1. Measurements of Inductance & Capacitance
Ms. Rokhaiya Banu, Asst. Professor
Department: B.E. Electrical Engineering
Subject: Electrical & Electronics Measurements
Semester: III
Teaching Aids Service by KRRC Information Section

2. Measurement of Inductance
Inductance can be measured by,
Maxwell Bridge
Anderson Bridge

3. AC Bridge

4. Derivation

5. AC bridge

6. Maxwell Inductance bridge

7. Maxwell Inductance Capacitance Bridge

8. Advantages
The frequency does not appear in the final expression of both equations, hence it is independent of frequency.
Maxwell's inductor capacitance bridge is very useful for the wide range of measurement of inductor at audio frequencies.

9. Disadvantages The variable standard capacitor is very expensive.
The bridge is limited to measurement of low quality coils (1 < Q < 10) and it is also unsuitable for low value of Q (i.e. Q < 1) from this we conclude that a Maxwell bridge is used suitable only for medium Q coils.

10. Anderson Bridge
This bridge is the modified version of Maxwell Inductance Capacitance Bridge.
Double balance can obtained by fixing the value of capacitance and changing the value of electrical resistance only.

11. Circuit diagram & Phasor Diagram

12. Derivation
In this circuit the unknown inductor is connected between the point a and b with electrical resistance r1 (which is pure resistive). The arms bc, cd and da consist of resistances r3, r4 and r2 respectively which are purely resistive.
A standard capacitor is connected in series with variable electrical resistance r and this combination is connected in parallel with cd.
A supply is connected between b and e. Now let us derive the expression for l1 and r1: At balance point, we have the following relations that holds good and they are: i1 = i3 and i2 = ic + i4
Now equating voltages drops we get,
Putting the value of ic in above equations, we get
The above equation (7) obtained is more complex that we have obtained in Maxwell bridge.

13. Advantages
It is very easy to obtain the balance point in Anderson's bridge as compared to Maxwell bridge in case of low quality factor coils.
There is no need of variable standard capacitor is required instead of thin a fixed value capacitor is used.
This bridge also gives accurate result for determination of capacitance in terms of inductance.

14. Disadvantages
The equations obtained for inductor in this bridge is more complex as complex as compared to Maxwell's bridge.
The addition of capacitor junction increases complexity as well as difficulty of shielding the bridge.
Considering all the advantages and disadvantages, Maxwell bridge is preferred over Anderson's bridge whenever use of variable capacitor is permissible.

15. Schering Bridge This bridge is used to measure to the
Capacitance of the capacitor
Dissipation factor and
Measurement of relative permittivity

16. Circuit diagram & Phasor diagram

17. Derivation

18. Derivation

19. Advantages The balance equation is independent of frequency.
It is used for measuring the insulating properties of electrical cables and equipments.

20. De Sauty's bridge
This bridge provide us the most suitable method for comparing the two values of capacitor if we neglect dielectric losses in the bridge circuit. The circuit of De Sauty's bridge is shown below.
Battery is applied between terminals marked as 1 and 4. The arm 1-2 consists of capacitor c1 (whose value is unknown) which carries current i1 as shown, arm consists of pure resistor (here pure resistor means we assuming it non inductive in nature), arm also consists of pure resistor and arm consists of standard capacitor whose value is already known to us. Let us derive the expression for capacitor c1 in terms of standard capacitor and resistors.

21. Let us derive the expression for capacitor c1 in terms of standard capacitor and resistors. At balance condition we have,
It implies that the value of capacitor is given by the expression
In order to obtain the balance point we must adjust the values of either r3 or r4 without disturbing any other element of the bridge. This is the most efficient method of comparing the two values of capacitor if all the dielectric losses are neglected from the circuit.

22. Now let us draw and study the phasor diagram of this bridge
Now let us draw and study the phasor diagram of this bridge. Phasor diagram of De Sauty bridge is shown below:
Let us mark the current drop across unknown capacitor as e1, voltage drop across the resistor r3 be e3, voltage drop across arm be e4 and voltage drop across arm be e2. At balance condition the current flows through path will be zero and also voltage drops e1 and e3 be equal to voltage drops e2 and e4 respectively. In order to draw the phasor diagram we have taken e3 (or e4) reference axis, e1 and e2 are shown at right angle to e1 (or e2).

23. Modify De Sauty's bridge, we want to have such a kind of bridge that will gives us accurate results for imperfect capacitors also. This modification is done by Grover. The modified circuit diagram is shown below:
Here Grover has introduced electrical resistances r1 and r2 as shown in above on arms and respectively, in order to include the dielectric losses. Also he has connected resistances R1 and R2 respectively in the arms and Let us derive the expression capacitor c1 whose value is unknown to us. Again we connected standard capacitor on the same arm as we have done in De Sauty's bridge. At balance point on equating the voltage drops we have:
On solving above equation we get

24. By making the phasor diagram we can calculate dissipation factor
By making the phasor diagram we can calculate dissipation factor. Phasor diagram for the above circuit is shown below
Let us mark δ1 and δ2 be phase angles of the capacitors c1 and c2 capacitors respectively.
From the phasor diagram
we have tan(δ1) = dissipation factor = ωc1r1 and similarly we have tan(δ2) = ωc2r2.
From equation (1) we have
on multiplying ω both sides we have
Therefore the final expression for the dissipation factor is written as
Hence if dissipation factor for one capacitor is known. However this method is gives quite inaccurate results for dissipation factor.

25. Q-Meter

26.  The ratio of the inductive reactance to the effective resistance of the coil is called the quality factor   or Q-factor of the coil.
So Q = XL / R = ωL / R
A high value of Q is always desirable as it means high inductive reactance and low resistance. A low value of Q indicates that the resistance component is relatively high and so there is a comparatively large loss of power.
The Q-meter is an instrument designed for the measurement of Q-factor of the coil as well as for the measurement of electrical properties of coils and capacitors. –
This instru­ment operates on the principle of series resonance i.e. at resonate condition of an ac series circuit voltage across the capacitor is equal to the applied voltage times of Q of the circuit. If the voltage applied across the circuit is kept-constant then voltmeter connected across the capacitor can be calibrated to indicate Q directly.