CHAPTER 5 DC AND AC BRIDGES

1. Introduction Bridge circuit (DC or AC) is 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 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

2. Wheatstone Bridge The Wheatstone bridge is an electrical bridge circuit used to measure resistance. This bridge consists of a galvanometer and TWO (2) parallel branches containing FOUR (4) resistors. 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 To operate the bridge, a voltage source is connected to two terminals of the bridge.

2. Wheatstone Bridge In the circuit at right, if 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 two resistors are fixed and the resistance of other one is adjusted until the current passing through the galvanometer decreases to zero”. Figure 5.1: Wheatstone Bridge Circuit

2. Wheatstone Bridge When no current flows through the galvanometer, the bridge is called in a balanced condition. A B C D Figure 5.2: A variable resistor; the amount of resistance between the connection terminals could be varied. Figure 5.1: Wheatstone Bridge Circuit

2. Wheatstone Bridge I1R1 = I2R2 I3R3 = I4R4 When the bridge is in balanced condition, we obtain, voltage drops across R1 and R2 are equal, I1R1 = I2R2 voltage drops across R3 and R4 are also equal, I3R3 = I4R4 A B C D (2.1) (2.2) Figure 5.1: Wheatstone Bridge Circuit

2. Wheatstone Bridge I1 = I3 I2 = I4 In this point of balance, we also obtain; I1 = I3 I2 = I4 Therefore, the ratio of two resistances in the known leg is equal to the ratio of the two in the unknown leg; (2.3) (2.4) Figure 5.1: Wheatstone Bridge Circuit or (2.5)

2. Wheatstone Bridge Example 1 Find Rx at the balance condition? Figure 5.3 Find Rx at the balance condition?

2. Wheatstone Bridge Sensitivity of the Wheatstone Bridge When the pointer of a galvanometer deflects towards right or left hand side, 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 (Cont…..) How to find the current value? Sensitivity S can be expressed in linear or angular units as follows: (2.6) How to find the current value? Figure 5.4.

2. Wheatstone Bridge Thevenin’s Theorem Thevenin’s theorem is an 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 (2.7) (2.8)

Fig. 5.6: Thevenin’s resistance circuit 2. Wheatstone Bridge A B C D (Cont…..) The difference in Va and Vb represents Thevenin’s equivalent voltage. That is, (2.9) 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 as shown in Figure 5.6. Fig. 5.6: Thevenin’s resistance circuit

2. Wheatstone Bridge (Cont…..) 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 Thevenin’s resistance as follows: Fig. 5.6: Thevenin’s resistance circuit (2.10)

2. Wheatstone Bridge (Cont…..) 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

2. Wheatstone Bridge (Cont…) If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is (2.11) Fig. 5.7: Thevenin’s equivalent circuit where Rg = the internal resistance in the galvanometer.

2. Wheatstone Bridge Example 2 E= 6 V Figure 5.8 : Unbalance Wheatstone Bridge R2 = 1.5 kΩ R1 = 1.5 kΩ 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 Wheatstone 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 Figure 5.9, the voltage at point a is given as (2.12) The voltage at point b is expressed as: (2.13) 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 (2.14)

2. 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 Figure 5.10: Resistance of a Wheatstone. or (2.15)

2. 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 resistances in low resistance measurement. 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 The resistor Rlc represents the resistance of the connecting leads from R2 to Rx (unknown resistance). The second set of ratio arms (Ra and Rb in figure) compensates for this relatively low lead-contact resistance. Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms

3. Kelvin Bridge When a null exists, the value for Rx is the same as that for the Wheatstone bridge, which is C or D If the galvanometer is connected to point B, the ratio of Rb to Ra must be equal to the ratio of R3 to R1. Therefore, Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms (3.1)

