# Maxwell Bridge – Measurement of Inductance

Maxwell Bridge is used for the measurement of Self-Inductance. Like other bridge, this method also works on balancing of bridge. Balancing of bridge is achieved when there is no current through the detector. Don’t worry we will be discussing everything in detail in this post.

Maxwell Bridge employs the comparison of test specimen variable standard inductor for the measurement of inductance. Basically there are two different bridges in Maxwell Bridge, one is Maxwell Inductance Bridge and second one is Maxwell Inductance-Capacitance Bridge.

**Maxwell Inductance Bridge**

The connection diagram and the phasor drawing of Maxwell Inductance Bridge is shown in figure below.

In the above circuit,

L_{1} = Unknown inductance having resistance R_{1}

L_{2} = Variable standard inductance with fixed resistance r_{2}

R_{2} = Variable resistance

R_{3} and R_{4} = Known resistance

As we know that, for a balanced bridge the multiplication of impedances of opposite arms must be equal.

Impedance of arm ab, Z_{1} = (R_{1}+jωL_{1})

Impedance of arm cd, Z_{2} = R_{4}

Impedance of arm ad, Z_{3} = (R_{2}+r_{2}+jωL_{2})

Impedance of arm bc, Z_{4} = R_{3}

Hence for balanced bridge,

Z_{1}Z_{2} =Z_{3}Z4

⇒(R_{1}+jωL_{1})xR_{4} = (R_{2}+r_{2}+jωL_{2})xR_{3}

⇒R_{1}R_{4}-R_{2}R_{3}-r_{2}R_{3}+jω(L_{1}R_{4}-L_{2}R_{3}) = 0

Equating real and imaginary part we get,

R_{1}R_{4}-R_{2}R_{3}-r_{2}R_{3} = 0 ……………(1)

and (L_{1}R_{4}-L_{2}R_{3}) = 0 ……………(2)

From (1),

R_{1}R_{4} = R_{2}R_{3}+r_{2}R_{3}

= R_{3}(R_{2}+r_{2})

Hence, **R _{1} = (R_{3}/R_{4})(R_{2}+r_{2})**

Now from (2),

L_{1}R_{4 }= L_{2}R_{3}

Hence, **L _{1 }= L_{2}R_{3 }/ R_{4}**

Thus unknown inductance L_{1} and its resistance R_{1} may be calculated.

Phasor diagram of Maxwell Inductance Bridge is shown below.

**Maxwell Inductance Capacitance Bridge:**

In this method, the unknown inductance is measured by comparison with standard known capacitance. The connection diagram of Maxwell Inductance Capacitance Bridge is shown below.

In the above diagram,

L_{1} = Unknown inductance with resistance R_{1}

C_{4} =variable standard capacitor

R_{2}, R_{3} & R_{4} = Known fixed resistance

Now,

Impedance of arm ab, Z_{1} = (R_{1}+jωL_{1})

Impedance of arm cd, Z_{2} = R_{4} / (1+jωC_{4}R_{4})

Impedance of arm ad, Z_{3} = R_{2}

Impedance of arm bc, Z_{4} = R_{3}

For bridge to be balance,

Z_{1}Z_{2} =Z_{3}Z4

⇒(R_{1}+jωL_{1})x [R_{4} / (1+jωC_{4}R_{4})] = R_{2}R_{3}

⇒R_{1}R_{4}-R_{2}R_{3} +jw(L_{1}R_{4}-R_{2}R_{3}C_{4}R_{4}) = 0

Equating real and imaginary parts we get,

**R _{1} = R_{2}R_{3} / R_{4}**

**and L _{1} = R_{2}R_{3}C_{4}**

The quality factor of inductor may also be calculated as

Q = ωL_{1}/R_{1}

= ωR_{2}R_{3}C_{4} / R_{1}

Since R_{4} = R_{2}R_{3}C_{4} / R_{1} , hence

**Q = ωC _{4}R_{4}**

The phasor diagram of Maxwell Inductance Capacitance Bridge is shown below.

**Advantage**

- The expression of inductance is independent of frequency.
- A wide range of inductance can be measured at power and audio frequencies.
- The expression for inductance is simple and can easily be calculated.

**Disadvantage**

Since Maxwell Inductance Capacitance Bridge uses variable standard capacitor, it is very expensive to get variable standard capacitor.

**Applicability of Maxwell Bridge:**

Maxwell Bridge is suitable for the measurement of inductance with medium vale of Q (1<Q<10). This method is not suitable for measurement of inductance with high value of quality factor Q. Since Q = wC_{4}R_{4}, we will need higher value of resistance R_{4} for measurement of high Q coil which is very expensive.