Summing voltages

[_ Six _]

Hey guys,

I'm looking for an easier way to sum two single phase voltages together. Both of different frequency, amplitude and phase angle.

V1+V2 = Asin(wt+a)+Bsin(wt+b) = Csin(wt+c)

I've tried a couple of trig identities (sin(A+B) and derivations of) but it doesn't look like the right answer. I've worked out the periodic time, frequency and phase angle of each signal. I know that the sum of two sine waves (is a sine wave) and that my first wave is 50hz and my second 100Hz with different amplitudes.

Is there a way I can convert the waves into the same frequency to make the maths easier or am I missing something?

It's a homework question which is why I haven't published any figures. I just need a push in the right direction! All of the textbooks only include one phase angle or use the graphical method.

Cheers!

Six

[Logarhythm]

Both of different frequency, amplitude and phase angle.
V1+V2 = Asin(wt+a)+Bsin(wt+b)

I'll try to help a bit more when I've got time to go through some real calculations, as it's a while since I've had to use trig identities to do anything real, but just a couple of initial thoughts that if nothing else should help avoid going down the wrong route with this.
If the frequencies are different then the angular frequency ω is also different for the two waves. I suspect your sum therefore looks more like the following:

V1+V2 = Asin(ω1t+a)+Bsin(ω2t+b).
(ω1 and ω2 are the two angular frequencies - don't think the forum software supports addition of subscript notation!)

Unfortunately this almost certainly makes things more fugly in terms of simple trig identities.
As I say, it's been a while since I've done much using this, so while there may be a way to simplify or rephrase that, it will require a bit of thought and some old-fashioned working out with ye olde pen and paper.

I know that the sum of two sine waves (is a sine wave)

Don't let this lead you astray - it's an easy misconception to acquire, as learning this stuff necessarily requires us all to start at the simpler end of things, but this is only true for certain specific cases. Many people would find this easiest to see visually - image below is a reasonable illustration of the idea, where two simple sines are added and the result is definitely not a single sine wave!
Depending on what you're studying, in due course you may move on to e.g. Fourier series, where you'll find that you can add sine waves to make e.g. a square wave

use the graphical method.

Presume this is a variation on drawing out and summing a vector representation of the signals?
If so, this is almost certainly going to be the easiest way to approach it.
Difficult to say for certain as obviously not familiar with course content / syllabus, but from the sound of things I'd approach it this way initially.
Sorry none of this is really a direct answer to your queries, but hopefully a bit of useful background!

Edit: Found a trig identity for your original formula just after I posted, in case it helps.
Tried representing in plain text here, but having started trying to type it out it's pretty much impossible to show what it actually means. Instead, try table 2 in this handy pdf

[dmills]

Your identity has the same w in both sines, so they are the same frequency, just different phases, if this is what you want then think in terms of simple geometry and polar form.

If you want different frequencies then you might do better to think in terms of polar coordinates, and Euler e^(ix) = cos(x) + i sin(x).

A sin(wt+p) + B sin(Wt+P) = Im [Ae^(i(wt+p)) + Be^(i(Wt+P))]
For cosine just take the real part instead.

For two signals having arbitrarily different frequencies there is no identity that is going to reduce it to less then the sum of two operations, if they are the same frequency with just different amplitude and phase then it is a simple matter of polar geometry to get to a single trig operation.

Where are you trying to end up with this (Is this part of a proof, or are you trying to optimise software, or what)?

This place really needs to support LaTex.

Regards, Dan.

[Hugh Robjohns]

I just love it when the boffins get down and dirty with their equations!

Nicely done gentlemen.

H

[_ Six _]

Thanks for the input gents! I'm studying EEE at HNC if that helps!

Yes, my apologies. I seem to have misrepresented the data. Apparently I'm only required to bung the lot into graphmatica and take measurements but I think that's a bit of a cop out.

So the numbers are fair game....

Two voltages are to ve added together. Determine the resultant waveform V3

V1= 10sin100(pi)t+((pi)/4)

V2 = 5sin(200(pi)t-(2(pi)/3))

That's the only information I've been given.

Here's my attempt at working it out mathematically... There are a few errors but I'll tidy that up once I know I'm headed in the right direction.

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-xpf1/v/t34.0-12/1509302_10153206271527829_910040113687820611_n.jpg?oh=9e9d61eac90c979eff50c63ac08bfcab&oe=553855BF&__gda__=1429789376_d7fbd7ff61c6447dbe64da733fa874cb

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-xft1/v/t34.0-12/11159944_10153206297462829_6584892221551090053_n.jpg?oh=e793f00522a686a704314ef09a9f77b9&oe=553883CF&__gda__=1429751287_1b08c913e6cca330ba64b5ec17ca1bd2

I've never seen this published as an identity but it looks like pythagoras could work and tan-1 gives you the phase angle.

Cheers!
https://fbcdn-sphotos-h-a.akamaihd.net/ ... 12adf48769

[_ Six _]

Your identity has the same w in both sines, so they are the same frequency, just different phases, if this is what you want then think in terms of simple geometry and polar form.

If you want different frequencies then you might do better to think in terms of polar coordinates, and Euler e^(ix) = cos(x) + i sin(x).

A sin(wt+p) + B sin(Wt+P) = Im [Ae^(i(wt+p)) + Be^(i(Wt+P))]
For cosine just take the real part instead.

For two signals having arbitrarily different frequencies there is no identity that is going to reduce it to less then the sum of two operations, if they are the same frequency with just different amplitude and phase then it is a simple matter of polar geometry to get to a single trig operation.

Where are you trying to end up with this (Is this part of a proof, or are you trying to optimise software, or what)?

