Electrical Engineering - Inductor & Capacitor in Parallel in AC Circuit

Author

Inductor & Capacitor in Parallel in AC Circuit

Chris Barrett

2007-04-17, 9:25 am

If I have the following AC circuit:

.. .. ..
||||--( V )---/\/\/-------((((()-------||------||||
'' '' ''

I can describe it with the following equations

V_L + V_R + V_C = V
L dq^2/dt^2 + R dq/dt + (1/C) q = V

I now have to deal with the following AC circuit:

.---((((()---.
.. | | .. ..
||||--( V )---/\/\/---| |---||------||||
'' | .. | '' ''
'----||------'
''

How do I treat the inductor and capacitor that are in parallel? My guess
is that I have a term representing the inductor and capacitor together,
but I'm not sure. How do I represent this with a differential, or
coupled differential equation?

Thanks for any help.

Salmon Egg

2007-04-17, 5:25 pm

On 4/17/07 6:15 AM, in article WN3Vh.5402$vE1.1504@newsfe24.lga, "Chris
Barrett" <"chrisbarret"@0123456789abcdefghijk113322.none> wrote:

> If I have the following AC circuit:
>
> .. .. ..
> ||||--( V )---/\/\/-------((((()-------||------||||
> '' '' ''
>
> I can describe it with the following equations
>
> V_L + V_R + V_C = V
> L dq^2/dt^2 + R dq/dt + (1/C) q = V
>
> I now have to deal with the following AC circuit:
>
>
> .---((((()---.
> .. | | .. ..
> ||||--( V )---/\/\/---| |---||------||||
> '' | .. | '' ''
> '----||------'
> ''
>
> How do I treat the inductor and capacitor that are in parallel? My guess
> is that I have a term representing the inductor and capacitor together,
> but I'm not sure. How do I represent this with a differential, or
> coupled differential equation?
>
> Thanks for any help.

At any frequency, calculate the impedance. For true (ideal) inductors and
capacitors, that impedance is going to be imaginary. For real ones, the
impedances will be complex. It is the complex impedances that combine for
series and parallel combinations in the same way resistors do at dc. But you
must use complex algebra.

There are operational forms for combining impedances. You start with pL and
1/pC as impedances for inductors and capacitors respectively and p is the
derivative operator. The combination is going to be a quotient of two
polynomials of p.

If you do not understand what I am saying, you are going to have to study
more. See Bode's book on feedback amplifiers.

Bill
-- Fermez le Bush--about two years to go.

Don Kelly

2007-04-18, 3:25 am

"Chris Barrett" <"chrisbarret"@0123456789abcdefghijk113322.none> wrote in
message news:WN3Vh.5402$vE1.1504@newsfe24.lga...
> If I have the following AC circuit:
>
> .. .. ..
> ||||--( V )---/\/\/-------((((()-------||------||||
> '' '' ''
>
> I can describe it with the following equations
>
> V_L + V_R + V_C = V
> L dq^2/dt^2 + R dq/dt + (1/C) q = V
>
> I now have to deal with the following AC circuit:
>
>
> .---((((()---. .. | |
> .. ..
> ||||--( V )---/\/\/---| |---||------||||
> '' | .. | '' ''
> '----||------'
> ''
>
> How do I treat the inductor and capacitor that are in parallel? My guess
> is that I have a term representing the inductor and capacitor together,
> but I'm not sure. How do I represent this with a differential, or coupled
> differential equation?
>
> Thanks for any help.

You have the basic KVL and KCL equations. Use them. It gets messy. In the
case of steady state AC you have the phasor approach which deals with a
frequency domain model rather than a time domain model in that the frequency
domain model leads. through solution of simultaneous linear equations to an
easy evaluation of what you are trying- solution of simultaneous
differential equations.
For transient conditions, it is messier-and the Heaviside operator which
Bill mentions (p=d/dt) is still used for machine modelling although the
closely related Laplace operator is more commonly used for control and
general transient analysis.
These both offer a reduction of simultaneous linear time domain differential
equations to simultaneous frequency domain linear algebraic equations-
allowing, as in the steady state AC analysis, computational advantages .

For your parallel L.C then the Laplace model represents this as sL in
parallel with 1/sC
Compare this to the steady state AC situation where you have jwL in parallel
with 1/jwC

Don Kelly dhky@shawcross.ca
remove the X to answer
----------------------------

Salmon Egg

2007-04-19, 3:25 am

On 4/17/07 9:59 PM, in article qAhVh.89604$DE1.38152@pd7urf2no, "Don Kelly"
<dhky@shaw.ca> wrote:

> For transient conditions, it is messier-and the Heaviside operator which
> Bill mentions (p=d/dt) is still used for machine modelling although the
> closely related Laplace operator is more commonly used for control and
> general transient analysis.

