XL and XC
Robin L. Gooch
a way to replace them with one type of reactance? Does anyone know the math?
John Popelish
>
> When capacitive and inductive reactances are connected in parallel is there
> a way to replace them with one type of reactance? Does anyone know the math?
Reactances are the absolute value of impedances, a way to deal with
these components in a simplified way, while avoiding the extra work of
doing the two dimensional math required when dealing with the full
description. Unfortunately, you loose some actual information when
using the short cut. About all you can do with reactance is predict
the magnitude of current if you know the magnitude of the voltage
across the component, or vice versa. This is essentially ohm's law
for DC components, duct taped to AC components.
The full version requires that you do all the math with two
dimensional numbers (called "complex math" to scare you away). Two
dimensions are necessary, because unlike resistors that only need a
single number to relate instantaneous voltage to instantaneous
current, inductors and capacitors deal not only with the present, but
the past, because they store energy. These properties can be captured
either by keeping track of a magnitude and a phase shift (two
dimensions on a polar coordinate system), or by keeping track of all
magnitudes in two parts that are at right angles (phase shifted by 90
degrees) with respect to each other (cartesian coordinates). Then you
have to learn all the rules of arithmetic over, so that addition,
subtraction, multiplication division, etc. can be performed in a
consistent way, while carrying these components through out the math.
Once you reach this point, the answer to your question about parallel
LC circuits and every other possible connection of those components
with resistors becomes more obvious.
I know all this sounds like learning an alien language including
unpronounceable syllables, but it is really fairly simple, though mind
numbingly tedious. And like any other kind of math, computers handle
it with easy grace. I have several pocket calculators that handle it,
and a program like Mathcad can do problems you would not want to
tackle if you had weeks of free time to devote. But you really need
to find a text book or tutorial on the subject and start at the
beginning.
you might try some web search strings like:
complex arithmetic tutorial
AC circuit analysis tutorial
--
John Popelish
William L. Bahn
XL = wL = 2pifL
XC = 1/(wC) = 1/(2pifC)
Is that what you are using for XL and XC? Sometimes people embed a minus
sign in the XC.
The corresponding impedances are:
ZL = j2pifL = jXL
ZC = 1/(jwC) = -j(1/(2pifC) = -jXC
Where j = sqrt(-1). You can see the minus sign I was referring to.
The beauty of working with impedances is that all of your resistor-based
equations are still valid.
So, for these two elements in parallel, you have:
1/ZT = 1/ZL + 1/ZC
or ZT = (ZL)(ZC)/(ZL+ZC)
Taking the latter:
ZT = (jXL)(-jXC)/(jXL-jXC) = (XL)(XC)/[j(XL-XC)]
ZT = -j(XL)(XC)/(XL-XC)
XT = (XL)(XC)/(XL-XC)
If XT > 0, then you will note that ZT looks like ZC and XT is capacitive.
If XT < 0, then you will note that ZT looks like XL and reactance is the
absolute value of XT and is inductive.
Robin L. Gooch wrote in message ...
Mike Mccarty Sr
disappointed by this post.
I agree with your attempt to make the concept approachable. It really
isn't all that complicated a thing. I think you missed the mark in a few
ways, though.
In article <39D95C51...@rica.net>,
John Popelish <jpop...@rica.net> wrote:
)"Robin L. Gooch" wrote:
)>
)> When capacitive and inductive reactances are connected in parallel is there
)> a way to replace them with one type of reactance? Does anyone know the math?
)
)Reactances are the absolute value of impedances, a way to deal with
)these components in a simplified way, while avoiding the extra work of
)doing the two dimensional math required when dealing with the full
)description. Unfortunately, you loose some actual information when
Untrue. Reactance is the imaginary component of impedance.
)using the short cut. About all you can do with reactance is predict
)the magnitude of current if you know the magnitude of the voltage
)across the component, or vice versa. This is essentially ohm's law
)for DC components, duct taped to AC components.
One cannot even do that, unless the entire impedance is essentially
reactive.
)The full version requires that you do all the math with two
)dimensional numbers (called "complex math" to scare you away). Two
That is not the reason it is called "complex".
