Circuit theory AC ?

Thread Starter


Joined Jul 21, 2008
I guess this belong's in the abstract forums.

But I am wondering what is the proof behind why kirchcoffs rules still apply to phasors.

What I mean is once you convert the sine voltage/ current functions into phasors. Why does all the laws still hold with working with phasors?

And why can you add phasors and translate back into a sine wave that would be equivalent to do the arithmetic with trig function identities always.

I hope you don't miss understand this question.

I know how to do the math but I don't fully understand the proof why the phasor kirchcoffs rules hold. I see no link between them.

One is a number (phasor ) that represents trig function (sin wave voltage ac ...etc)

The other is the actually equation for voltage , current etc etc...

Note I am looking for proof.
I can kind of see the translation between phasors to trig function and visa versa ( except it is not a one to one corospondences)

Anyway I will stop rambling on.
Thanks for any help.


Joined Mar 6, 2009
Text books devote many pages to the explanation and justification of the use of phasors and sinusoids.

In one case the analysis is in the time domain - sinusoidal maths - differential equations etc. In the other, the analysis is done in the frequency domain - where vectors / phasors & complex numbers are the tools used.

Provided the transformations between the two domains are linear then the principles of circuit analysis hold true in either case.

A key justification is that the derivatives and integrals of time based sinusoidal functions are linear with respect to frequency. This allows the user to deal with inductors and capacitors (energy storage elements). The only things that change are amplitudes and phases - and these are the same irrespective of the domain you work in.

Another key point to remember is that the system being analyzed is assumed to be in steady state.

If you are able to obtain a copy, there is an excellent chapter on this in "Linear Circuits" by Ronald E. Scott [Addison-Wesley 1960]

Thread Starter


Joined Jul 21, 2008
Covers the math of Maxwell's equations simplified to KCF/KVF in certain circumstances, such as closed circuit analysis with a voltage or current source.
Thanks for the google book , but I am fully aware of how to prove Max and KCF/KVF Theorems/laws. I want to know why they still hold in phasor land.

In one case the analysis is in the time domain - sinusoidal maths - differential equations etc. In the other, the analysis is done in the frequency domain - where vectors / phasors & complex numbers are the tools used.
I don't see this as much justification.
Since you are probably trying to make me understand it is like laplace or fourier transforms analogy. But remember some properties don't carry over like multiplication in laplace transforms you need to define convolution first...etc

But I see no reason why the KCF/KVF laws still hold...etc?
We are talking about converting a time function to a complex number.
How does this complex number poses the laws and the properties of the time sin voltage/current function.

For example

3sin(2t+30') + 2sin(2t-15')
the phasors are
3/sqrt(2)<_30' + 2/sqrt(2)<_-15' = 4.64/sqrt(2)<_12.2'

so converting back we get
3sin(2t+30') + 2sin(2t-15') = 4.62sin(2t +12.2') where ' means degrees and
<_ indicate angle

But I don't think we can use this for multipling and dividing sin functions

correct me if I am wrong but
3sin(2t+30') / 2sin(2t-15') or 3sin(2t+30') * 2sin(2t-15') cann't be simplified by converting to phasors and the back

You can only use phasor arithmetic on sums and differences of sin or cos functions of the same Frequency.

For instants sin(2t)sin(2t+90') = sin(4t)/2
But converting to phasors first <_0' <_90' = 1/2<_90'
Then back gives 1/2 sin( 2t + 90' ) = 1/2 cos(2t)

I can prove that addition and subtraction of sin/cos functions hold in phasor translation.

Don't think mult/division do unless I made a stupid mistake above for my counter example for mult. I believe you can do it for the special case where the phase is the same. That would yield that ohms laws still hold in phasor land for resistors.

But I have yet to figure out why the phasor domain can be used for showing ohms law for inductive / capactive reactance part.

It is a phase off by 90 so how does the phasor corospond to the trig ?
When I do it in trig I get
W L I sin(wt +O + 90)/ I sin(wt+O)
How do you get WL out of this I get WL cot(wt+O) ?

Maybe their are just considering magnitudes?

<_90 in phasor land can corrospond to sin(wt+90) where w is any thing so not an inverse type of conversion.

Like laplace or fourier ....
Last edited:


Joined Jul 31, 2008
if i were to guess, what would help here would be the properties of LAPlace and fourier transforms... such as linearity and the conv to multiplication in opposite domains etc... i am sorry i am not that much help but i am pretty certain that the proof will lie in the property that sine and cosine can be described using euler's definition. have you looked more into laplace and fourier transform properties?

Thread Starter


Joined Jul 21, 2008
have you looked more into laplace and fourier transform properties?
Yes , I was a mathematician for many years I taught this stuff

I can prove that translating from sin time domain to phasor frequency domain and back the properties of addition and subtraction carry over provided that we are working with the same frequency sin waves.

The proof is basically to write the sin function in terms of complex expotential functions ...etc

But for mult/div it does not hold since translating to phasor land for the
sin(wt ) /sin(wt + 90) gives <_0' / <_ 90' = <_ -90 => sin(wt -90 )
but clearly sin(wt ) /sin(wt + 90) = tan(wt)

My thing is I know the KVL /KCL hold when dealing with the trig functions.
But why can we use them in phasor land and why can we translate back into sine time domain.

When we have V/I = Z but if V, I where expressed as sin waves won't the Z which is called impedence be a trig function as well?

However in books I have seen I = resistance + reactants j
which this complex number can be written in polar form.
So I guess impedence is only given in the phasor domain.
I know resistance cann't change with time but reactants can change with time so I would think that impedance is a trig function or should vary with time.

Plus the fact what if you had a voltage and current that didn't have the same frequency or phase in your circuit diagram.

Then how would you analysis them by phasors?