Measure the Resistance via the Wheastone Bridge

1.The purpose:

i.              To understand the principle and characteristics of the Wheatstone bridge.

ii.              To understand the concept of sensitivity of bridge.

iii.              Learn the “exchange measure method”, with which to eliminate the system error.

2.The apparatus:

   Unknown resistances; DC power supply; galvanometer; resistor box; variable resistor; switch and other electrical components; electrical bridge.

3.Theories:                                                    

1)The principle and the structure of the Wheatstone bridge

 

When U_C=U_D

R_X=

2)The sensitivity of the Wheatstone bridge

Because of the sensitivity of the galvanometer is limited, the current through the galvanometer is not truly zero, and the galvanometer cannot efficiently measure it.

The sensitivity of the bridge is defined as the ratio between the deflection of the galvanometer, Dd, and the corresponding relative value of R₀.

S=

3)The “exchange measure method”.

Exchange the positions of R1 and R2, or R0 and Rx, and adjusting R0 to R0 ¢, the bridge can get a new balanced state，and               R_X= 

Combine this equation with R_X=, we can get:

R_X=

This equation has nothing to do with R₁ and R₂. In this way we can make the error only relate to the instrument error of R₀, which is the instrument error of resistor box.

4.The procedures:

1)     Use simply constructed bridge to measure the metal film resistor.

Connect the circuit and measure the resistance then exchange R₀ and R_X measure the resistance again.

The key points:

         i.              Find the balance point.

       ii.              Measure the sensitivity of the bridge.

    iii.              Exchange R₀ and R_X measure the resistance again.

Error analysis:

The uncertainty of R₀:

_apparatus=R_{0 apparatus}=0.1%R_o+0.005(K+1)

The error caused by the sensitivity of the bridge is:

Δ=R₀  (S is the sensitivity of the galvanometer)

U_Ro=

The relative uncertainty of R_X is:

= (N=1.0)

2)     Use box-bridge to measure the resistance.

         i.              Make the reading of galvanometer back to “0” via mechanical method.

       ii.              Open K, press G button, adjust M and keep the galvanometer balance.

    iii.              Connect R_X with X.

     iv.              Adjust the system and record the resistance R₀ which make the galvanometer balance.

Error analysis:

The limit of fundamental error is:

E_lim=α%(NR_o+)

The error caused by the sensitivity of the bridge box is:

Δ=NR₀

The relative uncertainty of R_X is:

U_Rx=

5.Data processing:

TEST ONE.

Use simply constructed bridge to measure the metal film resistor.

a)      For U=8V R_inner=100Ω

R_X==Ω=498.35Ω

S===277

The uncertainty of R₀:

_apparatus=R_{0 apparatus}=0.1%R_o+0.005(K+1)

=0.1%497.7Ω+0.005(4+1)=0.5227Ω

The error caused by the sensitivity of the bridge is:

Δ=R₀=497.7Ω=0.3594Ω

U_Ro=

==0.63434Ω

The relative uncertainty of R_X is:

=

==9.0123x10^-4

==9.0123x10^-4x498.35Ω=0.45Ω

R_X=498.350.45Ω

b)      For U=8V R_inner=2500Ω

R_X==Ω=499.05Ω

S===53.6

The uncertainty of R₀:

_apparatus=R_{0 apparatus}=0.1%R_o+0.005(K+1)

=0.1%498.1Ω+0.005(4+1)=0.5231Ω

The error caused by the sensitivity of the bridge is:

Δ=R₀=498.1Ω=1.859Ω

U_Ro=

==1.9312Ω

The relative uncertainty of R_X is:

=

==2.7415x10^-3

==2.7415x10^-3x499.05Ω=1.4Ω

R_X=499.01.4Ω

c)       For U=4V R_inner=100Ω

R_X==Ω=498.85Ω

S===135

The uncertainty of R₀:

_apparatus=R_{0 apparatus}=0.1%R_o+0.005(K+1)

=0.1%497.7Ω+0.005(4+1)=0.5227Ω

The error caused by the sensitivity of the bridge is:

Δ=R₀=497.7Ω=0.7373Ω

U_Ro=

==0.90378Ω

The relative uncertainty of R_X is:

=

==1.2840x10^-3

==1.2840x10^-3x498.85Ω=0.6Ω

R_X=498.80.6Ω

d)      For U=4V R_inner=2500Ω

R_X==Ω=499.15Ω

S===31.1

The uncertainty of R₀:

_apparatus=R_{0 apparatus}=0.1%R_o+0.005(K+1)

=0.1%498.2Ω+0.005(4+1)=0.5232Ω

The error caused by the sensitivity of the bridge is:

Δ=R₀=498.2Ω=3.010Ω

U_Ro=

==3.0551Ω

The relative uncertainty of R_X is:

=

==4.3362x10^-3

==4.3362x10^-3x499.15Ω=2.2Ω

R_X=499.22.2Ω

TEST TWO

Use box-bridge to measure the resistance.

A.      For resistor R_X1

S===17.4x10³

The limit of fundamental error is:

E_lim=α%(NR_o+)=0.1%x5120.4Ω+1Ω=6.1204Ω

The error caused by the sensitivity of the bridge box is:

Δ=NR₀=1x5120.4Ωx=0.058855Ω

The relative uncertainty of R_X is:

U_Rx==

   =6Ω

R_X=51206Ω

B.      For resistor R_X2

S===26.9x10³

The limit of fundamental error is:

E_lim=α%(NR_o+)=0.1%x10^-1x4983.8Ω+0.1Ω

=0.59838Ω (=10000, and I get it from

E_lim=α%(NR_o+)=₀+1)

The error caused by the sensitivity of the bridge box is:

Δ=NR₀=10^-1x4983.8Ωx=0.0037054Ω

The relative uncertainty of R_X is:

U_Rx=

=

   =0.6

R_X=498.46Ω

Discussion:

In the 1^st test, “use simply constructed bridge to measure the metal film resistor”. As we can see that the “exchange measure method” eliminated the error which caused by the uncertainty of R₁ and R₂.Thus the main error in this experiment is the instrumental error of the resistor box (U_Ro). The instrumental error is mainly caused byR_{0 apparatus}. The error caused by the sensitivity of the bridge (Δ) also contributes to the error U_Ro.

   From the form, we can get that R_{0 apparatus} plays an important role in the error when the inner resistance of the galvanometer is relative small or the voltage of the power is relative larger. On the other hand, the error caused by the sensitivity of the bridge (Δ) is the main error when the inner resistance of the galvanometer is big or the voltage of the power is small.

In the 2^nd test, “use box-bridge to measure the resistance”.

The main error is the fundamental error limit (E_lim) and the error caused by the sensitivity of the bridge box (Δ).

   We can easily get that the main error in “use box-bridge to measure the resistance” is the fundamental error limit (E_lim). And the error caused by the sensitivity of the bridge box (Δ) is so small that it can be neglected.

6.Conclusions:

From this experiment we can get that:

In the 1^st test, “use simply constructed bridge to measure the metal film resistor”

In the 2^nd test, “use box-bridge to measure the resistance”.

What’s more, when we use simply constructed bridge to measure the resistance, we’d better use a power supply with relative high voltage and also we should use a galvanometer with small inner resistance.

   While we use the box bridge, the error mainly relate to E_lim, and E_lim=α%(NR_o+), thus it is better to measure small resistor with box bridge.