CHAPTER 4
BRIDGE MEASUREMENTS
Dr. Gamal Sowilam Introduction
Precision measurements of component values have been made for many years using various forms of bridges. The simplest form of bridge is for the purpose of measuring resistance and is called Wheatstone bridge. There is entire group of ac bridge for measuring inductance, capacitance, admittance, conductance, and any of the impedance parameters. Bridge circuits are also frequently used in control circuit. When used in such applications. One arm of the bridge contains a resistive elements that is sensitive to the physical parameter (temperature, pressure, etc.) being control. . Some specialized measurements, such as impedance at high frequencies are still made with a bridge. . The bridge circuit still forms the backbone of some measurements and for the interfacing of transducers. Basic Operation . The bridge has four resistive arms, together with a source of emf (a battery), and a null detector, usually a galvanometer or other sensitive current meter. . The current through the galvanometer depends on the potential difference between points A and B. . The bridge is said to be balanced when the potential difference across the galvanometer is 0 V so that there is no current through the galvanometer. . The bridge is balanced when
I1 I3 and I 2 I4 I R I R 1 1 2 2 and I3 R3 I4 R4
I1R3 I 2 R4 . When the galvanometer current is zero, E E I1 I3 I I R R and 2 4 1 3 R2 R4 . Combining the equations above and simplifying,
R1 R2 R1R4 R2 R3 R3 R4 . Expression for balance of the Wheatstone bridge. If Rx R4 is unkown resistance
. Unknown resistor can be determined as
R2 Rx R3 R1
. Resistor R3 is called the standard arm and Resistor R2 and R1 are called the ratio arms.
R2 / R1 Sensitivity of the Wheatstone bridge S=millimeters/A or degrees/ A or radians/ A Therefore, if follows that total deflection D is: D=S*I Example 1: Determine the value of the unknown resistor, in the circuit of the following figure, assuming a null exists (the current through the galvanometer is zero. Solution: The bridge is balanced when: Measurement Errors
The main source of error: limiting errors of the three known resistors. Insufficient sensitivity of the null detector. Prevention: calculate the galvanometer current to determine whether or not the galvanometer has the required sensitivity to detect an unbalance condition. Changes in resistance of the bridge arms due to the heating effect of the current through the resistors. Prevention: The power dissipation in the bridge arms must be computed in advance and the current must be limited to a safe value. Thermal emf’s in the bridge circuit of the galvanometer circuit can also cause problems when low-value resistors are being measured. Prevention: the more sensitive galvanometers sometimes have copper coils and copper suspension systems to avoid having dissimilar metals in contact with one another and generating thermal emfs. Errors due to the resistance of leads and contacts exterior to the actual bridge circuit. Prevention: may be reduced by a Kelvin bridge. Thevenin Equivalent Circuit It is necessary to calculate the galvanometer circuit to determine whether or not the galvanometer has the required sensitivity to detect an unblance conditions. Different galvanometer not only may require different currents per unit deflection (current sensitivity), but also may have a difference internal resistance. Converting the Wheatstone bridge to its Thevenin equivalent circuit in order to find the current follows in the galvanometer: There are two steps must be taken: . Finding the equivalent voltage when the galvanometer is removed from the circuit (the open voltage between A and B of bridge). . Finding the equivalent resistance, with the battery replaced by its internal resistance (removing the voltage source and makes its side short circuit and removing current source makes its side open circuit). Calculate
Calculate Example 2 Calculate the current passes in the galvanometer of the following circuit. Solution: 1. Find Vth
R 2. Find rth Th
G VTh
3. Find IG Problem 1 The schematic diagram of a Wheatstone bridge with values of the elements is shown in figure. The battery voltage is 5 V and its internal resistance negligible. The galvanometer has a current sensitivity of 10 mm/A and an internal resistance of 100 Ω. Calculate the deflection of the galvanometer caused by the 5 Ω unbalance in arm BC. R1=100, R2 = 1000, R3=200 and R4=2005[ IG = 3.32A and Deflection=33.2mm]. Problem 2 The galvanometer of problem 1 is replaced by one with an internal resistance of 500 and a current sensitivity of 1 mm/A. Assuming that a deflection of 1 mm can be observed on galvanometer scale, determine if this new galvanometer is capable of detecting the 5munbalance in arm BC of figure. [ IG = 2.24A and Deflection=32.24mm].
5V D B 2. Digital Readout Bridges: Applications on dc Bridges Murray Loop
AC Bridges . The ac bridge is a natural outgrowth of the dc bridge and in its basic form consists of four bridge arms, source of excitation and a null detector. . The power source supplies an ac voltage to the bridge at the desired frequency. . For measurements at low frequencies, the power line may serve as the source of excitation; at higher frequencies, an oscillator generally supplies the excitation voltage. The null detector must respond to ac unbalance currents and in its cheapest (but very effective) form consists of a pair of headphones. . In other applications, the null detector may consist of an ac amplifier with an output meter, or an electron ray tube (tuning eye) indicator. 1. ac Wheatstone bridge
. The four bridge arms Z1, Z2, Z3, and Z4 are indicated as unspecified impedances and the detector is represented by headphones. . The balance condition in this ac bridge is reached when the detector response is zero, or indicates a null. . Balance adjustment to obtain a null response is made by varying one or more of the bridge arms. . The general equation for bridge balance is obtained by using complex notation for the impedances of the bridge circuit. (boldface type is used to indicate quantities in complex notation.) . The quantities may be impedances or admittance as well as voltages or currents. Balance of ac Wheatstone bridge . The condition for bridge balance requires that the potential difference from b to c be zero. . This will be the case when the voltage drop from a to b equals the voltage drop from a to c, in both magnitude and phase.
The voltage difference between a, b equal a,c
Vab Vac I1Z1 I 2 Z 2 The voltage difference between b, d equal c,d
Vbd Vcd I3 Z 3 I 4 Z 4 The current folows in bc 0,
I1 I3 and I 2 I4 Thus:
The above equations can be arranged as:
and
The balance conditions of bridge are: Example 3 The impedance of the basic ac bridge are given as follows: Z 30030o 1 Z2 150 o Z3 250 40 Z 4 unknown Determine the constants of the unknown arm. Solution From the balance condition of bridge If the frequency of source is 60Hz. Find the value of capactance
Example 4 The ac bridge in the figure is in balance with the following constants: arm AC, R = 450 ; arm AD, R = 300 in series with C = 0.4 F; arm BD, unknown; arm BC, R = 200 in series with L = 15.9 mH. The oscillator frequency is 1 kHz. Find the constants of arm BD. C
D Solution
From the balance condition of bridge 2. Maxwell Bridge To measure the inductive reactance
and The balance condition
Equation zzzz
Equating the real part of the equation zzzzz
Equating the imaginary part equation zzzz 3. Similar Angle bridge To measure the capacitive reactance
The balance condition
Equation xxx Equating the real part of the equation xxx Equating the imaginary part equation xxx
Note that the unknown impedance can be any impedance whose reactance is more capacitive than inductive. In other works, the unknown can be either an RC or RLC combination whose reactive component is negative. For this reason, the unknown resistance and capacitance obtained are referred as the equivalent series resistance and the equivalent series capacitance. THANK YOU FOR YOUR ATTENTION!!