5. Introduction HAPTE 5 D AND A BIDGE Bridge circuits, which are instruments for making comparison measurements, are widely used to measure resistance, inductance, capacitance, and impedance. Bridge circuits operate on a null indication principle. This means the indication is independent of the calibration of the indicating device or any characteristics of it. For this reason, very high degrees of accuracy can be achieved using the bridges. Two types of bridge are used in measurement: D bridge:. Wheatstone Bridge. Kelvin Bridge A bridge:. Similar Angle Bridge. Opposite Angle Bridge. Maxwell Bridge. Wein Bridge 5. adio Frequency Bridge 6. Schering Bridge 5. Wheatstone Bridge Wheatstone bridge is the basic dc bridge used for accurate measurement of resistance. The circuit diagram of Wheatstone bridge is shown in Figure 5.. 69
Figure 5.: Wheatsone bridge circuit The dc source, E is connected across the resistance network to provide a source of current through the resistance network. The sensitive current indicating meter or null detector usually a galvanometer is connected between the parallel branches to detect a condition of balance. When there is no current through the meter, the galvanometer pointer rests at 0 (midscale). urrent in one direction causes the pointer to deflect on one side and current in the opposite direction to otherwise. The bridge is balanced when there is no current through the galvanometer or the potential across the galvanometer is zero. At balance condition; I I () when the galvanometer to be zero, I E I + () I E I + () () and () into () 70
Ex + Ex + () (5) Example: Figure 5. consists of the following, kω, 5 kω, kω. Find the unknown resistance x. Assume a null exists. (Ans: 0 kω) Figure 5.: Wheatstone bridge circuit Solution: 7
5. Sensitivity of the Wheatsone Bridge When the bridge is in an unbalanced condition, current flows through the galvanometer, causing a deflection of its pointer. The amount of deflection is a function of the sensitivity of the galvanometer. Sensitivity can be thought of as deflection per unit current. A more sensitive galvanometer deflects by a greater amount for the same current. Deflection may be expressed in linear or angular units of measure, and sensitivity can be expressed: S milimeters deg rees µ A µ A radians µ A or Total deflection, D SxI 5. Unbalanced Wheatstone Bridge To determine the amount of deflection that would result for a particular degree of unbalance, the Thevenin s theorem can be applied. To find the current through the galvanometer, 7
Figure 5.: Unbalanced Wheatstone Bridge efer to Figure 5., the voltage at point a, a Ex E + and the voltage at point b, b Ex E + Use Voltage Divider so b a ab th Ex Ex E E E V + + therefore + + ab E E To find Thevenin s resistance, th The source, E with its internal impedance is short circuit (see Figure 5.) 7
Figure 5.: Thevenin s resistance // + th th + + // + + Therefore the Thevenin s equivalent circuit is Figure 5.5: Thevenin s equivalent circuit for unbalanced Wheatstone Bridge If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is Ig Vth th + g where g the internal resistance in the galvanometer 7
Example: alculate the current through the galvanometer in the circuit of Figure 5.6. Solution: Figure 5.6: Unbalanced Wheatstone bridge 75
5.5 Slightly Unbalanced Wheatstone bridge If three of the four resistors in a bridge are equal to and the fourth differs by 5% or less, we can develop an approximate but accurate expression for Thevenin s equivalent voltage and resistance. onsider the circuit in Fig 5.7. The voltage at point a is given as Figure 5.7: Slightly Unbalanced Wheatstone s bridge V a xe E + The point at point b is E V b + r xe + + r Thevenin s equivalent voltage is 76
V th V b V a + r + + r r xe xe + r If r is less 5% of or less, then the r term in the dominator can be neglected without introducing appreciable error. Therefore, Thevenin s voltage is E th Ex r r E The equivalent resistance can be calculated by replacing the voltage source with its internal impedance (for all practical purposes short circuit). The Thevenin s equivalent resistance is given by th x ( + r) + + + + r ( + r) + + r Again, if r is small compared to, r can be neglected. Therefore, th + Using these approximations, the Thevenin s equivalent circuit is shown in Figure 5.8. These approximate equations are about 98% accurate if r 0.05. Figure 5.8: An approximate Thevenin s equivalent circuit for a Wheatstone bridge 77
Example: Given a center zero 00 0 00µA movement having an internal resistance of 5Ω. alculate the current through the galvanometer given in Figure 5.9, by the approximation method. Solution: Figure 5.9: Slightly unbalanced Wheatstone bridge 78
5.6 Kelvin Bridge The 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. esistors in the range of Ω to approximately µω may be measured with high degree of accuracy using the Kelvin Bridge. Since the Kelvin Bridge uses a second set of the ratio as shown in Figure 5.0, it is sometimes referred to as the Kelvin double range. Figure 5.0: Basic Kelvin Bridge showing a second set of ratio arms The resistor lc shown in Figure 5.0 represents the lead and contact resistance present in the Wheatstone bridge. The second set of ratio arms ( a and b in Figure 5.0) compensates for this relatively low lead contact resistance. At balance the ratio of a to b must be equal to the ratio of to. It can be shown that, when a null exists, the value for x is the same as that for the Wheatstone bridge, which is x 79
This can be written as x Therefore when a Kelvin Bridge is balanced, we can say x b a Example: If in Figure 5.0, the ratio of a and b is 000Ω, is 5Ω and 0.5. What is the value of x. Solution: 80
