R1
C1
R3
~ Vi
Vd
Ru
R2
Lu
Bridge methods
ABE425 Engineering Measurement Systems
Agenda
Null methods (calibration and highly accurate measurements)
DC bridges (Resistance measurement)
• Wheatstone
Capacitor and inductance (coil) models
AC bridges (Inductance / capacitance measurement)
• Null-type Parallel-Resistance-Capacitance bridge for capacitance and
dissipation factor measurement
• Maxwell bridge for inductance (coil) and quality factor measurement
• Wien bridge
Approximate measurement of Inductance and Capacitance
Deflection methods (control systems)
Deflection-type Wheatstone bridge and non-linearity
Null-type DC Wheatstone bridges are used for
accurate resistance measurement
The bridge is balanced when the voltage Vd is adjusted to zero by
tuning R1 while R2 and R3 are known and kept constant.
The null-detector is usually some type of galvanometer
The unknown resistance value can then be computed using the values
of the other resistances
Since there are no inductances (coils) or capacitances, a DC source is
sufficient
This type of bridge is used for strain gage measurements
Rx
Vd
R1
Vi
R2
R3
Measurement procedure using Galvanometer and
decade resistor box
Unknown resistor
Rx
Decade Box
R1
Vd
Vi
R3
R2
Galvanometer
Vd 
R3
R2
Vi 
Vi
Rx  R2
R1  R3
 R2
R3 
Vd  Vi 


R

R
R

R
2
1
3 
 x
At balance:
R
Vd  0  Rx  R1 2
R3
Known, constant
Galvanometers are VERY SENSITIVE instruments to
detect zero current
D'Arsonval galvanometer
Thompson mirror type galvanometer (1880). Note the
‘antenna’ to compensate for electric/magnetic fields
Null-type AC Wheatstone bridge for impedance
measurement
The bridge is balanced when the voltage Vd is adjusted to zero
by tuning Z , Z or Z
1
Zx
Vd
2
3
Z1
Vi ~
Z2
Z x Z3  Z1Z2
Z3
Z1
Zx 
Z2
Z3
x  1  3  2
Capacitor and Inductance (coil) models
Capacitor model with Capacitance and dissipation resistance
Cx
Rx
Inductor (coil) model with Inductance and series resistance
Inductances have a quality factor
Ru
Q
Lu
 Lu
Ru
Null-type Parallel-Resistance-Capacitance bridge for
capacitance and dissipation factor measurement
1
jC x
Rx
Zx 

1
1  j Rx C x
Rx 
jC x
Rx
Cx
Rx C1
R1
Z 3  R3  Known, fixed
~ Vi
R2
Vd
R3
Z x Z3  Z1Z2
R 
Re : Rx  R1 2 
R3 
 Independent of 
R3 
Im : Cx  C1
R2 
1
jC1
R1
Z1 

1
1  j R1C1
R1 
jC1
R1
Z 2  R2  Known, fixed

Rx
R 
R1
 2

1  j RxCx R3  1  j R1C1 
Rx R3 1 jR1C1   R1R2 1 jRxCx 
Maxwell bridge to measure inductance, resistance and
quality factor of low quality coils (Q<10)
Z u  Ru  j Lu
R1
C1
R3
~ Vi
Vd
R2
Ru
Lu
Z1Zu  Z3Z2
R2 R3 

R1  Independent of 
Im : Lu  R2 R3C1 

Re : Ru 
Q
 Lu
Ru
  R1C1
1
jC1
R1
Z1 

1
1  j R1C1
R1 
jC1
R1
Z 2  R2 
 Known, fixed
Z 3  R3 


R1
R

j

L
 u
  R2 R3
u 
 1  j R1C1 
R1  Ru  jLu   R2 R3 1 jR1C1 
Hay bridge to measure inductance, resistance and
quality factor of high quality coils (Q>10)
Z x  Rx  j Lx
Z 3  R3 
Rx
R1
Lx
1  j R3C3
1

jC3
jC3
Z1  R1 
 Known, fixed
Z 2  R2 
 1  j R3C3 
  R1R2
j

C
3


 Rx  j Lx  
~ Vi
Vd
R2
C3
R3
Z x Z3  Z1Z2
 Rx  jLx 1 jR3C3   jR1R2C3
R1 R2C3
Lx 
1   2 R32C32
Q
 Lx
Rx
 2 R1R2 R3C32
Rx 
1   2 R32C32

1
 R3C3
Wien bridge for frequency measurement
Z 2  R2  Known, fixed
1
jC x
Rx
Zx 

1
1  j Rx C x
Rx 
jC x
Rx
R1
R2
Z 3  R3  Known, fixed
C1
~V
Z1  R1 
Vd
R3
Cx
Rx
2 
Z2 Z x  Z3Z1
1  j R1C1
1

jC1
jC1
1
R1C1RxCx
 1   2 R12C12 
Rx  R3  2
2 

R
R
C
1 2 1 

Cx 
R2C1
R3 1   2 R12C12


The coil characteristics inductance and series resistance can be
measured by equalizing the voltage across a variable resistor
and the coil itself
R
~ Vi
RL
L
V
Z L  RL  j L
ZR  R
V  i  RL  jL   iR
V
2
V  i  RL 2   L    iR


RL 2   L   R 2
2
L
1

R 2  RL 2
Series resistance of the
coil RL measured with a DVM
Approximate method of measuring capacitance
Measure the AC Voltages for a known input frequency
across resistor R and capacitor C
1
jC
ZR  R
ZC 
R
VR
1
C
VR  iR
VC  i
~ Vi
Cu
VC
VC
VR

1
 RC
C
1 VR
 R VC
Resistance measured with a DVM
Deflection type DC Wheatstone bridge
Rx
Vd
R1
Vi
R2
Vd 
R3
R3
R2
Vi 
Vi
Rx  R2
R1  R3
 R2
R3 
Vd  Vi 


R

R
R

R
2
1
3 
 x
Output (deflection) for R2, R3 = 1,000 Ohm showing significant
non-linearity
Output for R1 =2000, R2 =1000, R3=1000
Rx
2
Vd
R1
Vi
1.5
R2
1
Vd
Bridge balance
0.5
0
-0.5
-1
1000
1200
1400
1600
1800
2000
Rx
2200
2400
2600
2800
3000
R3
Output (deflection) for R2, R3 = 10,000 Ohm showing reduced
non-linearity
Output for R1 =2000, R2 =10000, R3=10000
Rx
2
Vd
R1
Vi
1.5
R2
1
Vd
Bridge balance
0.5
0
-0.5
-1
1000
1200
1400
1600
1800
2000
Rx
2200
2400
2600
2800
3000
R3
Measuring the drag coefficient of a sphere using a
compensation method
Electric current returns
sphere to original position
Air flow
pushes
sphere to the right
Drag coefficient ~ Electric current
Links
Schneider, N. 1904. Electrical instruments and testing. Spon and
Chamberlain, New York.
The End