Maybe I am missing something. I always thought that the commonly used
symbols p and s were both derivative operator symbols and pretty much
equivalent. IIRC Heavyside used the symbol p, as did Bode without much
explanation. More modern texts using Laplace transforms used s. Am I missing
something?

I also sat in on a course by Erdelyi in which he had fields (like fields of
numbers) of functions and derivatives intertwined in ways I have forgotten.
It was a formal way of dealing mathematically with Heavyside calculus but
without invoking Laplace transforms. There is some stuff about his methods
in Wikiepedia.

Bill
-- Fermez le Bush--about two years to go.

Don Kelly

2007-04-20, 3:25 am

--

Don Kelly dhky@shawcross.ca
remove the X to answer
----------------------------
"Salmon Egg" <salmonegg@sbcglobal.net> wrote in message
news:C24C2AD5.70F01%salmonegg@sbcglobal.net...
> On 4/17/07 9:59 PM, in article qAhVh.89604$DE1.38152@pd7urf2no, "Don
> Kelly"
> <dhky@shaw.ca> wrote:
>
>
> Maybe I am missing something. I always thought that the commonly used
> symbols p and s were both derivative operator symbols and pretty much
> equivalent. IIRC Heavyside used the symbol p, as did Bode without much
> explanation. More modern texts using Laplace transforms used s. Am I
> missing
> something?
>
> I also sat in on a course by Erdelyi in which he had fields (like fields
> of
> numbers) of functions and derivatives intertwined in ways I have
> forgotten.
> It was a formal way of dealing mathematically with Heavyside calculus but
> without invoking Laplace transforms. There is some stuff about his methods
> in Wikiepedia.
>
> Bill
> -- Fermez le Bush--about two years to go.
>
>

Don Kelly

2007-04-20, 3:25 am

----------------------------
"Salmon Egg" <salmonegg@sbcglobal.net> wrote in message
news:C24C2AD5.70F01%salmonegg@sbcglobal.net...
> On 4/17/07 9:59 PM, in article qAhVh.89604$DE1.38152@pd7urf2no, "Don
> Kelly"
> <dhky@shaw.ca> wrote:
>
>
> Maybe I am missing something. I always thought that the commonly used
> symbols p and s were both derivative operator symbols and pretty much
> equivalent. IIRC Heavyside used the symbol p, as did Bode without much
> explanation. More modern texts using Laplace transforms used s. Am I
> missing
> something?
>
> I also sat in on a course by Erdelyi in which he had fields (like fields
> of
> numbers) of functions and derivatives intertwined in ways I have
> forgotten.
> It was a formal way of dealing mathematically with Heavyside calculus but
> without invoking Laplace transforms. There is some stuff about his methods
> in Wikiepedia.
>
> Bill
> -- Fermez le Bush--about two years to go.
>

I don't think that you are missing anything. The Heaviside and Laplace
operators are both derivative operator symbols. Heaviside was covered well
in a series of articles in either AIEE or the British equivalent (IEE)- a
long time back-40's??. Laplace, for reasons that I knew and now don't
remember caught on while Heaviside didn't. Possibly something to do with
either initial conditions or the inverse transformation. The Heaviside
operator p was in vogue in the late 20's and early 30's where it was used
mainly as a symbolic operator p=d/dt in dealing with machines (particularly
transients in synchronous machines) and is still used in modern machine
texts in that sense (As the equations are generally non-linear- that is
about as far as it goes). Bode dates back to that time so that may be why he
also used "p"
Laplace, in engineering applications, appears to have become popular in the
'50's and was well suited to dealing with transients in general. Both
Heaviside and Laplace could be used for transfer functions or dealing with
characteristic equations but, and I may be wrong here, Laplace could handle
steady state situations better and common phasor analysis simply means
walking along the s=jw line in the complex frequency plane (of course it may
be that the mathematicians liked Laplace better).

