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ETEC 3501 Chapter 3 Simple Resistive Circuits 3.5 Measuring Voltage and Current • Ammeter – an instrument designed to measure current. It is placed in series with the circuit elements whose current is being measured. • Voltmeter – an instrument designed to measure voltage. It is placed in parallel with the with the circuit elements whose voltage is being measured. • Ideal ammeters and voltmeters have no effect on the circuit variables they are designed to measure. • That is an ideal ammeter has a resistance of zero Ohms and an ideal voltmeter has a resistance of infinite Ohms. • An ammeter and voltmeter connected in a circuit are shown in Figure 3.21. • The effects of an ideal ammeter and voltmeter are illustrated in Figure 3.22. Figure 3.21 An ammeter connected to measure the current in R1 and a voltmeter connected to measure the voltage across R2. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.22 A short-circuit model for the ideal ammeter, and an open-circuit model for the ideal voltmeter. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. • Digital meters – measure analog values using an analog-todigital (A/D) converter. Are typically accurate over a broad range of values. • Analog meters – based on d’ Arsonval movement (Figure 3.23). Accuracy is dependent on value of the elements being measured. • An analog ammeter consists of a d’ Arsonval movement in parallel with a resistor (Figure 3.24). • An analog voltmeter consists of a d’ Arsonval movement in series with a resistor (Figure 3.25). • A typical d’ Arsonval movement would be rated at 50 mV and 1 mA. So at those values the needle would be at full scale. Figure 3.23 A schematic diagram of a d’Arsonval meter movement. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.24 A dc ammeter circuit. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.25 A dc voltmeter circuit. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. • To measure a specific range of current or voltage, the value of parallel or series resistors must be sized accordingly. • These resistances as well as the resistance of the d’Arsonval movement affect the circuit being measured. • These effects are demonstrated in Example 3.5 and 3.6 Example 3.5 – Using a d’Arsonval Ammeter Example 3.6 – Using a d’Arsonval Voltmeter Assessment Problems (a) i = 1V / 100 Ohm = 10 mA (b) R m = 50 mV / 10 mA = 5 Ohm i = 1V/ 105 Ohm = 9.524 mA (a) v = 60V (75k/ (15k+75k)) = 50V (b) R m = 150k Ohm R’ = 75k || 150k = 50k Ohm v = 60V(50k/(15k + 50k)) = 46.15 V 3.6 Measuring Resistance – The Wheatstone Bridge • The Wheatstone bridge is used to precisely measure resistances of medium value (1 Ohm to 1M Ohm). • Accuracies of + 0.1 % are possible. • The bridge circuit is shown in Figure 3.26. • R1, R2 , R3 are known resistors and RX is the unknown resistor. • A d’Arsonval movement in the microampere range (galvanometer) is used for current detection. • The variable resistor (R3) is adjusted until zero current is indicated by the galvanometer. Figure 3.26 The Wheatstone bridge circuit. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. • RX can them be determined using Equation 3.33. • For the bridge to be balanced what must the voltage difference be between point a and point b of Figure 3.27 ? Figure 3.27 A balanced Wheatstone bridge (ig = 0). Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits • Looking further at the bridge circuit of Figure 3.26, if the galvanometer is replaced with an equivalent resistance Rm , it results in the circuit shown in Figure 3.28 • To reduce this bridge circuit to a single resistance across the battery, methods beyond simple series and parallel reduction must be employed. • The interconnecting resistors can be reduced to a single equivalent resistance by means of a delta-to-wye equivalent circuit. Figure 3.28 A resistive network generated by a Wheatstone bridge circuit. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.29 A ∆ configuration viewed as a π configuration. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.30 A Y structure viewed as a T structure. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.31 The ∆-to-Y transformation. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. • The terminals of a, b, and c shown in Figure 3.31 must have equivalent behavior in the delta configuration as well as the wye configuration to be equivalent circuits. • Equations 3.41 through 3.43 are derived using series and parallel circuit simplification. • Manipulation of Equations 3.41 through 3.43 yields the Yconnected resistors in terms of the delta-connected resistors. • Expressions for the three delta resistors in terms of Yconnected resistors are given in Equations 3.47 through 3.49 Example 3.7 – Applying a Delta-to-Wye Transform Assessment Problem Figure 3.36 Model of a defroster grid. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Figure 3.37 A simplified model of the defroster grid. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Electric Circuits, Ninth Edition James W. Nilsson • Susan A. Riedel Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.