# Other AC bridges

In the previous chapter, we discussed two AC bridges that can be used to measure inductance. In this chapter, let's talk about the following ** two AC bridges **.

- Schering Bridge
- Vienna Bridge

These two bridges can be used to measure respectively capacity and frequency.

## Schering Bridge

The Schering Bridge is an AC bridge with four arms, which are connected in a diamond shape or ** square shape ** , one arm of which consists of a single resistor, one arm consists of a series combination of resistor and capacitor, one arm consists of a single capacitor, and the other arm consists of 'a parallel combination of resistor and capacitor.

The AC detector and the AC voltage source are also used to find the value of the unknown impedance, so one of them is placed in a diagonale of the Schering Bridge and the other is placed in the other diagonal of the Schering Bridge.

The Schering bridge is used to measure the value of the capacitance. The ** electrical diagram ** of the Schering bridge is shown in the figure below.

In the above circuit, the arms AB, BC, CD and DA together form a diamond or ** square shape **. The AB arm is made up of a resistor, $ R_ {2} $. The BC arm consists of a series combination of resistors, $ R_ {4} $, and capacitors, $ C_ {4} $. The CD arm consists of a capacitor, $ C_ {3} $. The DA arm consists of a parallel combination of resistor, $ R_ {1} $, and capacitor, $ C_ {1} $.

Let $ Z_ {1} $, $ Z_ {2} $, $ Z_ {3} $ and $ Z_ {4} $ respectively be the impedances of the arms DA, AB, CD and BC. The ** values of these impedances ** will be

$ Z_ {1} = frac {R_ {1} left (frac {1} {j omega C_ {1}} right)} {R_ {1} + frac {1} {j omega C_ {1}}} $

$ Rightarrow Z_ {1} = frac {R_ {1}} {1 + j omega R_ {1} C_ {1}} $

$ Z_ {2} = R_ {2 } $

$ Z_ {3} = frac {1} {j omega C_ {3}} $

$ Z_ {4} = R_ {4} + frac {1} {j omega C_ {4}} $

$ Rightarrow Z_ {4} = frac {1 + j omega R_ {4} C_ {4}} {j omega C_ {4}} $

** Replacing ** these values impedance under the following balancing condition of the AC bridge.

$$ Z_ {4} = frac {Z_ {2} Z_ {3}} {Z_ {1}} $$

$$ frac {1 + j omega R_ { 4} C_ {4}} {j omega C_ {4}} = frac {R_ {2} left (frac {1} {j omega C_ {3}} right)} {frac {R_ {1}} {1 + j omega R_ {1} C_ {1}}} $$

$ Rightarrow frac {1 + j omega R_ {4} C_ {4}} {j omega C_ {4}} = frac {R_ {2} left (1 + j omega R_ {1} C_ {1} right)} {j omega R_ {1} C_ {3}} $

$ Rightarrow frac {1 + j omega R_ {4} C_ {4}} {C_ {4}}= frac {R_ {2} left (1 + j omega R_ {1} C_ {1} right)} {R_ {1} C_ {3}} $

$ Rightarrow frac {1} {C_ {4}} + j omega R_ {4} = frac {R_ {2}} {R_ {1} C_ {3}} + frac {j omega C_ {1} R_ {2}} {C_ {3}} $

By ** comparing ** the respective real and imaginary terms of the above equation, we will get

$ C_ {4} = frac {R_ {1} C_ {3}} {R_ {2}} $ Equation 1

$ R_ {4} = frac {C_ {1} R_ {2}} {C_ {3}} $ Equation 2

By replacing the values of $ R_ {1}, R_ {2} $ and $ C_ {3} $ in the equation 1, we will get the value of the capacitor, $ C_ {4} $. Likewise, by substituting the values of $ R_ {2}, C_ {1} $ and $ C_ {3} $ in equation 2, we will get the value of the resistor, $ R_ {4} $.

The ** advantage ** of the Schering bridge is that the resistance values, $ R_{4} $ and of the capacitor, $ C_ {4} $ are independent of the value of the frequency.

## Wien 's Bridge

** Wien ' s bridge ** is an AC bridge with four arms, which are connected in the shape of a diamond or a square shape. Of two arms consist of a single resistor, one arm consists of o f a parallel combination of resistor and capacitor and the other arm consists of a series combination of resistor and capacitor.

