# Inductor & Capacitor in Parallel in AC Circuit

Discussion in 'Electrical Engineering' started by Chris Barrett, Apr 17, 2007.

1. ### Chris BarrettGuest

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.

2. ### Don KellyGuest

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
remove the X to answer
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3. ### Don KellyGuest

--

Don Kelly
remove the X to answer
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4. ### Don KellyGuest

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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).

5. ### Don KellyGuest

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)