Ac Bridges and Measurement of Inductance and Capacitance:

Ac bridges and measurement of inductance and capacitance:

A bridge is a special class of instrument consisting of four arms, where three arms have known values and an unknown value of the remaining arm is calculated. The value of unknown arm is calculated in terms of known parameters of other three arms.

Normally, bridges are used for the measurement of passive circuit component such as Resistance, inductance, capacitance etc. Generally, Bridges are classified on the basis of input base, namely:

a.Dc bridge

b.Ac bridge

AC Bridges

AC Bridges consist of a source, balance detector, and four arms. In AC bridges, all the four arms consist of impedance. The AC bridges are formed by replacing the DC battery with an AC source and galvanometer by the detector of Wheatstone bridge.They are highly useful to find out inductance, capacitance, storage factor, dissipation factor etc. Now let us derive general expression for an AC bridge balance Figure given below shows AC bridge network,

Here Z₁, Z₂, Z₃, and Z₄ are the arms of the bridge. Now at the balance condition, the potential difference between b, and d must be zero. From this, when the voltage drop from a to d equals to drop from a to b both in magnitude and phase.

Thus, we have from figure e₁ = e₂ .

From equation 1, 2 and 3 we have Z₁.Z₄ = Z₂.Z₃ and when impedance are replaced by admittance, we have Y₁.Y₄ = Y₂.Y₃. Now consider the basic form of an AC bridge. Suppose we have bridge circuit as shown below,

In this circuit R₃ and R₄ are pure electrical resistance. Putting the value of Z₁, Z₂, Z₃ and Z₄ in the equation that we have derived above for AC bridge.

Now equating the real and imaginary parts we get

Following are the important conclusions that can be drawn from the above equations:

(a) We get two balanced equations that are obtained by equating real and imaginary parts this means that for an ac bridge both the relation (i.e.magnitude and phase) must be satisfied at the same time. Both the equations are said to be independent if and only if both equations contain the single variable element. This variable can inductor or resistor

(b) The above equations are independent of frequency that means we do not require an exact frequency of the source voltage and also the applied source voltage waveform need not be perfectly sinusoidal.

Maxwell’s Bridge(Maxwell’s inductor-capacitor bridge):

Accurate pure standard capacitors are more easily constructed than standard inductors. Consequently, it is desirable to be able to measure inductance in a bridge that uses a capacitance standard rather than an inductance standard. The Maxwell bridge (also known as the Maxwell-Wein bridge) is shown in Figure below. In this circuit, the standard capacitor C3 is connected in parallel with adjustable resistor R3. R1 is again an adjustable standard resistor. and R4 may also be made adjustable. L_s and R_s represent the inductor to be measured.

The Maxwell bridge is found to be most suitable for measuring coils with a low Q-factor (i.e., where wL_s is not much larger than R_s). The circuit diagram of Maxwell bridge is shown below

According to Maxwell’s bridge circuit, the fundamental ac bridge equivalent parameters are:

Substituting for all components in equation of ac bridge,

Equating the real components in above equation,

Equating the imaginary components in above equation,

So using above expressions we can measure unknown resistance R_s and unknown inductance L₄. The expression for quality factor is given by,

The above equation shows that for high quality factor , the value of R1 must be high(in the order of 10⁵ to 10⁶). The cost of such high value decade resistance is very high. So, it becomes impracticable for the measurement of the inductance of coil having high quality factor.

Advantage:

1. Balance equations are independent if C3 and R3 are variables.
2. Frequency does not affect the calculation of unknown variable.
3. Simple expressions for unknown L_s and R_s in terms of known bridge elements.
4. Maxwell‘s inductance-capacitance bridge is very useful for measurement of wide range of inductance at power and audio frequency.

Disadvantage:

1.The variable capacitor is expensive and less accurate. So fixed capacitor is sometimes used. In that case, balance adjustments are done by-

1. Varying R1 and R3, which is difficult.
2. Putting an additional resistance in series with inductance under measurement and varying this resistance and R3.

2.Limited to measurement of low Q coil (1< Q< 10) also unsuitable for very low Q (Q<1)

Note: Maxwell‘s bridge is suitable for measurement of only medium Q coils.

Hay-Inductance Bridge

It is the modification of Maxwell’s bridge .The Hay bridge circuit diagram shown in Figure below is similar to the Maxwell bridge , except that R3 and C3 are connected in series instead of parallel, and the unknown inductance is represented as a parallel LR circuit instead of a series circuit. The balance equations are found to be exact the same as those for the Maxwell bridge . It must be remembered, however, that the measured L_p and R_p are a parallel equivalent circuit. When the bridge in Figure below is balanced,