Applied Electricity I ELE 201 T.O Fajemilehin Course content • DC Bridges | AC Bridges • Resistance, Capacitance • Inductance, Transducers DC Bridges Fundamental concepts of Bridges Circuit In DC measurement circuits, the circuit configuration known as a bridge can be a very useful way to measure unknown values of resistance. As a review, the bridge circuit works as a pair of two-component voltage dividers connected across the same source voltage, with a null-detector meter movement connected between them to indicate a condition of "balance" at zero volts: Figure 1: Basic schematic diagram of standard bridges Bridge in balanced condition Any one of the four resistors in the bridge (Figure 1) can be the resistor of unknown value, and its value can be determined by a ratio of the other three, which are “calibrated” or whose resistances are known to a precise degree. When the bridge is in a balanced condition (zero voltage as indicated by the null detector), the ratio works out to be this: π 1 π 2 = π 3 which is equivalent to π 1 π 4 = π 3 π 2 π 4 Example What would be π 1 , if π 2 =2Ω, π 3 = 4Ω and π 4 =8Ω ? Practical implication of bridges One of the advantages of using a bridge circuit to measure resistance is that the voltage of the power source is irrelevant. Practically speaking, the higher the supply voltage, the easier it is to detect a condition of imbalance between the four resistors with the null detector, and thus the more sensitive it will be. A greater supply voltage leads to the possibility of increased measurement precision. However, there will be no fundamental error introduced as a result of a lesser or greater power supply voltage unlike other types of resistance measurement schemes. Principle of DC Bridges Types of DC Bridges 1. Wheatstone Bridge 2. Kelvin Bridge Wheatstone Bridge The Wheatstone bridge is the best-known bridge circuit and is used for measuring resistance. It is constructed from four resistors, one of which has an unknown value (Rx), one of which is variable (R2), and two of which are fixed and equal (R1 and R3) connected as the sides of a square. Two opposite corners of the square are connected to a source of electric current, such as a battery. A galvanometer is connected across the other two opposite corners. 3 Figure 2: Wheatstone bridge schematic diagram Measuring with the Wheatsone Bridge The variable resistor is adjusted until the galvanometer reads zero. When the voltage between point C and the negative side of the battery is equal to the voltage between point B and the negative side of the battery, the null detector will indicate zero and the bridge is said to be "balanced." It is then known that the ratio between the variable resistor and its neigbour is equal to the ratio between the unknown resistor and its neighbour, and this enables the value of the unknown resistor calculated: π 1 π π₯ = π 2 π 3 3 Figure 2: Wheatstone bridge schematic diagram π π₯ = unknown value of π 1 , π 3 = Fixed resistor π 2 = Variable resistor π= Galvanometer with high sensitivity 3 πΌ = Source Calculations from the Wheatstone bridge The bridge is balanced when no current flows through the galvanometer (πΌπ = 0) ππ΄π΅ = ππ΄πΆ or ππ΅π· = ππΆπ· Expanding: ππ΄π΅ = ππ΄πΆ π π π π +π 3 xI = π 1 π 2 +π 1 xI π π (π 1 + π 2 ) = π 1 (π π + π 3 ) π π π 1 + π π π 2 = π 1 π π + π 1 π 3 π π π 2 = π 1 π π + π 1 π 3 − π π π 1 π π π 2 = π 1 π 3 π π = π 1 π 3 π 2 Example 2 Given value R1 = 4KΩ, R2=2kΩ, R3=1.5KΩ and Rx = 3KΩ. Prove that the bridge is in balance condition. Solution: Example 3 Calculate Rx when the bridge is in balance condition Class Exercise Kelvin Bridge A Kelvin bridge (also called a Kelvin double bridge and in some countries Thomson bridge) is a measuring instrument invented by William Thomson, 1st Baron Kelvin. It is used to measure very low resistances (typically less than 1/10 of an ohm) Its operation is similar to the Wheatstone bridge except for the presence of additional resistors. These additional low value resistors and the internal configuration of the bridge are arranged to substantially reduce measurement errors introduced by voltage drops in the high current (low resistance) arm of the bridge. Figure 3: Kelvin bridge schematic diagram Rationale for the Kelvin Bridge The low-value resistors are represented by thick-line symbols, and the wires connecting them to the voltage source (carrying high current) are likewise drawn thickly in the schematic. This bridge will be explained by beginning with a standard Wheatstone bridge set up for measuring low resistance, and evolving it step-by-step into its final form in an effort to overcome certain challenges encountered in the standard Wheatstone configuration. If we were to use a standard Wheatstone bridge to measure low resistance, it would look something like this (Figure 4): Figure 4: Using a standard Wheatstone bridge to measure low resistance Challenge with using Wheatsone bridge for low resistance When the null detector indicates zero voltage, we know that the bridge is balanced