--

Don Kelly dhky@shawcross.ca
remove the X to answer

Salmon Egg

2007-04-20, 3:25 am

On 4/19/07 8:21 PM, in article OkWVh.100269$aG1.51668@pd7urf3no, "Don Kelly"
<dhky@shaw.ca> wrote:

> I don't think that you are missing anything. The Heaviside and Laplace
> operators are both derivative operator symbols. Heaviside was covered well
> in a series of articles in either AIEE or the British equivalent (IEE)- a
> long time back-40's??. Laplace, for reasons that I knew and now don't
> remember caught on while Heaviside didn't. Possibly something to do with
> either initial conditions or the inverse transformation. The Heaviside
> operator p was in vogue in the late 20's and early 30's where it was used
> mainly as a symbolic operator p=d/dt in dealing with machines (particularly
> transients in synchronous machines) and is still used in modern machine
> texts in that sense (As the equations are generally non-linear- that is
> about as far as it goes). Bode dates back to that time so that may be why he
> also used "p"
> Laplace, in engineering applications, appears to have become popular in the
> '50's and was well suited to dealing with transients in general. Both
> Heaviside and Laplace could be used for transfer functions or dealing with
> characteristic equations but, and I may be wrong here, Laplace could handle
> steady state situations better and common phasor analysis simply means
> walking along the s=jw line in the complex frequency plane (of course it may
> be that the mathematicians liked Laplace better).

It should also be pointed out that Fourier transforms can also be used for
an operational calculus. There, jw is the differential operator. Also, do
not forget that most DE books use D as the differential operator. I learned
operational calculus primarily using Laplace transforms.

Operational methods are mostly useful for linear Des with constant
coefficients. I have worked some problems variable coefficients. In some
cases, taking a transform for a different kind of DE leads to a DE for the
transform of a lower order. Also, often taking a transform of a PDE will get
you an ODE. I have also worked on some problems where you take a double
transform. That seems to be more common for FTs especially when applied to
optical images where two dimensions are involved.

I do not know how far Heavyside went in solving heat problems or diffusion
problems using operational calculus. The classic problem that Heavyside
might have tackled would be for a twisted telephone pair for which series
resistance and distributed shunt capacitance predominate. This leads to a
diffusion equation with terms containing sqrt(p). With modern transform
theory, inverting such functions is relatively easy.

Bill
-- Fermez le Bush--about two years to go.

Don Kelly

2007-04-21, 3:25 am

"Salmon Egg" <salmonegg@sbcglobal.net> wrote in message
news:C24D92DA.71784%salmonegg@sbcglobal.net...
> On 4/19/07 8:21 PM, in article OkWVh.100269$aG1.51668@pd7urf3no, "Don
> Kelly"
> <dhky@shaw.ca> wrote:
>
>
> It should also be pointed out that Fourier transforms can also be used for
> an operational calculus. There, jw is the differential operator. Also, do
> not forget that most DE books use D as the differential operator. I
> learned
> operational calculus primarily using Laplace transforms.
>
> Operational methods are mostly useful for linear Des with constant
> coefficients. I have worked some problems variable coefficients. In some
> cases, taking a transform for a different kind of DE leads to a DE for the
> transform of a lower order. Also, often taking a transform of a PDE will
> get
> you an ODE. I have also worked on some problems where you take a double
> transform. That seems to be more common for FTs especially when applied to
> optical images where two dimensions are involved.
>
> I do not know how far Heavyside went in solving heat problems or diffusion
> problems using operational calculus. The classic problem that Heavyside
> might have tackled would be for a twisted telephone pair for which series
> resistance and distributed shunt capacitance predominate. This leads to a
> diffusion equation with terms containing sqrt(p). With modern transform
> theory, inverting such functions is relatively easy.
>
> Bill
> -- Fermez le Bush--about two years to go.
>

I really don't know how far Heaviside went. I had a copy of Bode's book at
one time but where it went, along with some others is lost in the past. I
know I passed on the original IEEE publication of Fortescue's symmetrical
component paper to a person who would value it and preserve it and deserved
to have it.
The "p" notation, is still used in many machine texts simply is inherited
from early nomenclature and is not, in fact, a transformation, nor
considered as such, except in cases which can be linearised. The early
nomenclature was in the time that complex number theory had not really made
its mark on circuit analysis - "j" simply treated as a shorthand for a 90
degree phase shift akin to comsideration of vectors in a 2 dimensional
world. ("i" taken granted, "j" at 90 degrees and "k" ignored.
One hell of a lot was pulled into EE education in the '50's -e.g. in '55 I
met Laplace in a graduate math course (and was frustrated in trying to apply
it) and in 57-58 it was in a 3rd year EE text (admittedly without much of
the contour integration material met in '55-later added)
--

Don Kelly dhky@shawcross.ca
remove the X to answer
----------------------------

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