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Re: Impedance, Mechanical, Electrical



Original poster: Yurtle Turtle <yurtle_t@xxxxxxxxx>

I think it's fantastic that you are trying to put
complex concepts into laymans terms. I'd like to
suggest one "tweak". I've always thought that to
describe a capacitor as "storing charge" didn't make
sense. It stores energy, not charge. Energy and charge
are different.

Of course, that may confuse them even more, since
we're all taught by "expert" books and professors that
a cap stores "charge".

Adam

--- Tesla list <tesla@xxxxxxxxxx> wrote:

> Original poster: "Jared Dwarshuis"
> <jdwarshuis@xxxxxxxxx>
>
>
> Impedance, Mechanical, Electrical
>
>
>
> I wrote this to help the novice coiler understand
> a rather difficult topic. There is nothing
> familiar or intuitive about the mathematical
> approach to understanding impedance. Brains are
> not "wired from the factory" for grasping second
> order differential equations and imaginary
> numbers. Such stuff takes practice, and faith!
> Fortunately there are tangible,
>
> " hands on"  physical objects that can be related
> to. This will help us to resolve the mystery.
>
>
>
> So why should I care about this topic, where is the
> impetus ?
>
>
>
> Always a valid question!
>
>
>
> Hmmm, well??.
>
>
>
> A Tesla coil should have a power factor correction
> for the  inductive "load"
>
> A Tesla coil might need radio frequency suppression,
> a "choke"
>
> Optimal transfer of energy to the load will require
> careful impedance matching.
>
>
>
> You need to be armed with some theoretical
> weaponry, or none of this will ever make any sense.
>
>
????????.???????????????????????????????????????????????????????????????
>
>
>
> When we have a mass tied to the end of a spring
> we have an object that behaves mathematically
> very much like a series inductor and capacitor.
>
>
>
> Capacitors are like springs. Theoretically you
> could store charge forever just like you could
> keep a spring under tension forever. Both cases
> concern potential energy.
>
>
>
> Inductors are very much like a mass. Lenz law and
> Newton second law of motion are essentially
> equivalent. Both cases concern forms of kinetic
> energy. (energy of motion!)
>
>
>
> Since the lumped model of  LCR and dampened
> mass/spring share the exact same energy
> equations, they also share the same equations for
> impedance. The amplitude equations which describe
> how much mass and spring bob up and down along
> the X axis  are very old, I don't know who first
> wrote them. Here I will demonstrate that the
> familiar impedance formula can actually be
> derived from mass spring amplitude equations.
>
>
>
> M dv/dt + CV +KX = F e(iwt)       (where C is Kg/s)
>
>
>
> L  di/dt + RI  + Q/C = V e(iwt)      (where this C
> is capacitance)
>
>
>
> Let:
>
>
>
> X(t) = A e i(wt-phi)
>
>
>
> Q(t) = Ae i(wt-phi)
>
>
>
> Then:
>
>
>
> M d2/dt2 A e i(wt-phi)  + C d/dt A e i(wt-phi) + K A
> e i(wt-phi) = F e iwt
>
>
>
> Where:  M = Kg   C = Kg/s   K = Kg/ss  F = Newton
>
>
>
> L d2/dt2 A e i(wt-phi)  + R d/dt A e i(wt-phi) + 1/C
> A e i(wt-phi) = V e iwt
>
>
>
> Where:     L = Kg (m/coul)sqrd   R = Kg/s
> (m/coul)sqrd
>
>                 1/C = Kg/ss (m/coul)sqrd  V = N
> (m/coul)
>
>
>
> Equating the real and imaginary parts of the
> previous equations, squaring both sides, using
> the trigonometric identity:   (sine phi) sqrd + (cos
> phi)sqrd = 1
>
>
>
> We get:
>
>
>
> X = F / ((K- M(w)sqrd)sqrd + Csqrd(w)sqrd))^1/2
>
>
>
> Q = V / ((1/C- L(w)sqrd)sqrd + Rsqrd(w)sqrd))^1/2
>
>
>
> (w) can be pulled from the denominator.
>
>
>
> Then:
>
>
>
> X(w) = Velocity = F/ [  ( K/(w) ­ M(w) )sqrd + Csqrd
> ]^1/2
>
>
>
> Q(w) = Current = V/ [  ((w)L ­ 1/(w)C )sqrd + Rsqrd
> ]^1/2
>
>
>
> We recognize impedance in the denominator for both
> systems
>
>
>
> On the surface it may seem a bit odd that X
> corresponds to Q, but in reality both are just
> describing displacements from an equilibrium
> value. In the case of a dampened mass/spring, the
> equilibrium is simply where the device comes to
> rest.
>
>
>
>   So far a lot of math. But what does it all mean??
>
>
>
> Lets look at the easiest part first, namely
> resistance. (named C in the mechanical world.)
>
>
>
> C in the mechanical world has units of kg/s. When
> we push a box across a uniform floor at a
> constant velocity we can describe this with a
> simple force equation.   Force = velocity x
> C    The units are:  Kg m/ ss = m/s x kg/s   (C
> describes how slippery the floor is)
>
>
>
> R in the electrical world is analogous. Voltage
> takes the roll of force. Current becomes velocity
> (moving charge) and R takes the roll of C.
>
>
>
> *C and R are both linear in these models, not
> always true in either the electric or mechanical
> worlds.*
>
>
>
>
?????????????????????????............................................................................................................................................................................................
>
>
>
>
>
>
>
> Now an informal look at reactance!
>
>
>
> Let us imagine a smooth block attached to a
> spring which is at the other end anchored to a
> wall such that our smooth block can slide on a
> slightly rough floor.
>
>
>
> We can always wiggle and jiggle the contraption
> to find exactly when to push, so as to build up
> pushes for maximum jiggle.
>
>
>
> Why is that?
>
>
>
> Well suppose the block is really massive but
> attached to a bitty joke of a spring.
>
>
>
> Question?
>
>
>
> Since it is attached to a bitty spring is it ok to
> give it a good kick?
>
>
>
> Answer!
>
>
>
> No! You will break your foot, nearly all of the
> force is going into moving the large mass
>
>
>
> Ok! Now push the contraption slowly. Did you will
> find that nearly all the force went into the
> collapsing the spring?
>
>
>
> This is the essence of reactance!
>
>
>
> When we found our maximum jiggle, (likely without
> realizing it) we had found the perfect compromise
> point between pushing slowly and giving it a good
> kick. And of course since we are talking about
> different amounts of time between pushes, we are
> actually talking about frequency. (And so it is!,
> reactance is frequency dependant.)
>
>
>
> We have not exhausted the topic, there is also
> the matter of phase angle. Also there are several
> other forms of impedance that we have not even
> talked about. But enough for now.
>
>
> End.
>
> Jared Dwarshuis   w/07
>
>
>




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