kill(all)$

electro(chem) , measurement with bridge ' Weathstone Bridge',part 4:version 1.0
peter.vlasschaert@gmail.com,12/01/2020

rem: other parts , see : website : electrical or electrochemistry

load packages

load(lrats)$

1e) definition : j : sqrt(-1)
===================

j:%i;

Impedance Bridge circuit
====================
Z = R+j*X ( = cartesian representation )

Z =Z1,Z2,Z3,Z4,Z5

Thevenin Theorema ( = Thevenin equivalent)
==================================

That any real source can be replaced = ideal potential source (=VTh) & 
resistor (Req =RTh) in series.

Z5 : load of the circuit

How to find Thevenin equivalent ?
==========================

1e)  find Req
==========
a)  Change  E by short circuit
b)  Remove load Z5 between terminal A,B , Figure 1

result : Z1//Z3 + Z2//Z4  ( look between two terminals A,B)
legend :  // = parallel circuit
                + = serial   circuit 

p1: Req = (Z1*Z3)/(Z1+Z3) + (Z2*Z4)/(Z2+Z4);

2e)  find VTh
==========
a) remove load
b) ideal voltage source

How potential find potential @ point A ( = voltage divider, Z3 against  Z1+Z3 ), E[A]
How potential find potential @ point A ( = voltage divider, Z4 against  Z2+Z4 ), E[B]
ref : ground ( = connect neg terminal of  E)

p2a: V[A] = E *Z3/(Z1+Z3);
p2b: V[B] = E *Z4/(Z2+Z4);

voltage :differences.(Between terminals)
===============================

p2c:V[Th] = V[B]-V[A];

p2d:subst([p2a,p2b],p2c);

p2d1:ratsimp(p2d);

p2d2:lhs(p2d1) = factorout(num(rhs(p2d1)),E)/denom(rhs(p2d1));

p2d3:expand(p2d2);

p2d4: VTh =rhs(p2d3);

current : trough circuit Figure2
=======================

p3a: I = V[Th]/(Req+Z5);
p3b: subst([p2d,p1],p3a);
p3c:ratsimp(p3b);

see : figure 2
==========
voltage divider  Z5  against  Req

p4:E[AB] = V[Th]*Z5/(Req+Z5);

Brigde balanced : current  : i5 =0, Figure1
================================
Null reading of current : i5,meaning that V[A] = V[B]

p5:rhs(p2a)= rhs(p2b);

p5a:p5/E;

p5b:rhs(p5a)-lhs(p5a)=0;

p5c:ratsimp(p5b);

p5d :num(lhs(p5c))=0;

suppose : Z1 ?

p5e:solve(p5d,Z1)[1];

balanced : current (i5 = 0)
see : Figurev1

p5e:p5e/Z3;

found also when VTh = 0,p2d1

When : Zi=Ri  Then  : Weathstone -Bridge.