and that the ratios π π π π and π π π π are equivalent. Knowing the values of π π , π π and π π therefore is expected to provide us with the necessary data to solve for π π . However, that is not always the case… Challenge: the connections and connecting wires between Ra and Rx also possess some resistance, and this ‘stray’ resistance (indicated in Figure 5 as ‘Ewire’) may be substantial compared to the low resistances of Ra and Rx. These ‘stray’ resistances will drop substantial voltage, given the high current through them, and thus affect the null detector's indication and thus the balance of the bridge: Figure 5: Stay resistances that may occur if Wheatstone bridge is used to measure low resistance Solution to measuring low resistance Since we don't want to measure these stray wire and connection resistances, but only measure π π , we must find some way to connect the null detector so that it will not be influenced by voltage dropped across them. If we connect the null detector and π π π π ratio arms directly across the ends of π π and π π , this gets us closer to a practical solution Figure 6: Evolving towards a practical solution for the stray resistances that may occur if Wheatstone bridge is used to measure low resistance Resolving the remaining stray resistances Now the top two Ewire voltage drops are of no effect to the null detector, and do not influence the accuracy of Rx's resistance measurement. However, the two remaining Ewire voltage drops (in red – Figure 6) will cause problems, as the wire connecting the lower end of Ra with the top end of Rx is now shunting across those two voltage drops, and will conduct substantial current, introducing stray voltage drops along its own length as well. Fact: The left side of the null detector must connect to the two near ends of Ra and Rx in order to avoid introducing those Ewire voltage drops into the null detector's loop. Any direct wire connecting those ends of Ra and Rx will itself carry substantial current and create more stray. Solution: … next slide Figure 7: Resolving the two stray resistances that may still occur if Wheatstone bridge is used to measure low resistance Solving for the stray resistances Make the connecting path between the lower end of Ra and the upper end of Rx substantially resistive: Insert two new resistors (π π and π π ) so that the ratio from upper to lower is the same ratio as the two arms on the other side of the null detector. Note: π π and π π are proportional to π π and π π The rheostat arm resistor π π is adjusted until the null detector indicates balance, and then we can say that Therefore: π π = π π Figure 8 : Kevin Double bridge π π π π π π π π = π π π π Simplifying the equation… The actual balance equation of the Kelvin Double bridge is as follows (π π€πππ is the resistance of the thick, connecting wire between the low-resistance standard π π and the test resistance π π ): So long as the ratio between π π and π π is equal to the ratio between π π and π π , the balance equation is π π no more complex than that of a regular Wheatstone bridge, with π = π π π π π The last term in the equation (the π π€πππ part) will be zero, cancelling the effects of all resistances except π π , π π , π π and π π Bridge circuits rely on sensitive null-voltage meters to compare two voltages for equality Review A Wheatstone bridge can be used to measure resistance by comparing unknown resistor against precision resistors of known value A Kelvin Double bridge is a variant of the Wheatstone bridge used for measuring very low resistances. Its additional complexity over the basic Wheatstone design is necessary for avoiding errors incurred by stray resistances along the current path between the low-resistance standard and the resistance being measured. Questions AC Bridges Fundamentals of AC Bridges AC bridges are similar to Wheatstone bridge in which D.C. source is replaced by an A.C. source and galvanometer with null detector (usually a headphone). The resistors of bridge are replaced with combination of resistor, inductor and capacitors (i.e. impedances). These bridges are used to determine the unknown capacitance/inductance of capacitor/inductor. The working of these bridges is also based on Ohm’s and Kirchoff’s law. Principle and Balance Condition An ac bridge having impedances π1 , π2 , π3 , π4 in its arms is shown in the Figure 1 When balance of the bridge is obtained the terminals A and C are said to be at same potential with respect to point B i.e. πΈπ΄π΅ = πΈπ΅πΆ πΌ1 π1 =πΌ2 π2 where πΈ π1 +π3 πΌ1 = πΈ π2 +π4 and πΌ2 = πΌ3 Figure 1 AC Bridge πΌ4 Principle and Balance Condition II Substituting πΌ1 and πΌ2 … πΈ π π1 +π3 1 = πΈ π π2 +π4 2 π1 π = 2 π1 +π3 π2 +π4 π1 π2 + π4 = π2 (π1 + π3 ) πΌ3 π1 π2 + π1 π4 = π2 π1 + π2 π3 π1 π4 = π2 π3 Figure 1 AC Bridge general equation for balance of ac bridge πΌ4 General equation for balance (admittances) Similar to the impedances, we can also write the general equation in terms of admittances (Y): π1 π4 = π2 π3 If the impedance is written in the form