Introductions about filters

General properties of a RCLM two-terminals

A RCLM two-terminals is a passive device only built by resistors (R), capacitors (C), inductors(L) and ideal transformers (M).

We call the terminals A and B; the voltage from A to B is called v(t), and the current entering to A is called i(t). If v(t) and i(t) are Laplace-transformable, exists the relation:

If the two-terminals is defined in this way: "WITH I ASSIGNED, IS OBTAINED ONLY ONE VALUE OF V", we have:

The structure of Z(s) is always corresponding to a ratio of polynomials like this:

 

Properties of a two-terminals defined in this way:

• For s real, the integrals who define I(s) and V(s) are real: so Z(s) is real.
  
  

• The two-terminals is passive.In Alternating Current, the entering active power must be greater or egual to 0. In this case:
  
  
  
  
  
  
  
  
  This demonstrate that the two-terminals is passive and, in alternate current, the real part of Z(jw) must be positive or null.

• If some internal components of the two-terminals are inictially charged, the voltage v₀(t) will evolve by free frequencies that are poles of Z(s). The poles of Z(s) can be zeroes of q(s) or, if n > m, a pole to infinite.
  v₀(t) can't grow indefinitely if the two-terminals is passive, so:
  
  - the poles must be in the negative half of the complex plane or on the imaginary axys.
  - the ones on the imaginary axys must be simple (with multiplicity =1).

Last revision: March 22, 2008