I've been working with phasors and s-domain analysis for a little bit
now. My mathematical skills aren't the best and I'm not seeing how
these two analysis techniques are linked together.
I analyzed the simple RC circuit here:

formatting link

Is my math at the end correct in finding the sinusoidal response?
There should be an s^3 term in the denominator?
How do we manage to find frequency response so much easier using
Eulers formulas?
I'm not asking how to do it with Euler's, I'm asking to understand
intuitively how the two are linked. Is there a certain proof out
there that is commonly used to make this "click" ?
Thanks.

----------------------------
--------------------
As a start: [w/(s^2+w^2)] *(1/(sRC+1) does not give the expression that you
have for Vout/Vin (algebra error) in the s domain. Assuming T=RC the time
domain response would be a transient term + the steady state sinusoidal with
a phase shift. The phasor analysis only gives information as to the latter.
For your information
1/(1+Ts)(1+s^2/w^2) transforms to (T*w^2*e^-t/T)/(1+(Tw)^2)
+[w*sin(wt-phi)]/(1+(Tw)^2)^0.5
where phi= arctan (Tw)
Compare the result of the sinusoidal value to the phasor result.
Don Kelly snipped-for-privacy@shawcross.ca
remove the X to answer

n message
Thanks for the reply Don. Can you point out the algebra error for
me?
I used [w/(s^2+w^2)] because this is the Laplace transform of
sin(w*t).
From your answer it looks like I should have used something else
here?

Both the Laplace and the phasor (Euler) approaches are frequency domain
solutions to time domain problems (essentially linear differential
equations) The Laplace works in a complex frequency domain where s= sigma
+jw and leads to a complete solution to problems such as the one you
propose, giving the transient response, as well as steady state DC and AC
responses (including multiple frequencies.
The phasor model considers the steady state response at one particular
frequency corresponding to a single point on the sigma=0 axis.
Consider a series RLC circuit with 0 initial conditions and a voltage
applied at t=0
v(t)=Ri(t) +Ldi(t)/dt +(1/C) integral of i(t) dt and knowing v(t) we can
solve for i(t)
If we use Laplace this results in
V(s)=(R+sL +1/sC)I(s) or I(s)= V(s)/(R+sL+1/SC) and knowing v(t) we can
find V(s) and solve for I(s) THEN convert back to the time domain through
the inverse transform.
We will get a complete solution for all times >t=0.
IF the applied voltage v(t) is a sinusoid, and we are only interested in the
steady state response we can simply substitute jw for s.
V(jw)=(R+jwL +1/jwC)I(jw)
There is a bit more math background here but we don't need to go into it at
present.
The point of both is that it is easier to solve algebraic equations using
complex numbers than it is to solve differential equations- particularly
when you may have simultaneous equations. Neither phasor not Laplace
voltages and currents exist except mathematically but they are convenient.
The phasor approach is easier than the Laplace because you are looking for
only a specific part of the total solution.
In your particular case, you are applying a sinusoid to a circuit which has
a transfer function
Vo/Vi =1/(1+sT) for any actual signal (within limits).
In particular, for a sinusoid of any particular frequency, Vo/Vi=1/(1+jwT)in
steady state.

----------------------------
Thanks for the reply Don. Can you point out the algebra error for
me?
I used [w/(s^2+w^2)] because this is the Laplace transform of
sin(w*t).
From your answer it looks like I should have used something else
here?
What you have done, mathematically correctly, is to expand the denominator
so it is expressed as a cubic. This complicates things as you do need the
roots of this. you already have one root from sT+1 s=-and the others from
s^2+w^2 =(s+jw)(s-jw)
You can then write Vo/Vi =w/[(s+jw)(s-jw)(s+1/T)] where T is RC
You can then use partial fractions to expand this in three terms
K1/(s+jw) +K2/(s-jw) +K3(s+1/T)
K2= the complex conjugate of K1 and K3 is real
Note that K1=w(s+jw)/(s+jw)(s-jw)(s+1/T) evaluated at s=-jw
K3 =w(s^2+w^2) evaluated at s=-1/T
Alternatively you can use
(As+B)/(s^2+w^2) +C(s+1/T) and expand the numerator and solve for A,B C
considering that the terms in s, s^2 are 0 and the other term =w
As for not getting a sinusoid from mathcad- the Laplace will give the
transient as well as the Steady state results. The transient in this case is
exponentially decaying and the steady state is a phase shifted sinusoid.
I'm sorry to not get back to you earlier but my Oct 10 reply was the last I
got out before my mail server went out and it has just come back on.

Dear Professor:
I got the Schaum's Electric Circuits, Basic Electrical
Engineering and Lyshevski's Electromechanical Systems,Electric
Machines and Applied Mechatronics books per your
recommendations. I can understand the first few chapters of
the Schaums books but Lyshevski starts something like "derive
the differential equation for ....." The preface says it is a
textbook for one or two semester graduate classes.
Apparently the rust that has accumulated on my brain over the
last 28 years is much more substantial than the original
estimate. I went to Barns and Nobel and found this book which
is helping to get through the several inches of loose rust
that breaks off easily.
Forgotten Algebra. by Barbara Lee Bleau. ISBN 10: 0 7641
2008-5

formatting link

When I complete the forgotten lessons, hopefully I will begin
to remember a little more about differential equations than
which pretty girls were in the class.
A Devoted Fan
Peace
Dawg

----------------------------
The key block here is ""graduate classes" I am sorry for the reference which
looked good to me.
Go back and look for texts such as Fitzgerald, Kingley and Umans . Basic
machines are covered well- even though the higher level approaches are not.
Another that looks good is Electric machinery by Chapman (at least the part
presented on line looks not too difficult) or one by Charles Gross.
Unfortunately calculus is involved and this is beyond forgotten algebra.

