CHAPTER 5 DC AND AC BRIDGES

1. Introduction Bridge circuits (DC & AC) are an instrument to measure resistance, inductance, capacitance and impedance. Operate on a null-indication principle. This means the indication is independent of the calibration of the indicating device or any characteristics of it. # Very high degrees of accuracy can be achieved using the bridges. Used in control circuits. # One arm of the bridge contains a resistive element that is sensitive to the physical parameter (temperature, pressure, etc.) being controlled.

1. Introduction TWO (2) TYPES of bridge circuits are used in measurement: 1) DC bridge: a) Wheatstone Bridge b) Kelvin Bridge 2) AC bridge: a) Similar Angle Bridge b) Opposite Angle Bridge/Hay Bridge c) Maxwell Bridge d) Wein Bridge e) Radio Frequency Bridge f) Schering Bridge

DIRECT-CURRENT (DC)BRIDGE

Figure 5.1: Wheatstone Bridge Circuit The Wheatstone bridge is an electrical bridge circuit used to measure resistance. It consists of a voltage source and a galvanometer that connects two parallel branches, containing four resistors. Figure 5.1: Wheatstone Bridge Circuit One parallel branch contains one known resistance and one unknown; the other parallel branch contains resistors of known resistances.

Figure 5.1: Wheatstone Bridge Circuit In the circuit at right, R4 is the unknown resistance; R1, R2 and R3 are resistors of known resistance where the resistance of R3 is adjustable. How to determine the resistance of the unknown resistor, R4? “The resistances of the other three are adjusted and balanced until the current passing through the galvanometer decreases to zero”. Figure 5.1: Wheatstone Bridge Circuit

Figure 5.1: Wheatstone Bridge Circuit R3 is varied until voltage between the two midpoints (B and D) will be zero and no current will flow through the galvanometer. A B C D Figure 5.1: Wheatstone Bridge Circuit Figure 5.2: A variable resistor; the amount of resistance between the connection terminals could be varied.

Figure 5.1: Wheatstone Bridge Circuit When the bridge is in balance condition (no current flows through galvanometer G), we obtain; voltage drop across R1 and R2 is equal, I1R1 = I2R2 voltage drop across R3 and R4 is I3R3 = I4R4 Figure 5.1: Wheatstone Bridge Circuit

Figure 5.1: Wheatstone Bridge Circuit In this point of balance, we also obtain; I1 = I3 and I2 = I4 Therefore, the ratio of two resistances in the known leg is equal to the ratio of the two in the unknown leg; Figure 5.1: Wheatstone Bridge Circuit

2. Wheatstone Bridge Example 1 Figure 5.3 Find Rx?

2. Wheatstone Bridge Sensitivity of the Wheatstone Bridge When the pointer of a bridge galvanometer deflects to right or to left direction, this means that current is flowing through the galvanometer and the bridge is called in an unbalanced condition. The amount of deflection is a function of the sensitivity of the galvanometer. For the same current, greater deflection of pointer indicates more sensitive a galvanometer. Figure 5.4.

2. Wheatstone Bridge Sensitivity of the Wheatstone Bridge (Cont…) Sensitivity S can be expressed in units of: How to find the current value? Figure 5.4.

Fig. 5.5: Thevenin’s equivalent voltage 2. Wheatstone Bridge Thevenin’s Theorem Thevenin’s theorem is a approach used to determine the current flowing through the galvanometer. Thevenin’s equivalent voltage is found by removing the galvanometer from the bridge circuit and computing the open-circuit voltage between terminals a and b. Fig. 5.5: Thevenin’s equivalent voltage Applying the voltage divider equation, we express the voltage at point a and b, respectively, as

Fig. 5.6: Thevenin’s resistance 2. Wheatstone Bridge Thevenin’s Theorem (Cont…) The difference in Va and Vb represents Thevenin’s equivalent voltage. That is, Fig. 5.5: Wheatstone bridge with the galvanometer removed Thevenin’s equivalent resistance is found by replacing the voltage source with its internal resistance, Rb. Since Rb is assumed to be very low (Rb ≈ 0 Ω), we can redraw the bridge as shown in Fig. 5.6 to facilitate computation of the equivalent resistance as follows: Fig. 5.6: Thevenin’s resistance

2. Wheatstone Bridge Thevenin’s Theorem (Cont…) Fig. 5.6: Thevenin’s resistance If the values of Thevenin’s equivalent voltage and resistance have been known, the Wheatstone bridge circuit in Fig. 5.5 can be changed with Thevenin’s equivalent circuit as shown in Fig. 5.7, Fig. 5.5: Wheatstone bridge circuit Fig. 5.7: Thevenin’s equivalent circuit

Fig. 5.7: Thevenin’s equivalent circuit 2. Wheatstone Bridge Thevenin’s Theorem (Cont…) If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is Fig. 5.7: Thevenin’s equivalent circuit where Rg = the internal resistance in the galvanometer

2. Wheatstone Bridge Example 2 R2 = 1.5 kΩ R1 = 1.5 kΩ Rg = 150 Ω E= 6 V Figure 5.8 : Unbalance Wheatstone Bridge Rg = 150 Ω R3 = 3 kΩ R4 = 7.8 kΩ Calculate the current through the galvanometer ?

Slightly Unbalanced Wheatstone Bridge If three of the four resistors in a bridge are equal to R and the fourth differs by 5% or less, we can develop an approximate but accurate expression for Thevenin’s equivalent voltage and resistance. Consider the circuit in Fig- 5.9, the voltage at point a is given as The voltage at point b is expressed as Figure 5.9: Wheatstone Bridge with three equal arms

2. Wheatstone Bridge Slightly Unbalanced Wheatstone Bridge (Cont…) Thevenin’s equivalent voltage is the difference in this voltage If ∆r is 5% of R or less, Thevenin equivalent voltage can be simplified to be

Figure 5.10: Resistance of a Wheatstone. 2. Wheatstone Bridge Slightly Unbalanced Wheatstone Bridge (Cont…) Thevenin’s equivalent resistance can be calculated by replacing the voltage source with its internal resistance and redrawing the circuit as shown in Figure 5.10. Thevenin’s equivalent resistance is now given as o R R + Δr If ∆r is small compared to R, the equation simplifies to or Figure 5.10: Resistance of a Wheatstone.

2. Wheatstone Bridge Slightly Unbalanced Wheatstone Bridge (Cont…) We can draw the Thevenin equivalent circuit as shown in Figure 5.11 Figure 5.11: Approximate Thevenin’s equivalent circuit for a Wheatstone bridge containing three equal resistors and a fourth resistor differing by 5% or less

3. Kelvin Bridge Kelvin bridge is a modified version of the Wheatstone bridge. The purpose of the modification is to eliminate the effects of contact and lead resistance when measuring unknown low resistances. The measurement with a high degree of accuracy can be done using the Kelvin bridge for resistors in the range of 1 Ω to approximately 1 µΩ. Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms Since the Kelvin bridge uses a second set of ratio arms (Ra and Rb, it is sometimes referred to as the Kelvin double bridge.

3. Kelvin Bridge Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms The resistor Rlc represents the lead and contact resistance present in the Wheatstone bridge. The second set of ratio arms (Ra and Rb in figure) compensates for this relatively low lead-contact resistance.

3. Kelvin Bridge When a null exists, the value for Rx is the same as that for the Wheatstone bridge, which is or At balance the ratio of Rb to Ra must be equal to the ratio of R3 to R1. Therefore,