of π = ∠ππ, then the general equation can be written as: (π1 ∠π1 )(π4 ∠π4 ) = (π2 ∠π2 )(π3 ∠π3 ) π1 π4 ∠(π1 + π4 ) = π2 π3 ∠(π2 + π3 ) (7) Balance conditions Equation (7) shows two balance conditions: The first condition is that the magnitudes of the impedances satisfy the relationship shown by equation (5). It states that “The products of the magnitudes of impedances of the opposite arms must be equal”. The second condition requires that the phase angles of the impedances satisfy the relationship: ∠π1 + ∠π4 = ∠π2 + ∠π3 It states that “The sum of the phase angles of the opposite arms must be equal”. The capacitance (C) of capacitor, inductance (L) of an inductor and frequency (f) of ac source can be measured with AC bridge. AC bridges for measurement of C Applications for AC Bridges (1) De-sauty bridge (2) Weins bridge (Series) (3) Schering bridge AC bridges for measurement of L (4) Anderson bridge (5) Maxwell inductance bridge (6) Maxwell L/C bridge or Maxwell-Weins bridge (7) Hay bridge (8) Owen’s bridge (9) Heavisible –Campbell equal ratio bridge AC bridges for measurement of f (10)Robinson bridge (11)Weins bridge (parallel Measurement of Capacitance Schering Bridges A Schering Bridge is generally used to determine the unknown capacitance of a capacitor. However, it is sometimes also used to measure the insulating or dielectric properties of materials. The circuit arrangements for the bridge are given below in Figure 2 Figure 2 Schering Bridge Measurement of Capacitance II Schering Bridges Here, the ratio arm has the capacitor πΆ1 in parallel with resistor π 1 The balance condition for the bridge is given by the equation: (8) ππ = π2 π3 π1 The impedances of the arms are given by: 1 π1 1 1 1 = π1 = π + ππΆ = π + jωπΆ1 ; π2 = π 2 1 1 −π (9) 1 π π3 = −jππΆ3 = ωπΆ ; ππ = π π − jππΆπ = π π − ωπΆ ; 3 π (10) Figure 2 Schering Bridge Measurement of Capacitance III Schering Bridges Substituting the values of (9) and (10) into (8) The balance condition for the bridge is given by the equation: π π − π ωπΆπ = π 2 ( −π 1 )( ωπΆπ π 1 + jωπΆ1 ) (11) Which becomes: π π − π ωπΆπ = π 2 πΆ1 πΆ3 − ππ 2 ωπΆ3 π 1 Figure 2 Schering Bridge Combining the real and imaginary parts π π = π 2 πΆ1 πΆ3 and πΆπ = πΆ3 π 1 π 1 The power factor for the series RC combination is defined as the cosine of the phase angle of the circuit. It is given by: πππ€ππ ππππ‘ππ = π π ππ = ππΆπ π π Similarly, the dissipation factor of a series RC circuit is defined as the cotangent of the phase angle of the circuit. It is given by: π· = ππΆ1 π 1 Measurement of Inductance Maxwell Bridge The Maxwell bridge is used to measure an unknown inductance in terms of a known capacitance. The circuit for the Maxwell Bridge is shown in Figure 3: Figure 3 Maxwell Bridge Maxwell Bridge As shown in Figure 3, the ratio arm has parallel combination of resistance R1 and a capacitance C1. The unknown arm contains unknown inductance Lx and resistance Rx. The balance condition of the bridge can be given by: π1 ππ = π2 π3 ππ = π2 π3 π1 1 Where π1 = π (admittance) 1 The bridge is balanced by first adjusting π 3 for inductive balance and then by adjusting π 1 for resistive balance. Adjustment of π 1 disturbs the inductive balance and π 3 needs to be modified and accordingly π 1 also is modified. Balancing Maxwell Bridge The process gives a slow convergence and the balance slowly shifts towards the actual null point. This type of balance is called the ‘sliding balance’. The actual null point is obtained after few adjustments. From Figure 3 we can write the impedance of the arms as: Combining the real and imaginary parts Equating the real parts: π π = π 2 π 3 π 1 Equating the imaginary parts: πΏπ = π 2 π 3 πΆ1 Limitations of the Maxwell Bridge it is not suitable for the determination of the self-inductance of coils having quality factor Q > 10 or Q < 1. It is suitable for the coils having Q values in the range 1<Q<10 This is because for high Q coils the phase angle is very nearly 90 degree(positive). This requires a capacitor having phase angle very nearly equal to 90 degree (positive). This situation demands that π 1 should be very large, which is impractical. Measurement of Frequency Wein Bridge A Wien bridge is generally used to measure the frequency of the source. It is also extensively used in other applications like, harmonic distortion analyser, in audio and HF oscillators as the frequency determining element, etc. As shown in Figure 4, the Wien bridge consists of a series RC combination in one arm and a parallel RC combination in the adjoining arm. Figure 4 Wein Bridge Wein Bridge Balancing condition: π2 π3 = π1 π4 π2 = π1 π4 (1/π3 )= π1 π4 π3 (12) Now the impedances of the arms are given by the following equations: (13) (14) Balancing the Wein Bridge Substituting (13) and (14) into (12) Balancing the Wein Bridge II Using the above equation, we can determine the frequency of the source. Also, the bridge can be calibrated directly in terms of frequency Questions References DC and AC Bridges in Textbooks, Online tutorials
