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A
Hay
Bridge
is an AC bridge circuit used for measuring an unknown
inductance
by balancing the loads of its four arms, one of which contains the
unknown inductance. One of the arms of a Hay Bridge has a capacitor
of known characteristics, which is the principal component used for
determining the unknown inductance value. Figure 1 below shows a
diagram of the Hay Bridge.
Figure
1. The Hay Bridge
As shown
in Figure 1, one arm of the Hay bridge consists of a capacitor in
series with a resistor (C1 and R2) and another arm consists of an
inductor L1 in series with a resistor (L1 and R4). The other two
arms simply contain a resistor each (R1 and R3). The values of
R1and R3 are known, and R2 and C1 are both adjustable. The unknown
values are those of L1 and R4.
Like
other bridge circuits, the measuring ability of a Hay Bridge
depends on 'balancing' the circuit. Balancing the circuit in Figure
1 means
adjusting R2 and C1 until the current through the
ammeter between points A and B becomes zero. This happens when
the voltages at points A and B are equal. When the Hay Bridge
is balanced, it follows that Z1/R1 = R3/Z2 wherein Z1 is the impedance
of the arm containing C1 and R2 while Z2 is the impedance of the arm
containing L1 and R4. Thus, Z1 = R2 + 1/(2πfC) while Z2 = R4 +
2πfL1.
Mathematically, when the
bridge is balanced,
[R2 +
1/(2πfC1)] / R1 = R3 / [R4 + 2πfL1]; or
[R4 + 2πfL1]
= R3R1 / [R2 + 1/(2πfC1)]; or
R3R1 =
R2R4 + 2πfL1R2 + R4/2πfC1 + L1/C1.
When the
bridge is balanced, the reactive components are equal, so
2πfL1R2 =
R4/2πfC1, or
R4 =
(2πf)2L1R2C1.
Substituting R4, one comes up with the following equation:
R3R1 =
(R2+1/2πfC1)((2πf)2L1R2C1) + 2πfL1R2 + L1/C1; or
L1 =
R3R1C1 / (2πf)2R22C12 + 4πfC1R2 + 1); or
L1 =
R3R1C1 / [1 + (2πfR2C1)2] after dropping the reactive
components of the equation since the bridge is balanced.
Thus, the
equations for L1 and R4 for the Hay Bridge in Figure 1 when it is balanced are:
L1 =
R3R1C1 / [1 + (2πfR2C1)2]; and
R4 =
(2πfC1)2R2R3R1 / [1 + (2πfR2C1)2]
Note that
the balancing of a Hay Bridge is frequency-dependent.
See Also:
Bridge Circuits;
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