)dimensions are necessary, because unlike resistors that only need a
)single number to relate instantaneous voltage to instantaneous
)current, inductors and capacitors deal not only with the present, but
)the past, because they store energy. These properties can be captured
)either by keeping track of a magnitude and a phase shift (two
)dimensions on a polar coordinate system), or by keeping track of all
They do indeed store energy, but that is irrelevant for the use of
impedance. Impedance is a steady-state concept. If you want to take
into account the memory of these devices and compute instantaneous
voltages and currents (as opposed to steady state voltages and
currents) then one needs to use differential equations and find the
particular solutions to them. The concept of impedance is related to
the homogeneous solutions, not the particular solutions, and are limit
case behavior as time grows beyond all bounds, not transient behavior.
Impedance is pretty much useless for transient behavior.
[snip]
)I know all this sounds like learning an alien language including
)unpronounceable syllables, but it is really fairly simple, though mind
)numbingly tedious. And like any other kind of math, computers handle
It sounds either grossly glossed over, or just incorrect to me.
)it with easy grace. I have several pocket calculators that handle it,
)and a program like Mathcad can do problems you would not want to
)tackle if you had weeks of free time to devote. But you really need
)to find a text book or tutorial on the subject and start at the
)beginning.
Excellent advice.
--
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I can explain it for you, but I can't understand it for you.
I don't speak for Alcatel <- They make me say that.
John Popelish
>
> Usually, you are on the mark, John, but I found myself somewhat
> disappointed by this post.
Thanks for the help. It was a rush job.
> I agree with your attempt to make the concept approachable. It really
> isn't all that complicated a thing. I think you missed the mark in a few
> ways, though.
>
> In article <39D95C51...@rica.net>,
> John Popelish <jpop...@rica.net> wrote:
> )"Robin L. Gooch" wrote:
> )>
> )> When capacitive and inductive reactances are connected in parallel is there
> )> a way to replace them with one type of reactance? Does anyone know the math?
> )
> )Reactances are the absolute value of impedances, a way to deal with
> )these components in a simplified way, while avoiding the extra work of
> )doing the two dimensional math required when dealing with the full
> )description. Unfortunately, you loose some actual information when
>
> Untrue. Reactance is the imaginary component of impedance.
Well, perhaps the magnitude of the imaginary component. But only if
you include the sign, which is commonly left out. And aren't both the
same for an ideal inductor or capacitor?
> )using the short cut. About all you can do with reactance is predict
> )the magnitude of current if you know the magnitude of the voltage
> )across the component, or vice versa. This is essentially ohm's law
> )for DC components, duct taped to AC components.
>
> One cannot even do that, unless the entire impedance is essentially
> reactive.
One can do it, if one is willing to be wrong by a factor of 1/ the
square root of two. :) (more if resonance is involved)
> )The full version requires that you do all the math with two
> )dimensional numbers (called "complex math" to scare you away). Two
>
> That is not the reason it is called "complex".
That was a joke.
> )dimensions are necessary, because unlike resistors that only need a
> )single number to relate instantaneous voltage to instantaneous
> )current, inductors and capacitors deal not only with the present, but
> )the past, because they store energy. These properties can be captured
> )either by keeping track of a magnitude and a phase shift (two
> )dimensions on a polar coordinate system), or by keeping track of all
>
> They do indeed store energy, but that is irrelevant for the use of
> impedance. Impedance is a steady-state concept. If you want to take
> into account the memory of these devices and compute instantaneous
> voltages and currents (as opposed to steady state voltages and
> currents) then one needs to use differential equations and find the
> particular solutions to them. The concept of impedance is related to
> the homogeneous solutions, not the particular solutions, and are limit
> case behavior as time grows beyond all bounds, not transient behavior.
> Impedance is pretty much useless for transient behavior.
I mentioned nothing about transient behavior. Phasors (complex
representations of AC quantities) always carry an implication of
continuous, single frequency operation. But the second dimension is
where the energy storage information is kept track of.
> [snip]
>
> )I know all this sounds like learning an alien language including
> )unpronounceable syllables, but it is really fairly simple, though mind
> )numbingly tedious. And like any other kind of math, computers handle
>
> It sounds either grossly glossed over, or just incorrect to me.
My apologies.
> )it with easy grace. I have several pocket calculators that handle it,
> )and a program like Mathcad can do problems you would not want to
> )tackle if you had weeks of free time to devote. But you really need
> )to find a text book or tutorial on the subject and start at the
> )beginning.
>
> Excellent advice.
--
John Popelish