5.7 A Bridges A bridges are used to measure inductance and capacitances and all A bridge circuits are based on the Wheatstone bridge. The general ac bridge circuit consists of impedances, an A voltage source, and detector as shown in Figure 5.. In A bridge circuit, the impedances can be either pure resistance or complex impedances. Figure 5.: General A bridge circuit These circuits find other applications in many communication system and complex electronic circuits. A bridge circuits are commonly used for shifting phase, providing feedback paths for oscillators and amplifiers, filtering out undesired signals, and measuring the frequency of audio signals. The operation of the bridge depends on the fact that when certain specific circuit conditions apply, the detector current comes zero. This is known as the null or balanced condition. Since the zero current means that are is no voltage difference across detector, the bridge circuit may be redrawn as in Figure 5.. The voltages at point a and b and from point a to c must be equal now. 8
Figure : Equivalent of balanced ac bridge circuit I Z I Z () Similarly, the voltages from point a and b and point d to point c must also be equal, therefore I Z I Z () equation () divide by equation () Z Z Z Z Example: The impedances of the A bridge in Figure 5. are given as follows: Z 00 0 Z 50 0 0 0 Ω Ω Z 50 0 Z x Z 0 Ω unknown Determine the constants of the unknown arm. 8
Solution: Example: Given the A bridge of Figure 5. is in balance; find the components of the unknown arms Z x. Figure 5.: A bridge in balance 8
5.8 Similar Angle Bridge The similar angle bridge (refer Figure 5.) is used to measure the impedance of a capacitive circuit. This bridge is sometimes called the capacitance comparison bridge of the series resistance capacitance bridge. The impedance of the arms of this bridge can be written as Z Z Z jx c Z x jx cx At balance condition, Figure 5.: Similar angle bridge ( x jx cx ) ( jx ) x x Example: A similar angle bridge is used to measure capacitive impedance at a frequency of khz. The bridge constants at balance are 8
00µF 50kΩ 0kΩ 00kΩ Find the equivalent series circuit of the unknown impedance. Solution: 5.9 Maxwell Bridge It is possible to determine an unknown inductance with capacitance standards. The bridge sometimes is called a Maxwell-Wein Bridge. Using capacitance as a standard has several advantages. apacitance is influenced to a lesser degree by external fields and capacitors set up virtually no external field. Furthermore, capacitors are small and inexpensive. The Maxwell Bridge is shown in Figure 5.5. 85
Figure 5.5: Maxwell Bridge The impedance of the arms of the bridge are: Z + jω Z Z Z x + jx Lx At balance, x L x Example: A Maxwell bridge is used to measure inductive impedance. The bridge constants at balance are 0.0µF 70kΩ 5.kΩ 00kΩ Find the series equivalent resistance and inductance. Solution: 86
5.0 Opposite Angle Bridge The Opposite Angle Bridge or Hay Bridge (see Figure 5.6) is used to measure the resistance and inductance of coils in which the resistance is small fraction of the reactance X L, that is a coil having a high Q, meaning a Q greater than 0. The symbol Q designates the ratio of X L to for coil. Otherwise the Maxwell Bridge is used for measuring low Q coils (Q < 0). 87
Figure 5.6: Opposite Angle Bridge or Hay Bridge x ω + ω L x + ω For Opposite Angle Bridge, it can be seen that the balance conditions depends on the frequency at which the measurement is made. Example: Find the series equivalent inductance and resistance of the network that causes an opposite angle bridge to null with the following component values: ω 000 rad/s 0kΩ kω kω µf Find x and L x. Solution: 88
5. Wein Bridge The Wein Bridge shown in Figure 5.7 has a series combination in one arm and a parallel combination in the adjoining arm. It is designed to measure frequency (extensively as a feedback arrangement for a circuit). It can also be used for the measurement of an unknown capacitor with great accuracy. Figure 5.7: Wien bridge Z jx c Z Z Z jx c a) The equivalent parallel components At balance condition: Z Z Z Z () ( jx c) jx () 89
) ( jx jx c c jx X j jx X j c c c c + j j + + ω ω setting both the real and imaginary parts to zero + () ω ω () solving equation () for : ω ω (5) and substituting equation (5) into equation (): + ω (6) 90 substituting equation (6) into equation (5) ) ( + ω (7) b) The equivalent series component solving equation () for ω (5b) Substituting equation (5b) into equation ()
+ ω (6b) Substituting equation (6b) into equation (5b) + ω (7b) 5. adio Frequency Bridge The radio frequency bridge shown in Figure 5.8 is often used in laboratories to measure the impedance of both capacitance and inductive circuits at higher frequencies. Figure 5.8: adio frequency bridge The measurement technique used with this bridge is known as the substituting technique. The bridge is first balanced with the Z x terminals shorted. After the values of and are noted, the unknown impedance is inserted at the Z x terminals, where Z x x ± jx x. ebalancing the bridge gives new values of and, which can be used to determine the unknown impedance. x ' ( ) () 9
X x ( ) () ω ' X x can be either capacitive or inductive. If >, and thus / < /, then X x is negative, indicating a capacitive reactance. Therefore x () ωx x However, if <, and thus / > /, then X x is positive and inductive and X x L x () ω The unknown impedance is represented by x ± jx x, which indicates a series connected circuit. Thus, equation () and () apply to the equivalent series components of the unknown impedance. 5. Schering Bridge A very important bridge used for the precession measurement of capacitors and their insulating properties is Schering Bridge. Its circuit arrangement is given in Figure 5.9. The standard capacitor is high quality mica capacitor (low-loss) for general measurements or an air capacitor (having stable value and a very small electric field) for insulation measurement. For balance condition: Figure 5.9: Schering Bridge 9
Z Z x Z Z Z Z Z x Z Z Y Z where Z x x j/ω x Z Z -j/ω Y / + j/ω x x Example: Find the equivalent series elements for the unknown impedance of the Schering Bridge network whose impedance measurements are to be made at null. 70kΩ 0.0µF 00kΩ 0.µF Solution: 9