Hello again,
As Carl was saying, we can not always take RMS readings of current and voltage and multiply to get the
power. For sine waves this works if we also know the phase angle but if we dont know the phase angle that
wont help.
Of course here we need to talk about pulsed waves not sine waves anyway. That's because in power
converter circuits we always see pulsed waveforms, and often pulsed waveforms with ramping tops or just
ramping waveforms. Pulsed waveforms with ramping tops in power converters are really the result of an
exponential function, but because they are so short relative to the time constants of the circuit they
look like true ramping top pulse waves so we can almost always get away with approximating them as ramped tops rather than exponential tops. At the very least however, we can delve into the theory assuming
ramping tops which greatly simplifies the math yet still allows us to investigate this behavior when we
want to understand average power and RMS.
We dont have sine waves here, so we start by going back to the definition of the average power and the
true RMS values of ramping top pulsed waves.
First, the ramping top pulsed wave looks like a pulse but instead of a square flat top it ramps either up
or down. So the main body of the pulse is just a pulse, but it has on top a slant that goes either up or
down. Because we only have to deal with one cycle, we can simplify the mathematical definition of this
pulse to a large extent with:
i(t)=(i2-i1)*t/t1+i1
for the current pulse, and:
v(t)=(v2-v1)*t/t1+v1
for the voltage pulse. The valid time t for both of these is 0<=t<=t1 where t1 is the period where the
pulse turns off (goes to zero) and it is assumed to start at t=0. v1 is the amplitude of the voltage
pulse at the start of the pulse (t=0) and v2 is the amplitude at the end of the pulse (t=t1). Similarly,
i1 is the amplitude of the current pulse at t=0 and i2 the amplitude at the end of the pulse at t=t1.
Next we need a few definitions. We'll first look at RMS vs Power and see what we find. We can also look
at average values and see what turns up.
Definition for RMS:
RMS(f)=sqrt(1/Tp*integrate(f^2,t,0,t1))
[LATEX]\[RMS(f)=\sqrt{\frac{1}{Tp}\,\int_{0}^{t1}{f}^{2}dt}\][/LATEX]
Definition for Power:
Power(i,v)=1/Tp*integrate(i*v,t,0,t1)
[LATEX]Power(i,v)=\frac{1}{Tp}\,\int_{0}^{t1}i\,v\,dt[/LATEX]
In these, f, i, and v are all functions of time. f will be either current or voltage here, and i is
current and v is voltage. The pulses always start at t=0 and end at t=t1, and the total period is Tp. It
is also assumed that the pulses are both 'on' for the same time (0 to t1) and off for the same time (t1 to
Tp). This is the common scenario in a power converter. If one pulse happens to be longer than the other,
simply limit it's pulse length to the shorter one's time because there is no power there.
What we'd like to do is compare the attempt to calculate power using RMS values to the actual calculation
of power using the definition of power. We have some assumed waveforms which are the ramping pulsed waves above. Using those waves we have in short:
P1=RMS(i(t))*RMS(v(t))
P2=Power(i(t),v(t))
and we want to set P1=P2 so we can compare the two calculations.
Doing the calculations, we come up with the two powers:
P1=(sqrt(i2^2+i1*i2+i1^2)*t1*sqrt(v2^2+v1*v2+v1^2))/(3*Tp)
P2=((2*i2+i1)*t1*v2+(i2+2*i1)*t1*v1)/(6*Tp)
Just to note, the second one is correct and can be used for these calculations using a scope to measure
the pulse start amplitudes and end amplitudes.
Now we set them equal to each other:
P1=P2
or:
(sqrt(i2^2+i1*i2+i1^2)*t1*sqrt(v2^2+v1*v2+v1^2))/(3*Tp)=((2*i2+i1)*t1*v2+(i2+2*i1)*t1*v1)/(6*Tp)
To simplify this, we just multiply both sides by 6*Tp and then square both sides, etc., until we end up
with this:
4*(i2^2+i1*i2+i1^2)*(v2^2+v1*v2+v1^2)=(2*i2*v2+i1*v2+i2*v1+2*i1*v1)^2
Now we want to compare so we can subtract the left side from both sides and we get:
3*i1^2*v2^2-6*i1*i2*v1*v2+3*i2^2*v1^2=0
and then we factor that to get:
3*(i1*v2-i2*v1)^2=0
then divide both sides by 3 to get:
(i1*v2-i2*v1)^2=0
So the solution is:
i1*v2=i2*v1
This last result is the condition that has to be met in order for the RMS(current) times RMS(voltage) to
equal the true average power calculation. In words this is:
"The current starting amplitude times the voltage ending amplitude has to be equal to the ending current
amplitude times the starting voltage amplitude".
If this condition is not met then it will not do any good to measure the RMS values. But keep in mind
this is for ramping pulsed waves or just ramping waves. If the pulses are just regular rectangular pulses
then there are less stringent conditions. In general though it is not hard to use the algebraic
simplification of the power in pulsed ramping current and voltage waves from above:
P=P2=((2*i2+i1)*t1*v2+(i2+2*i1)*t1*v1)/(6*Tp)
and if one of them is just a flat top pulse then i2=i1 (or v2=v1).
The required measurements are:
i1: the amplitude at the start of the current wave,
i2: the amplitude at the end of the current wave,
v1: the amplitude at the start of the voltage wave,
v2: the amplitude at the end of the voltage wave,
t1: the time of the end of the shortest pulse,
Tp: the total period,
and the start of both pulses are assumed to be at t=0. If this is not the case, make the start equal to the pulse start time that comes last and shorten the other pulse by the same amount that is the difference between the start and t=0. Of course the pulses must overlap at some place or else there would be zero power.
and this has to be done twice, once for input and once for output. The efficiency can then be calculated
by dividing output power over input power as usual.
Just to note again, if one or both of them is a flat top pulse (ratther than ramping) then we just set the
second variable to be equal to the first variable and that takes away the ramping time part. So for
example if we had a 1.5v voltage pulse that was constant then we set v1=1.5 and v2=1.5 so they are both
the same. This is also true if the voltage (or current) is not a wave but is constant. Sometimes the
input and/or output voltage of the converter is quite constant but this formula still covers that.
What else we could do...
We could look for a simple ratio to see if there is a quick way to convert from RMS1*RMS2 into Power.
We could compare multiplying the average values of the waves to the true average power calculation.
This second one might be interesting because a common way used to estimate power is to multiply the
average values of the two waves together. I'll see if i can get to that next, but it's almost the same
idea except we use the definition of average value:
AVG(f)=1/Tp*integrate(f,t,0,t1)
and see what happens when we multiply the average of the current and the average of the voltage.
Simple example:
i1=1, i2=2, v1=2, v2=1, t1=2, Tp=3. This is for a ramping current pulse that ramps up from 1 amp to 2 amps, and the voltage ramps down from 2 amps to 1 amp, the two pulses are 'on' for 2 seconds and off for 1 second so the total period is 3 seconds, and this wave repeats indefinitely. The result is:
P=1.4444 watts
Comments welcome.