductance of the coil circuit. Remember 
>> that
>> the power factor of the current delivered by the capacitor to the 
>> coil is
>> close to zero and it doesn't actually supply any power, just the 
>> current
>> that the inductance was planning to drain from the wall.
>> 
>> 
>> I hope that many people had a good read and maybe some fun too =)
>> 
>> 
>> Darren Freeman
>> 
>> 
>> PS I'm studying Electrical/Electronic Engineering and everything 
>> I've said
>> I'm 100% confident of..
>> 
>> 
>> 
>
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