----------------------------
------------
I wrote " You can then write Vo/Vi =w/[(s+jw)(s-jw)(s+1/T)] where T is RC"
Correction:
Vo/Vi =1/(Ts+1) where T=RC"
Vo =Vo/Vi) =Vo [1/(1+sT)]={w/(s^2+w^2)(1+sT)}
so the solution that you get will be for Vo(s) leading to vo(t).

Thanks Don, I'm working my way through this.
Before performing the partial fraction decomposition you wrote s^2+w^2
=3D (s+jw)(s-jw)
I intuitivey think to write (s+w)(s-w) ... Why does the imaginary term
end up in this expression?
David
n message
in
.
froms^2+w^2 =3D(s+jw)(s-jw)

----------------------------
Thanks Don, I'm working my way through this.
Before performing the partial fraction decomposition you wrote s^2+w^2
= (s+jw)(s-jw)
I intuitivey think to write (s+w)(s-w) ... Why does the imaginary term
end up in this expression?
David
------------
(s+w)(s-w) = s^2+ws-ws-w^2 =s^2-w^2
(s+jw)(s-jw) =s^2 +jsw -jsw +w^2 = s^2+w^2
-------
You could avoid complex numbers by using the alternative approach
As+B/(s^2+w^2) but it will be just as much work.
For some development :
The Laplace transform of e^-at =1/(s+a)
so that of e^jwt =1/(s-jw) and that of e^-jwt =1/(s+jw)
consider that sin(wt) = (1/2j) [e^jwt -e^-jwt] and work it out to get the
Laplace transform for
sin(wt)
--
Don Kelly snipped-for-privacy@shawcross.ca
remove the X to answer

Re: Linking s-domain and phasor analysis...
Group: alt.engineering.electrical Date: Thu, Oct 16, 2008, 7:45pm
(EDT-3) From: snipped-for-privacy@yahoo.com (davidd31415)
Thanks Don, I'm working my way through this.
Before performing the partial fraction decomposition you wrote s^2+w^2
=3D (s+jw)(s-jw)
I intuitivey think to write (s+w)(s-w) ... Why does the imaginary term
end up in this expression?
David
----------------------------"davidd31415" wrote
in message
news: snipped-for-privacy@m32g2000hsf.googlegroups.com...
----------------------------"davidd31415" wrote
in
message
news: snipped-for-privacy@u65g2000hsc.googlegroups.com... =
I've been working with phasors ands-domainanalysis for a little bit now.
My mathematical skills aren't the best and I'm not seeing how these two
analysis techniques are linked together.
I analyzed the simple RC circuit here:

formatting link

Is my math at the end correct in finding the sinusoidal response? There
should be ans^3 term in the denominator?
How do we manage to find frequency response so much easier using Eulers
formulas?
I'm not asking how to do it with Euler's, I'm asking to understand
intuitively how the two are linked. Is there a certain proof out there
that is commonly used to make this "click" ?
Thanks.
--------------------
As a start: [w/(s^2+w^2)] *(1/(sRC+1) does not give the expression that
you
have for Vout/Vin (algebra error) in thesdomain.
Thanks for the reply Don. =A0Can you point out the algebra error for me?
I used [w/(s^2+w^2)] because this is the Laplace transform of sin(w*t).
From your answer it looks like I should have used something else here?
What you have done, mathematically correctly, is to expand the
denominator so it is expressed as a cubic. This complicates things as
you do need the roots of this. you already have one root from sT+1
=A0s=3D-and the others froms^2+w^2 =3D(s+jw)(s-jw) You can then write Vo/V=
i
=3Dw/[(s+jw)(s-jw)(s+1/T)] where T is RC You can then use partial
fractions to expand this in three terms K1/(s+jw) +K2/(s-jw) +K3(s+1/T)
K2=3D the complex conjugate of K1 and K3 is real Note that
K1=3Dw(s+jw)/(s+jw)(s-jw)(s+1/T) evaluated ats=3D-jw K3 =3Dw(s^2+w^2)
evaluated ats=3D-1/T
Alternatively you can use
(As+B)/(s^2+w^2) +C(s+1/T) and expand the numerator and solve for A,B C
considering that the terms ins,s^2 are 0 and the other term =3Dw
As for not getting a sinusoid from mathcad- the Laplace will give the
transient as well as the Steady state results. The transient in this
case is exponentially decaying and the steady state is a phase shifted
sinusoid.
I'm sorry to not get back to you earlier but my Oct 10 reply was the
last I got out before my mail server went out and it has just come back
on.

I am a bit reluctant to respond because I think the Ray on Web TV might
really be a Roy in disguise.
Your intuition is wrong. (s+w)(s-w) = s^2 - w^2. Rely less upon your
intuition and more on mathematical accuracy.
Bill

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