This place really needs to support LaTex.

Regards, Dan.

Cheers Mr Mills!

The second part of the question was a power equation derivation which I managed okay.. A product rather than the sum of...

I'll have another go at it using logs.

Cheers!

https://fbcdn-sphotos-h-a.akamaihd.net/ ... 29418b14e1

[Logarhythm]

Here's my attempt at working it out mathematically
<snipped to remove several pages of remarkably neat calculations>

Be careful with your trig identities - it does indeed appear that you have two separate values of ω, i.e. ω1=100π and ω2=200π.
Unfortunately this means that the handy identity for Asin(ωt+α)+Bsin(ωt+β) doesn't apply in this situation...

For some reason this thread has gone a bit screwy on my PC - the "quote" and "reply" buttons have disappeared from all the posts (I'm writing this using "reply all" at the bottom), but also can't see the second page of your working. If you can try re-posting it I'll have a look through in full.
(Or alternatively, if everyone else can see both it and the usual forum buttons then presume it's just me having problems, but feel free to PM me a link to the image I can't see!).

[Ellis Sutehall]

Can I remind everyone (yet again) if you're going to be embed images please make them sensible sizes as large images break threads. I've edited this images to remove the embed and just have external links. Thanks.

[Logarhythm]

Thanks Ellis.

Six - could you relink the images if you'd still like input on this please?
Removing the embedded images seems to have fixed the thread, but the links now just seem to give an HTTP403 forbidden error.

[_ Six _]

Hi Logarhythm,

It would appear that the original question was a printing error. The actual question is the sum of two voltages of the SAME frequency! I think I can manage that one okay.

The original question I was attempting to tackle is:

V1= 10sin(100(pi)t + (pi/4))

V2= 5sin(200(pi)t - (2pi/3)

V1+V2=V3 No other information was provided.

I tried it using trig identities and it didn't work out. My answer was 5(root)5sinwt-0.32. I didn't know what to do with omega t because they had different values.

I'm looking at Eulers identity at the moment. I've covered complex numbers so I'm okay with j or i notation. I'll have another go at it. I don't like to be defeated!

We're covering fourier series so it works out well in the grand scheme of things.

Cheers for your input!

My apologies for the attachments error!

[dmills]

If they are the same frequency, then just draw a phasor diagram and all will be made instantly clear (Draw the two vectors, linked end to end, starting at the origin, the answer has the magnitude and phase shift of the point at the end of the second vector, now solve the resulting triangle).

Your solution for the case where the frequencies differ is nonsense, sorry, you cannot do that (But the double angle formula may be worth a look in the particular case of 100 and 200 pi).

It is worth seeking to develop a way to mentally graph these things, makes it real obvious when you have a sign wrong somewhere.

Regards, Dan.

[Logarhythm]

It would appear that the original question was a printing error. The actual question is the sum of two voltages of the SAME frequency!

Makes much more sense based on the formulae they've given you. The one you had just won't work for different values of f/ω - makes it significantly more complicated!

I'm looking at Eulers identity at the moment.

Maths porn alert
I remember deriving this for the first time - remains a particularly favourite bit of maths, and still think it's one of the most beautiful things I've ever seen!
Enjoy it if you can

It is worth seeking to develop a way to mentally graph these things, makes it real obvious when you have a sign wrong somewhere.

Great advice (again) from Dan. Just need to do loads of practice until you start to get a feel for how the maths "behaves" - some find visual representation helpful, although becomes a bit more abstract if/when you have to move into more than three dimensions...(not sure if that applies for EEE though).
Anyway, enjoy the course

[_ Six _]

Cheers Gents!

I'm quoting from Communication Engineering Principals - Otung (P97)

Sinusoids of different frequencies

When two or more sinusoids of different frequencies are added together the resultant waveform is no longer a single sinusoid as in the previous cases. There is no general analytical method for obtaining the resultant waveform in this case

Except: If the sinusoidal signals are harmonically related (ie integer multiples of one of the signals)then the resultant waveform is a periodic signal.

No wonder I was banging my head against the wall!

The maths teacher won't touch it! haha! He said the last time he did anything that heavy was 1972

[Logarhythm]

^^Really glad you posted that little update - I was frustrated at not being able to see a solution immediately (and still not having come up with one, two days later) and figured I must be rustier than I'd realised. Somewhat reassured that no-one else has either!

Dan may be on to something with the double angle formula in the specific case of ω1=100π and ω2=200π, but the phase difference makes it something of a pig. He may alternatively just be trolling to see if there is an idiot out there, perhaps with a username like "Logarhythm", who'll end up spending a chunk of his weekend trying to work it out...

[dmills]

although becomes a bit more abstract if/when you have to move into more than three dimensions...(not sure if that applies for EEE though).

Ooh, finite dimensional vector spaces, fun for all the family.
Configuration and phase spaces are useful for some controls theory stuff which is usually an EE subject.

Interestingly, 3 dimensions is one of the ones for which a division algebra does not exist, which is probably why the affine transforms in a 3 dimensional space are 4 by 4 matrices.

If your maths teacher is that much of a wuss, ask them to show you the maths behind the modulation spectrum of an FM signal sometime, Bessel functions as far as the eye can see.....

I could see how the phase shift could make double angle a bit of a pain, no idea if it works but thought it worth suggesting.

Incidentally www.mathworld.com is a great timesink on a wet evening.

Regards, Dan
(Who only has A level maths from 20 some years back, so more then slightly rusty).

[Logarhythm]

ask them to show you the maths behind the modulation spectrum of an FM signal sometime, Bessel functions as far as the eye can see.....