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Re: resistance in an LRC circuit used to calculate time constant

Hi Alfred,

> Original Poster: "Alfred C. Erpel" <alfred-at-erpel-dot-com> 
> 
> Hi Malcolm,
> 
>     thanks for your response and please see comments interspersed below.
> 
> 
> > Original Poster: "Malcolm Watts" <malcolm.watts-at-wnp.ac.nz>
> >
> > Hi Alfred,
> 
> 
> > > Original Poster: "Alfred C. Erpel" <alfred-at-erpel-dot-com>
> > >
> > > Hello,
> > >
> > >
> > >     An LRC circuit has three components of resistance; the internal
> resistance
> > > of the inductor, the internal resistance of the capacitor, and
> resistance in
> > > the wiring connecting the inductor and capacitor.
> > > The resonant frequency of this LRC circuit is 1 / [2 * PI * SQRT(L * C)]
> > > regardless of the total resistance in the circuit.
> 
> 
> > Not strictly true.  In fact it is 1/[2*PI*SQRT{L*C + (R^2/4L^2)}]  or
> > something pretty close. R has to be factored in because... consider
> > the case where it is very large for instance.
> 
> 
>     None of my reference books include the "(R^2/4L^2)" section you cite in
> the equation above. Is this a special case scenario, or does it apply to all
> resonant circuits? Also, in using this formula, would you assign some value
> to the R of the spark gap in the circuit?

Firstly, the formula applies to all. Most people ignore it as being fine 
detail and you can see why if you plug a range of resistance figures 
into (try 0, 200 Ohms, 2kOhms, 2MOhms and see what happens). 
However, it is strictly true. The gap can be treated as a voltage 
source and therefore be ignored (unless one gets very picky). The 
reason is that gap conduction voltage is rather more constant over a 
range of currents (consider a sinusoidal signal)  than the voltage 
across a resistance would be. 
> 
> > > a)   The time constant of a capacitor is C * R.
> > > b)   The time constant of an inductor is L / R.
> 
> 
> > Both of those assume that R is present in the circuit.
> 
>     This seems to be a good assumption in the real world arena of tesla
> coiling.

Indeed - I was pointing out that a pure capacitance doesn't have a 
time constant - R must be present. Nit picking sorry. 
 
> > >     In the context of this resonant circuit, when you calculate the time
> > > constants of each device, how do you figure R?  Is R just the resistance
> > > internal to the device (inductor or capacitor) or do you add up the
> total R for
> > > the circuit (all three components) to determine R for the equations
> above? How
> > > do you account for the R in the circuit external to both devices?
> 
> 
> > R is simply the ESR (effective series resistance) of the resonant
> > circuit and encompasses all circuit resistances suitably modelled as
> > a single resistor.   Note that you can derive the value of Q required
> > for critical damping from the formula given above (turns out to be
> > 0.5).
> 
> I don't know enough to do this derivation yet. If it is easy enough, please
> explain.

It can be most easily quantified by measurement: 

 ESR = SQRT(L/C)/Q 

You know L and C - measure Q (bandwidth measurement) and you 
can calculate ESR. Trying to predict it is a SPICE nightmare. All 
components contribute to it and it varies due to materials, physical 
construction, dimensions, coupling to external onjects and so on.

Regards,
Malcolm