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Re: A question about LCR circuits



Subject: Re: A question about LCR circuits
  Date: Mon, 12 May 1997 07:18:05 +0000
  From: "John H. Couture" <couturejh-at-worldnet.att-dot-net>
    To: Tesla List <tesla-at-pupman-dot-com>


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 = ((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