Download PHYSICS 2 INSTRUMENTATION

BRIDGES AND MESH EQUATION METHOD
An easy way to tackle the analysis of multiloop circuits is to use the method called
the mesh equation method or Maxwell’s Method.
1. Pick closed loops called mesh currents or loop currents.
2. Apply Kirchhoff Voltage Law to each loop, being very careful with your sign
convention to give a set of simultaneous equations.
3. Solve the simultaneous equations to find the loop currents and then the current
through each component and the voltage across each component.
To illustrate the mesh loop method we will consider a generalised bridge circuit as
shown below.
Z1
Z4
Z3
i2
i3
Z5
Z2
i1
vS
 Z 2  Z5  i1  Z 2 i2  Z5 i3  vS
 Z 2 i1   Z1  Z 2  Z3  i2  Z3 i3  0
 Z5 i1  Z3 i2   Z3  Z 4  Z5  i3  0
or in matrix form
 Z2
 Z5
 Z 2  Z5
  i1   vS 

   
Z1  Z 2  Z3
 Z3
 Z2
  i2    0 
 Z
 Z3
Z3  Z 4  Z5   i3   0 
 5
Z I = V
I = Z-1 V
Therefore to solve the simultaneous equation we simply have to multiply the vector
V by the inverse of the matrix Z. This can be done in both MATLAB and MS
EXCEL using their matrix commands.
Bridges
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MATLAB COMMANDS
Consider the matrix equation A x = b
Then the matrix x can be found using the following commands
x=A\b
or
x = b’ / A’
where ’ gives the transpose of the matrix.
Using these commands it is not necessary to find the inverse of the matrix. However,
the inverse is given by the command inv(A).
WHEATSTONE BRIDGE
The Wheatstone bridge is a specific circuit that is used for measuring resistances and
has varied applications in instrumentation systems. There are two basic modes of
bridge operation. In one mode the bridge can be used to determine the value of an
unknown resistance to a high degree of accuracy by comparing it with an accurately
known resistance. The value of the unknown resistance is measured by varying the
resistance of one of three other resistors in the bridge circuit to obtain a balanced
condition in which the bridge has zero output voltage, that is, a voltage “null”.
In the other mode, the bridge is in an unbalanced state and the value of an unknown
resistance is determined from the value of the bridge output voltage. This is
sometimes referred to an “off-null” operation. If a resistance type transducer, for
example a thermistor, light dependent resistor or a strain gauge is used as the
unknown resistance, then the bridge output will depend on the transducer resistance.
The output voltage can be calibrated directly in terms of the measured variable
(temperature, light, expansion, …)
One of the key components of a weighing system is an instrument called a strain
guage which is often used to measure small amounts of deformation. It consists of a
series of parallel, high resistance wire or foil elements and is mounted on a smooth
surface. This strain guage is then mounted on a metal beam that has a bucket hanging
from it and it forms one of the arms of a Wheatstone bridge.
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A variation of the resistive type Wheatstone bridge is the reactive-type or ac bridge.
This type of bridge is used to measure capacitance and inductive values.
MATLAB FILE
%m260af.m
%31 jan 01
%mesh equation method
%generalised bridge
%data for bridge impedances and emf
z(1) = 25.28;
z(2)= 10;
z(3) = 0;
z(4) = 2528;
z(5) = 1000;
vs = 1.0;
%impedance matrix
mz(1,1) = z(2)+z(5);
mz(1,2) = -z(2);
mz(1,3) = -z(5);
mz(2,1) = -z(2);
mz(2,2) = z(1)+z(2)+z(3);
mz(2,3) = -z(3);
mz(3,1) = -z(5);
mz(3,2) = -z(3);
mz(3,3) = z(3)+z(4)+z(5);
%emf
v(1)
v(2)
v(3)
matrix
= vs;
= 0;
= 0;
%current matrix
i = mz \ v';
%v' need to convert v from a row vector into a column
vector
%current through each resistance
ir(1) = i(2);
ir(2) = i(1)-i(2);
ir(3) = i(2)-i(3);
ir(4) = i(3);
ir(5) = i(1)-i(3);
%potential difference across each resistance
pd = ir .* z;
%m260ag.m
%31 jan 01
%mesh equation method
%generalised bridge
%very tranducer resistance z5
%detect voltages changes across z3
clear;
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%data for bridge impedances and emf
rtmin = 99;
rtmax = 101;
num = 21;
dr = (rtmax-rtmin)/(num-1);
rt = rtmin : dr : rtmax ;
%transducer resistance
z(1) = 100;
z(2)= 100;
z(3) = 1e5;
z(4) = 100;
vs = 1.0;
%loop to vary transducer resistance
for c = 1 : num
z(5) = rt(c);
%impedance matrix
mz(1,1) = z(2)+z(5);
mz(1,2) = -z(2);
mz(1,3) = -z(5);
mz(2,1) = -z(2);
mz(2,2) = z(1)+z(2)+z(3);
mz(2,3) = -z(3);
mz(3,1) = -z(5);
mz(3,2) = -z(3);
mz(3,3) = z(3)+z(4)+z(5);
%emf
v(1)
v(2)
v(3)
matrix
= vs;
= 0;
= 0;
%current matrix
i = mz \ v';
%v' need to convert v from a row vector into a column
vector
%current through each resistance
ir(1) = i(2);
ir(2) = i(1)-i(2);
ir(3) = i(2)-i(3);
ir(4) = i(3);
ir(5) = i(1)-i(3);
%potential difference across each resistance
pd = ir .* z;
vd(c) = pd(3);
end
figure(1);
plot(rt,vd*1000,'ob');
grid on
title('WHEATSTONE BRIDGE')
xlabel('tranducer resistance R5 (ohms)')
ylabel('output voltage V3mV)')
Bridges
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WHEATSTONE BRIDGE
3
2
output voltage V3 (mV)
1
0
-1
-2
-3
99
Bridges
99.2
99.4
99.6
99.8
100
100.2
tranducer resistance R5 (ohms)
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100.4
100.6
100.8
101
5