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Re: Parallel and Series LCR Circuit Qs
Tesla list wrote:
>
> Original poster: ghub005-at-xtra.co.nz
> It seems that the equivalence between linear voltage and current
> sources only applies to circuits with a non-zero internal impedance.
Zero internal impedance results in a limit, that tends to the correct
value.
> An interesting (and unrelated) observation; although in the general
> case there is no discernable external difference between the
> Thevenin and Norton equivalent circuits, there is an interesting
> difference in the internal source conditions - because the current
> through Z and the power that Z dissipates are different in each
> circuit! e.g. If no load is attached to a linear source then I = 0 inside
> the Thevenin equivalent so no power is dissipated in Z. But in the
> Norton equivent a current flows through Z, so Z is dissipating some
> power (assuming it has a resistive component).
But if you add the power dissipated in the resistor with the power
dissipated in the source (that is negative), both models result in
the same values.
> Here is my reasoning for the circuit.
>
> First transform the series combination of the right-hand inductor and
> the 9 ohm resistor into a parallel combination:
Oops...
Your drawing shows 10 Ohms. With 9 Ohms the result is really 1 Ohm.
> So the circuit is resistive.
It is.
> BTW I got the idea for this transform from a network theorem called
> the two terminal equivalence theorem:
>
> "Any passive linear network can be represented, at any one
> frequency, by either a series combination of a single resistance with
> a single reactance, or a parallel combination of a single
> conductance with a single susceptance"
Correct, as impedances and admittances for sinusoidal signals
can be treated as complex numbers. The equivalence involving Q is
very interesting.
> My TC modelling code just models the secondary as a large network
> (of linear components) and reduces it down to an equivalent circuit
> using the impedance operators. Then it evaluates the response of
> the equivalent circuit across an angular frequency range (in equal
> logarithmic steps). Once I have the numerical data, I just feed it into
> the Matlab graphing program to produce plots of the transfer
> function - including the Bode and Nyquist plots.
A valid method, but this will give you only impedances or admittances.
If you made your model as a ladder network, or a cascade or blocks
with two ports, there are several tricks to quickly obtain transfer
functions too (someone interested?). A general-purpose linear circuit
analyzer is not very difficult to set up too.
> I will, of course, have to develop a more sophisticated model for the
> primary circuit. One that accounts for the non-linearity of the SG.
> Then I will need to add a function that models the coupling between
> the primary and secondary networks.
The coupling is easy. For sinusoidal signals a transformer can be
modeled with the equations:
V1 = jw*L1*I1 + jw*M*I2
V2 = jw*M*I1 + jw*L2*I2
Where V1, V2, I1, and I2 are the input and output voltages and
currents, in phasor notation, L1 and L2 are the inductances,
M is the mutual inductance, w is the frequency in radians/second,
and j is the usual sqrt(-1).
If you add nonlinearities of any kind, forget about phasor analysis.
You will have to use differential equations and numerical integrations.
> Of course all this can be done in PSpice (and probably more
> accurately too), but I am enjoying working through the various
> circuit elements. It is, of course, a very simple (naive?) way of
> analysing a TC. But it is giving me a feel for what happens when the
> component values are changed etc.
>
> I will post my Matlab code, and TC models, when I have some more
> useful results.
It's perfectly possible to do everything that PSpice does in
Matlab (maybe a bit slowly).
Take a look at my somewhat overcomplicated Tesla coil simulator:
ftp://coe.ufrj.br/pub/acmq/teslasim.zip
Antonio Carlos M. de Queiroz