The AC detector and the AC voltage source are also needed to find the value of the frequency. Therefore, one of these two is placed in a diagonal of the Vienna Bridge and the other is placed in the other diagonal of the Vienna Bridge.

The ** electrical diagram ** of the Vienna Bridge is shown in the figure below.

In the above circuit, the arms AB, BC, CD and DA together form a diamond or ** square shape **. The arms, AB and BC are made up of resistors, respectively $ R_ {2} $ and $ R_ {4} $. The arm, CD consists of a combination parallele of resistance, $ R_ {3} $ and of capacitor, $ C_ {3} $. The arm, DA consists of a series combination of resistors, $ R_ {1} $ and capacitors, $ C_ {1} $.

Let $ Z_ {1}, Z_ {2}, Z_ {3} $ and $ Z_ {4} $ respectively be the impedances of the arms DA, AB, CD and BC. The ** values of these impedances ** will be

$$ Z_ {1} = R_ {1} + frac {1} {j omega C_ {1}} $ $

$$ Rightarrow Z_ {1} = frac {1 + j omega R_ {1} C_ {1}} {j omega C_ {1}} $$

$ Z_ {2} = R_ {2} $

$$ Z_ {3} = frac {R_ {3} left (frac {1} {j omega C_ {3}} right)} {R_ {3} + frac {1} {j omega C_ {3}}} $ $

$$ Rightarrow Z_ {3} = frac {R_ {3}} {1 + j omega R_ {3} C_ {3}} $$

$ Z_ {4} = R_ {4} $

** Replace ** these impedance values under the following balancing condition of AC bridge.

$$ Z_ {1} Z_ {4} = Z_ {2} Z_ {3} $$

$$ left (frac {1 + j omega R_ {1} C_ {1}} {j omega C_ {1}} right) R_ {4} = R_ {2} left (frac {R_ {3}} {1 + j omega R_ {3} C_ {3}} right) $ $

$ Rightarrow left (1 + j omega R_ {1} C_ {1} right) left (1 + j omega R_ {3} C_ { 3} right) R_ {4} = j omega C_ {1} R_ {2} R_ {3} $

$ Right left arrow (1 + j omega R_ {3} C_ {3} + j omega R_ {1} C_ {1} - omega ^ {2} R_ {1} R_ {3} C _ {1} C_ {3} right) R_ {4} = j omega C_ {1} R_ {2} R_ {3} $

$ Right arrow R_ {4} left (omega ^ {2} R_ {1} R_ {3} C_ {1} C_ {3} right) + j omega R_ {4} left (R_ {3} C_ {3} + R_ {1} C_ {1} right) = j omega C_ {1} R_ {2} R_ {3} $

** Match ** the respective ** real terms ** of the above equation.

$$ R_ {4} left (1- omega ^ {2} R_ {1} R_ {3} C_ {1} C_ {3} right) = 0 $$

$ Rightarrow 1- omega ^ {2} R_ {1} R_ {3} C_ {1} C_ {3} = 0 $

$ Rightarrow 1 = omega ^ {2} R_ {1} R_ {3} C_ {1} C_ {3} $

$ omega = frac {1} {sqrt {R_ {1} R_ {3} C_ {1} C_ {3}}} $

** Substitute **, $ omega = 2 pi f $ in equation above.

$$ Rightarrow 2 pi f = frac {1} {sqrt {R_ {1} R_ {3} C_ {1} C_ {3}}} $$

$ Rightarrow f = frac {1} {2 pi sqrt {R_ {1} R_ {3} C_ {1} C_ {3}}} $

We can find the frequency value, $ f $ of the AC voltage source by substituting the values of $ R_ {1}, R_ {3}, C_ {1} $ and $ C_ { 3} $ on top of the equation.

If $ R_ {1} = R_ {3} = R $ and $ C_ {1} = C_ {3} = C $, then we can find the value of the frequency, $ f $ of AC voltage source using the following formula.

$$ f = frac {1} {2 pi RC} $$

Wein's bridge is mainly used to find the ** frequency value ** of the AF range.