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A Reference for Overdamped Networks (Systems) in Electronics

10/29/2017 3:21 PM

I am a Mathematician and I am looking for some good reference about overdamped networks in electronics (the electrical circuit system). It must have some theory about overdamped networks and some concrete examples which include elementary derivations of the corresponding system of equations governing the time behaviour of the electrical system written in the matrix-vector form:

Lï+Ri+Di=O

where L, R, D are nxn matrices, i is the n-vector of current, for n>1 or (more important to me) very large n.

Some details:

The natural modes of a linear passive time-invariant n-port are
given by the Lagrange’s equilibrium equation in matrix form which may be
obtained by writing Kirchoff’s loop equations for all n dynamically independent loops of the system.
Lï+Ri+Di=O. (1)
The operators L,R and D≠ 0 represent the effective loop inductance, resistance and elastance respectively; and i is an n-vector loop charges (1).
The loop current n-vector is produced by the small initial conditions
prevailing at the time t = 0. No sustained source is applied.

Overdamped system is one in which all three matrices L, R and D are symmetric positive definite and all the eigenvalues λ of the quadratic eigenvalue problem (λ2* L+λ* R+D)x=0, x≠ 0, are real and negative.

One good workbook is

Dare A. Wells, Schaum’s Outline of Theory and Problems of Lagrangian Dynamics, 1967., Chapter 15,

but I need a book with the corresponding theory about overdamped networks and some examples like in the mentioned Schaum’s Outline.

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#1

Re: A reference for Overdamped Networks (systems) in Electronics

10/29/2017 3:52 PM
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#4

Re: A reference for Overdamped Networks (systems) in Electronics

10/30/2017 9:15 AM

I have seen the proposed pdf, but I need differential equations of the second order; something like (6) in Coupled Spring-Mass Systems and below in matrix formulation in the above pdf, p. 534 (this is example without the dampers); more precisely:

I need some examples like this one in the pictures (this is the book in which there are no further reference):,

And some theory about overdamped systems must be involved in the desidered reference...

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#8

Re: A reference for Overdamped Networks (systems) in Electronics

10/30/2017 6:24 PM

I figured out how to obtain the system of diff. equations of the second order in the current ι(t) (by using dQ/dt=ι(t), where Q is the charge) in 536-537 p. of Gustafson's pdf , but the corresponding matrices are not positive definite ( there are no capacitors in these examples), which means that these two examples are not the examples of overdamped electrcal circuits...

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#10

Re: A reference for Overdamped Networks (systems) in Electronics

10/31/2017 10:25 AM

There usually needs some capacitance (even stray capacitance) in order to stabilize such a network, although a pure oscillatory circuit is with minimal pure resistance if the Q value (not charge) is to be desirably high.

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#2

Re: A reference for Overdamped Networks (systems) in Electronics

10/29/2017 3:53 PM

Electronic Engineers don't generally solve these kinds of problems in the time domain, but simplify circuit analysis by converting linear differential equations to algebraic equations using the Laplace transform, where the solution is of the form e-st, where s = σ + jω is complex.

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#5

Re: A reference for Overdamped Networks (systems) in Electronics

10/30/2017 9:22 AM

Thank you for your comment; I just need to find some examples from the application of the Math; where the matrices in (1) appear (like in the Coupled Spring-Mass-Dampers Systems; in electrical networks,...), and further I work only with these three matrices

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#3

Re: A Reference for Overdamped Networks (Systems) in Electronics

10/30/2017 8:51 AM

My mistake in (1): CORRECT IS Lï+Ri+Dι=O, WHERE
ι is the n-vector of current; i corresponds to the first derivative (d/dt of ι), ï corresponds to the second derivative of ι.

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#6

Re: A Reference for Overdamped Networks (Systems) in Electronics

10/30/2017 12:04 PM

I do not recall this concept of elasticity in an electric circuit, is it related to capacitance?

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#7

Re: A Reference for Overdamped Networks (Systems) in Electronics

10/30/2017 5:10 PM

Yes, the term elasticity was from an very old paper of F. M. Reza " Overdamped and Underdamped Linear Dynamical Systems"

In above images this corresponds to the inverse capacitance matrix.

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#9

Re: A Reference for Overdamped Networks (Systems) in Electronics

10/31/2017 10:17 AM

Look into analog filter design theory. Here is an Analog Devices reference on filter design. That is a good reference but it glosses over the difference between ideal and real operational amplifiers. (They are an operational amplifier company.)

In my experience, the use of over damping in a network is more of a concern to make sure component variations due to tolerance and drift over time and temperature don't induce a meta stable condition.

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