ALTERNATING-CURRENT (AC)BRIDGES

4. AC Bridge Introduction In general, AC bridge has a similar circuit design as DC bridge, except that the bridge arms are impedances as shown in Figure 5.13. The impedances can be either pure resistances or complex impedances (resistance + inductance or resistance + capacitance). Therefore, AC bridges are used to measure inductance and capacitance. Some impedance bridge circuits are frequency-sensitive while others are not. The frequency-sensitive types may be used as frequency measurement devices if all component values are accurately known. Fig 5.13: General AC bridge circuit

4. AC Bridge Introduction The usefulness of AC bridge circuit is not restricted to the measurement of an unknown impedance. These circuits find other application in many communication systems and complex electronic circuits, such as for: shifting phase, providing feedback paths for oscillators or amplifiers; filtering out undesired signals; measuring the frequency of audio signals. Fig 5.13: General AC bridge circuit

4. AC Bridge Introduction AC bridge is excited by an AC source and its galvanometer is replaced by a detector. The detector can be a sensitive electromechanical meter movements, oscilloscopes, headphones, or any other device capable of registering very small AC voltage levels. AC bridge circuits work on the same basic principle as DC bridge circuits: that a balanced ratio of impedances (rather than resistances) will result in a “balanced” condition as indicated by the null-detector. Fig 5.13: General AC bridge circuit

4. AC Bridge Introduction When an AC bridge is in null or balanced condition, the detector current becomes zero. This means that there is no voltage difference across the detector and the bridge circuit in Figure 5.13 can be redrawn as in Figure 5.14. Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit

4. AC Bridge Introduction The dash line in the figure indicates that there is no potential difference and no current between points b and c. The voltages from point a to point b and from point a to point c must be equal, which allows us to obtain: (4.1) Similarly, the voltages from point d to point b and point d to point c must also be equal, leading to: Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit (4.2)

4. AC Bridge Introduction Dividing Eq. 4.1 by Eq. 4.2, we obtain: which can also be written as (4.3) Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit

4. AC Bridge Introduction If the impedance is written in the form Z = Z∟θ where Z represents the magnitude and θ the phase angle of the complex impedance, Eq. 19 can be written in the form or If the impedance is written in the form Z = Z θ where Z represents the magnitude and θ the phase angle of the complex impedance, Eq. 20 can be written in the form (4.4) Eq. 4.4 shows two conditions when ac bridge is balanced; First condition shows that the products of the magnitudes of the opposite arms must be equal: Z1Z4 = Z2Z3 Second condition shows that the sum of the phase angles of the opposite arms is equal: ∟θ1+ ∟θ4 = ∟θ2+ ∟θ3

Fig. 5-15: Similar angle bridge Similar-angle bridge is an AC bridge used to measure the impedance of a capacitive circuit. This bridge is sometimes known as the capacitance comparison bridge or series resistance capacitance bridge. The following are some components used to construct a similar-angle bridge: R1 = a variable resistor. R2 = a standard resistor. R3 = an added variable resistor needed to balance the bridge. Rx = an unknown resistor used to indicate the small leakage resistance of the capacitor. R2 C3 = a known standard capacitor in series with R3. Cx = an unknown capacitor. Fig. 5-15: Similar angle bridge

Fig. 5-15: Similar-angle bridge By referring to Figure 5.15, the impedance of the arms of this bridge can be written as R2 Fig. 5-15: Similar-angle bridge

5. Similar-Angle Bridge The condition for balance of the bridge is Two complex quantities are equal when both real and imaginary terms are equal. Therefore, or

5. Similar-Angle Bridge (5.1) and, or (5.2)

6. Maxwell Bridge Maxwell bridge is an ac bridge used to measure an unknown inductance in terms of a known capacitance. This bridge is sometimes called a Maxwell-Wien Bridge. Using capacitance as a standard has several advantages due to: Capacitance of capacitor is influenced by less external fields. Capacitor has small size. Capacitor is low cost. Fig. 5-15: Maxwell Bridge

