A.C Bridges

A.C Bridges
                                       Measurement of inductance, capacitance may be made conveniently and accurately by employing a.c bridge networks. An a.c bridge, in its basic form, consists of four arms, a source of excitation, and a balance detector. In an a.c bride each of the four arms, is an impedance, and the battery and the galvanometer of wheatstone bridge are replaced respectively by an a.c source and a detector   sensitive to small alternating potential differences (fig 1)

The usefulness of a.c bridge circuits is not restricted to the measurement of unknown impedances and associated parameters like inductance, capacitance, storage factor, dissipation factor e.t.c. hese circuit find other applications in communication systems and complex electronic circuits. A.C bridge circuits are commonly used for phase shifting, providing feedback paths for oscillators and amplifiers, filtering out undesirable signals and measuring the frequency of audio signals.
Sources and Detectors: for measurement at low frequency the power line may act as the sources of supply to the bridge currents.

For the higher frequency electronics oscillators aree universally used as bridge source supplies. These oscillators have the advantage that the frequency is constant, easily adjustable, & determinable with accuracy. The waveform is very close to a sine wave, and their power output is sufficient for most bridge measurement.

The detectors commonly used for a.c bridges are :
(i) headphones
(ii) vibration galvanometers and
(iii) tunable amplifier detectors.
Headphones are widely used as detectors at frequencies of 250Hz and over upto 3Khz or 4KHz. They are most sensitive detectors for this frequency range. Vibration galvanometers are extremely useful for power and low audio frequency ranges. They are manufactured to work at various frequencies ranging from 5Hz to 1000Hz but most commonly used below 200Hz as below this
frequency they are more sensitive than the head phones.

E1 = E2 ------------------ (1)
I1Z1 = I2Z2------------- (2)
Also at balance
I1 = I3 =    E           ----------------- (3)
Z1 +Z3
& I2 = I4 =   E            ----------------(4)

Z2 + Z4
Substituting of equs (3) & (4) into (2) gives
Z1Z4 = Z2Z3-----------------------(5)
This represent the basic equation for balance of an a.c bridge or when using admittance instead of impedances.

Y1Y4 = Y2Y3------------------ (6)
This equation is useful when dealing with parallel elements while equation (5) is convenient to use when dealing with series elements of a bridge.
Equations (5) state that the product of impedances of one pair opposite arms must equal the product of impedances of the other pair of opposite arms expressed in complex notation. This means that both magnitudes and the phase angles of the impedances must be taken into account.
Considering the polar form, the impedance can be written as Z = Z<, where Z represents the magnitude and  represents the phase angle in the form

(Z1<1) (Z4<4) = (Z2 <2) Z3 <3--------------(7)
Thus for balance, we must have

Z1 Z4< 1+4 = Z2Z3 <2 + 3-------------------(8)
Eqn. (8) show that two conditions must be satisfied simultaneously when balancing an a.c bridge.
The first condition is that the magnitude of impedance satisfied the relations

Z1Z4 = Z2Z3----------------------- (9)
i.e the products of the magnitudes of the opposite arms must be equal the second condition is that phase angles of impedances satisfying the relationship
<1+4 = < 2 + 3
i.e the sum of the phase angles of the opposite arms must be equal.
The phase angles are positive for inductance and negative for capacitive impedance.
If we work in terms of rectangular co-ordinates we have
Z1 = R1 + jX1,
Z2 = R2 + jX2,
Z3 = R3 + jX3
Z4 = R4 + jX4 thus alternative notations is the representation of impedance as the sum of a real tern and a complex term, where R is the resistance and x is the reactance.
The magnitude of the impedance is them  and the phase  is
Thus from the equation (5) for balance

Z1 Z4 = Z2 Z3
(R1 + jX1) (R4 + jX4) = (R2 + jX2) (R3 + jX3)
R1R4 + jR1X4 + jX1R4 –X1X4 = R2R3 + JX3R2 + jX2R3 – X2X3
R1R4 - X1X4 + j(R1X4 + X1R4) = R2R3 – X2X3 + j(X2R3 + X3R2)----- (v)
Equation (ii) is a complex equation and a complex equation is satisfied only if real and imaginary parts of each side of the equations are separately equal. Thus, for balance
R1R4 – X1X4 = R2R3 – X2X3 --------- (12)
X1R4 – X4R1 = X2R3 + X3R2--------- (13)
Thus there are two independent condition for balance and both of them must be satisfied for the bridge

For bridge to balance:

The following important conclusions must be considered.
(1) Two balance equations are always obtained for an a.c bridge circuit. This follows the fact that for balance in an a.c bridge, both magnitude and phase relationships must be satisfied. This requires that real and imaginary terms must be separated, which give two equations to be satisfied for balance.
(2) The two balance equations enable us to know two unknown quantities. The two quantities are usually a resistance and an inductances or a capacitance
(3) In order to satisfy both conditions for balance and for convenience of manipulation, the bridge must contain two variable elements in its configuration.
 (4) In this bridge circuit equations are independent of frequency. This is often a considerable advantage in an a.c bridge, for the exact value of the source frequency need not to be known.

Common a.c bridges
Capacitance Bridges

De Santy bridges
The De santy bridge is commonly seen as a means of comparing two capacitance, due to the fact that the bridge has maximum sensitivity when the two capacitors in the adjacent arms are equal. Through this method is quite simple but it is limited by the impossibility of obtaining a perfect balance if the capacitors use are not air capacitors, thus the need to avoid dielectric loss is very important. The figure( 5.14.) above shows de santy bridge 

At balance
Schering bridge
The Schering bridge is used for the measurement of capacitor, in terms of a pure capacitance in series with resistance and is generally used for the capacitors with very low dissipation factor

Tunable amplifiers detector are the most versatile of the detectors. The transistor amplifier can be turned electrically and thus can be made to respond to a narrow band with at the bridge frequency. The output of the amplifier is fed to a pointer types instrument. This detector can be used, over a frequency range of 10Hz to 100KHz.

General Equation for Bridge Balance
Fig 1.Shows a basic a.c bridge. The four arms of this bridge are impedances Z1, Z2, Z3 and Z4
The conditions for balance of bridges requires that there should be no current through the detector. This requires that p.d between points b and d should be zero. This will be the case when the voltage drop from a to b equals to voltage drop from a to d, both in magnitude and phase. In complex notation we can, thus write

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