Finding Period Of Voltage With Multiple Frequencies

[QwertyXP]

How do you solve questions of the type:

Find the period of the voltage
a) 3cos(2500t) + 4(7500t + pi/2)
b) a generic waveform: a*cos(x*t)+b*cos(y*t + theta)

I saw questions similar to the above while going through a book. I attempted such questions when I was in college, but don't know/remember how to solve them. I also spent quite some time looking it up on google but didn't see anything useful.

Thank you!

 

[NascentOxygen]

How do you solve questions of the type:

Find the period of the voltage
a) 3cos(2500t) + 4(7500t + pi/2)

Try sketching these two sinusoids, then do a graphical summation. Do this for half a dozen cycles at least.

 

[QwertyXP]

I think I have the answer. I plotted lots of graphs on WolframAlpha and it seems to me that the frequency of the resulting waveform (sum of 2 different functions) is the Highest Common Factor of the frequencies of the individual functions.

If the frequencies are not whole numbers, for example, in the case 3cos(2500t) + 4cos(7500t + pi/2) the frequencies would be 2500/(2*pi) and 7500/(2*pi), we would multiply the frequencies by the smallest possible number (call it 'x') that makes them integers. x=pi in this case. The frequencies above then become 1250 and 3750, respectively. Then we would find their HCF (=1250), and divide it by the same number (x). Hence the period of 3cos(2500t) + 4cos(7500t + pi/2) is pi/1250.

It's great to have solved it, but can anybody give an explanation why it works like this?

 

[NascentOxygen]

For the summation to be periodic, its time period τ must contain n complete cycles of f₁ and m complete cycles of f₂, where n,m ∈ ℕ.

 

[Baluncore]

The frequency of the fundamental is the greatest common divisor of the harmonic frequencies present.

Firstly consider integer angular frequencies of a=2500 and b=7500 rad/sec.
Remove all common factors as c = 2500. Then a/c = 1 and b/c = 3.
During a period of 2Pi/c radians, there will be 1 cycle of “wave a” for every 3 cycles of “wave b”.
The fundamental is c = a, with a 3rd harmonic of b.

Secondly consider integer angular frequencies of a=5000 and b=7500 rad/sec.
Remove all common factors as c = 2500. Then a/c = 2 and b/c = 3.
During a period of 2Pi/c radians, there will be 2 cycles of “wave a” for every 3 cycles of “wave b”.
The frequency of the fundamental is therefore again c = 2500 radians per second.

In this case the fundamental c is not actually present since neither a/c, nor b/c is 1.
Wave a is the 2nd harmonic, and wave b the 3rd harmonic of the missing fundamental.
But the combined wave still repeats with the period of the missing fundamental.

In both cases the period of the combined wave fundamental will therefore be 2Pi / c = 0.00251327 seconds.