# Wheatstone Bridge: Substitution Resistance Formula Derivation?

• bagasme
In summary, the conversation discusses the formula for finding substitution resistance in a Wheatstone bridge. If the cross products of certain resistances are the same, the galvanometer can be omitted and the series-parallel principle can be used. If the cross products are different, the circuit must be modified and new resistances are determined using a set of equations. The conversation also mentions the lack of explanation or derivation for the formula and suggests searching for "Delta Wye Transform derivation" to find the necessary information. The conversation also briefly touches on the derivation for formula a, which can be found through web searches.
bagasme
Hello,

In high school, I had been taught about finding substitution resistance from Wheatstone bridge.

The formula:

a. If the cross product of ##R1## and ##R3## is same as ##R2## and ##R4##, the galvanometer in the middle (##R_5##) can be omitted and use series-parallel principle to solve for the substitution resistance.

b. If instead the cross products are different, modify the circuit to the following diagram below,

and determine new resistances (##R_a##, ##R_b##, & ##R_c##) by:

\begin{align} R_a &= \frac {R_1 \cdot R_2} {R_1 + R_2 + R_5} \nonumber \\ R_b &= \frac {R_1 \cdot R_5} {R_1 + R_2 + R_5} \nonumber \\ R_c &= \frac {R_2 \cdot R_5} {R_1 + R_2 + R_5} \nonumber \end{align}
Then, use new resistance to solve for substitution resistance.

However, there isn't any explanation or derivation of the formula AFAIK (even on my textbook).

So what derivation that lead me to the formulas above?

Bagas

A web search for "Delta Wye Transform derivation" will turn up what you need.

But what about derivation of formula a. ?

bagasme said:
But what about derivation of formula a. ?
I'm not sure what you mean. What is "formula a"?

There are examples of the derivation of the transformations on the web that can be readily found.

'a' can probably be derived, but (by inspection):

R5 may be omitted from the analysis if there is no current flow through it - it isn't 'doing' anything
there is no current flow through R5 if (and only if) the voltages at either end of R5 are equal
the voltages are equal when R1/R4 = R2/R3 (or R1R3 = R2R4)
.

bagasme

## What is a Wheatstone Bridge?

A Wheatstone Bridge is a type of electrical circuit used to measure unknown electrical resistances. It consists of four resistors arranged in a diamond shape, with a voltage source connected to the top and bottom points of the diamond and a galvanometer connected to the two remaining points.

## What is the purpose of a Wheatstone Bridge?

The purpose of a Wheatstone Bridge is to measure an unknown resistance by comparing it to known resistances. When the bridge is balanced, meaning the galvanometer reads zero, the ratio of the known resistances can be used to calculate the value of the unknown resistance.

## How does the substitution resistance formula work?

The substitution resistance formula, also known as the Wheatstone Bridge formula, is used to calculate the value of the unknown resistance in a balanced Wheatstone Bridge. It states that the ratio of the two known resistances (R1 and R2) is equal to the ratio of the two unknown resistances (Rx and Ry). This can be expressed as Rx/Ry = R1/R2, or Rx = Ry(R1/R2).

## What is the derivation of the substitution resistance formula?

The substitution resistance formula can be derived using Kirchhoff's laws and Ohm's law. By applying Kirchhoff's voltage law to the top and bottom loops of the Wheatstone Bridge, and Kirchhoff's current law to the two side branches, a system of equations can be formed. By solving these equations, the substitution resistance formula can be derived.

## What are the limitations of the Wheatstone Bridge?

The Wheatstone Bridge is most accurate when the unknown resistance is similar in magnitude to the known resistances. Additionally, the Wheatstone Bridge is sensitive to changes in temperature and can be affected by the resistance of the connecting wires. It also assumes a linear relationship between voltage and current, which may not always be the case.

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