6. Maxwell Bridge The impedance of the arms of the bridge can be written as Fig. 5-15: Maxwell Bridge

6. Maxwell Bridge The general equation for bridge balance is Fig. 5-15: Maxwell Bridge

6. Maxwell Bridge Equating real terms and imaginary terms we have (6.1) (6.2) Fig. 5-15: Maxwell Bridge

7. Opposite-Angle Bridge Opposite-angle bridge is an AC bridge for measurement of inductance. To construct this bridge can be done by replacing the standard capacitor of the similar-angle with an inductor as shown in Figure 5-14. Opposite-angle bridge is sometimes known as a Hay Bridge. It differs from Maxwell bridge by having a resistor R1 in series with a standard capacitor C1. The impedance of the arms of the bridge can be written as Fig. 5-16: Opposite-angle bridge

7. Opposite-Angle Bridge At balance: Z1Zx = Z2Z3, and substituting the values in the balance equation we obtain Equating the real and imaginary terms we have (7.1) and (7.2)

7. Opposite-Angle Bridge Solving for Rx we have, Rx = ω2LxC1R1. Substituting for Rx in Eq.7.2, Multiplying both sides by C1 we get

7. Opposite-Angle Bridge Therefore, (7.3) Substituting for Lx in Eq.7.3 into Eq.7.2, we obtain (7.4) The term ω in the expression for both Lx and Rx indicates that the bridge is frequency sensitive.

8. Wien Bridge The Wien bridge is an ac bridge having a series RC combination in one arm and a parallel combination in the adjoining arm. In its basic form, Wien’s bridge is designed to measure either the equivalent-parallel components or the equivalent-series components of an impedance. The impedance of the arms of this bridge can be written as: Z1 = R1 Z2 = R2 Fig 5-17: Wien Bridge

8. Wien Bridge Fig 5-17: Wien Bridge The impedance of the parallel arm is The impedance of the series arm is Fig 5-17: Wien Bridge

8. Wien Bridge Using the bridge balance equation, Z1Z4 = Z2Z3 we obtain: Equivalent parallel components (8.1) (8.2) Equivalent series components (8.3) (8.4)

8. Wien Bridge Knowing the equivalent series and parallel components, Wien’s bridge can also be used for the measurement of a frequency. (8.5)

9. Radio-Frequency Bridge The radio-frequency bridge is an ac bridge used to measure the impedance of both capacitance and inductance circuits at high frequency. For determination of impedance: This bridge is first balanced with the Zx shorted. After the values of C1 and C4 are noted, the unknown impedance is inserted at the Zx terminals, where Zx = Rx ± jXx. Rebalancing the bridge gives new values of C1 and C4, which can be used to determine the unknown impedance by the following formulas: (9.1) (9.2)

9. Radio-Frequency Bridge Notice that Xx can be either capacitive or inductive. If C’4 > C4, and thus 1/C’4 < 1/C4, then Xx is negative, indicating a capacitive reactance. Therefore, (9.3) However, if C’4 < C4, and thus 1/C’4 > 1/C4, then Xx is positive and inductive and (9.4) Thus, once the magnitude and sign of Xx are known, the value of inductance or capacitance can be found.

10. Schering Bridge Schering bridge is a very important AC bridge used for precision measurement of capacitors and their insulating properties. Its basic circuit arrangement given in Figure 5-19 shows that arm 1 contains a parallel combination of a resistor and a capacitor. The standard capacitor C3 is a high quality mica capacitor for general measurements, or an air capacitor for insulation measurements. A high quality mica capacitor has very low losses (no resistance) and an air capacitor has a very stable value and a very small electric field. Fig 5-19: Schering bridge

10. Schering Bridge The impedance of the arms of the Schering bridge can be written as Fig 5-19: Schering bridge

10. Schering Bridge Substituting these values into general balance equation gives:

10. Schering Bridge Equating the real and imaginary terms, we find that (10.1) (10.2)