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Re: A question about LCR circuits
Subject: Re: A question about LCR circuits
Date: Mon, 12 May 1997 07:28:06 +0000
From: "John H. Couture" <couturejh-at-worldnet.att-dot-net>
To: Tesla List <tesla-at-pupman-dot-com>
Corrected copy - JC
>Date: Sun, 11 May 1997 23:21:57
>To: Tesla List <tesla-at-pupman-dot-com>
>From: "John H. Couture" <couturejh-at-worldnet.att-dot-net>
>Subject: Re: A question about LCR circuits
>
>At 06:08 AM 5/9/97 +0000, you wrote:
>>Subject: A question about LCR circuits
>> Date: Fri, 9 May 1997 11:22:38 +1200
>> From: "Malcolm Watts" <MALCOLM-at-directorate.wnp.ac.nz>
>>Organization: Wellington Polytechnic, NZ
>> To: tesla-at-pupman-dot-com
>>
>>
>>Hello all,
>> Since this relates to resonant circuits, I thought this
>>might be of interest to list members. I posted a version of this on
>>SCI-TESLA about 5 months ago. I was explaining the significance of Q
>>to some list members and relating it to oscillations in mechanical
>>and electrical systems.
>>
>>FYI,
>> Here is the mathematical derivation for the critical value for
>>Q in a "resonant" system - you will see why I have highlighted
>>resonant in a moment. We start by saying we want to find the value of
>>Q at which the frequency of a system has dropped to 0 Hz. I start
>>with an electrical system since I am familiar with the necessary
>>equations. However, this is true for all systems which can store
>>energy.
>>
>>The natural (unforced) resonant frequency of an LCR circuit is:
>>f = SQRT( 1/LC - R^2/4L^2 )
>>
>>Now equate frequency to 0, square both sides of the equation, and
>>algebraically re-arrange to give: 1/LC = R^2/4L^2
>>
>>Multiply both sides by L^2 gives: L/C = R^2/4
>>
>>[Now introduce Q into the equation using the identity:
>>Q = SQRT(L/C)/R (this is easily derived using Q = wL/R = 1/wCR)
>>Squaring both sides and multiplying each side by R^2 gives:
>>Q^2 x R^2 = L/C]
>>
>>Substituting for L/C in our original equation gives: Q^2 x R^2 = R^2/4
>>
>>Dividing each side by R^2 gives Q^2 = 1/4 which => Q = 1/2 = 0.5
>>(negative square root discarded).
>>
>>This is the value of Q, at and below which the circuit can no longer
>>ring. In other words, a circuit with this value of Q loses energy at
>>the same rate as it can take it up.
>>
>>Think that's all OK but comments welcomed as always,
>>Malcolm
>>
>>----------------------------------------------------------
>
> Malcolm -
>
> I do not believe that Q = sqrt(L/C)/R = cos A where A equals the angle
of the vector parameters R, X, and Z.
>
> Also the R does not equal the DC or AC resistance. The R equals an
effective resistance which is a higher value and difficult to determine.
>
> The Q = tan A = X/R is correct. The impedance of an RCL circuit is
>sqrt(L/C) only if the effective resistance is small enough to be neglected.
>
> The sqrt(L/C)/R = cos A cannot, therefore, equal X/R = tan A or the Q
factor.
>
> When the circuit frequency drops to zero HZ the R is so large that it
cancels out the 1/LC quantity and the Q factor approaches zero. When the
R
is small enough to be neglected the Q factor approches infinity.
>
> The critical damping point is when R = 2 sqrt(L/C) . When R is less
than this value the circuit will oscillate. When greater the circuit
will be
aperiotic or will not oscillate.
>
> The circuit will oscillate with Q below .5 if the R is less than
> 2 sqrt(L/C) and X is less than R/2 .
>
> John Couture
>