1/15/2012. Overview. 06-Basic Laws Part 3 Text: Chapter Voltage Drop. Voltage Drop. Multi-Resistor Circuits. Voltage Drop

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1 /5/0 Overview 0Basic Laws Part Text: Chapter 58 ECEG 0 Electric Circuits oltage Drop oltage Divider Series esistors Parallel esistors DeltaWye Conversion oltage Drop The voltage difference across the terminals of a resistor is called the voltage drop f the current is negative, then it is a voltage rise oltage drop is caused by the current flowing the resistor and by Ohm s Law is ual to x Higher oltage Lower oltage oltage Drop The current through a resistor can be quickly found by: Where and are referenced to the same node, and > Dr Louie Dr Louie oltage Drop For example, the current through the resistor can be computed as: A Multiesistor Circuits Now back to multiresistor circuits What is the voltage across each resistor? KL: 0 = KCL: = Ohm s Law: = = 0 5 est of circuit = 0 Dr Louie 5 Dr Louie

2 /5/0 Multiesistor Circuits Multiesistor Circuits Using 0 = = = = = Gives From this simple example, note: esistors are in series Proportion of applied voltage across a resistor, is the same proportion of the resistor to the total resistance : 0% of total resistance and 0% of the voltage ( = ) : 0% of total resistance and 0% of the voltage ( =) 0 Dr Louie 7 Dr Louie 8 oltage Divider The voltage across each resistor is More generally, the voltage across the nth resistor of N resistors in series is: Find,, n n N Dr Louie 9 Dr Louie 0 Series esistances Find,, 5 5 What is the current? = = = 075A = 5 = = 075A = 5 = = 075A Dr Louie Dr Louie

3 /5/0 Series esistances Find an uivalent circuit with one resistor ( ) so that the same current flows Series esistances By Ohm s Law: = 075 = Note = =?? Dr Louie Dr Louie Series esistances More generally: the uivalent resistance ( ) of any number (N) of resistors in series is the sum of the individual resistances ( n ) N n n Series esistances Consider a 00 m wire whose resistivity is Ohm per 5m We can model this in many ways For example: one 00m wire = or = = or = = two 50m wires one 5m wire connected endtoend connected endtoend with a 75 m wire Dr Louie 5 Dr Louie Series esistances The uivalent resistance of series connected resistors is the sum of their resistances Compute the uivalent resistance between the terminals of the following circuits Open circuit additional resistors N = = N Dr Louie 7 Dr Louie 8

4 /5/0 Compute the uivalent resistance between the terminals of the following circuits Open circuit Series esistance When adding resistances in series the overall resistance increases Less current flows than if there is only one resistance infinite 05 Dr Louie 9 Dr Louie 0 Find the voltage across each resistor, and the power supplied by the voltage source 8 Ω Ω Find the voltage across each resistor, and the power supplied by the voltage source Ω Ω Dr Louie Dr Louie Find the voltage across each resistor, and the power supplied by the voltage source The voltmeter reads What is? Ps 7W or s s s A P 7W 8 s Ω Ω 0Ω Dr Louie Dr Louie

5 /5/0 The voltmeter reads What is? 0 0 or by inspection 0Ω Find, and the voltage across the dependent voltage source 8Ω Ω Dr Louie 5 Dr Louie First find, then the voltage across the independent source d d A d Parallel esistance Assume two wires are connected as terminal a terminal b What is the circuit model of this? 8Ω Ω terminal a terminal b Dr Louie 7 Dr Louie 8 Find the fraction of that flows through each resistor 0 By KCL = By KL 0 = = By Ohm s Law = = 0 Dr Louie 9 Dr Louie 0 5

6 /5/0 Combining = 0 = = = = Yields Describes how current divides Note: 0 f we divide the numerators and denominators by, then: G G G 0 G G G Dr Louie Dr Louie From this simple example: esistors are in parallel Greater current flows through the resistor with lower resistance Proportion of current through a resistor is proportional to the conductance of that resistor to the total conductance 0 percent of total through the Ohm resistor 0 percent of total through Ohm resistor Parallel esistance We now know how the current divides, but how much current flows? 0 0 0( ) 7A Therefore G G (sum of conductances) 0 Dr Louie Dr Louie Note: Parallel esistance The uivalent resistance of two parallel resistors is their product divided by their sum The uivalent resistance of N resistors in parallel is G G GN G N The current through the nth resistor of N resistors in parallel is: 0 n n N Gn G G G N Dr Louie 5 Dr Louie

7 /5/0 Parallel esistance When adding resistances in parallel the uivalent resistance decreases esistance always be less than the smallest individual resistance More total current flows Symbol is used in an uation to designate a parallel relationship between elements : The uivalent resistance is always smaller than the smallest individual resistance : a 00 Ohm, 0 Ohm and Ohm resistors are placed in parallel What is the uivalent resistance? Dr Louie 7 Dr Louie 8 The uivalent resistance is always smaller than the smallest individual resistance : a 00 Ohm, 0 Ohm and Ohm resistors are placed in parallel What is the uivalent resistance? Find the current through each resistor, and the power supplied by the current source 5A 05 Dr Louie 9 Dr Louie 0 Find the current through each resistor, and the power supplied by the current source A 5 A 05 5 A 5A 05 Find the current through each resistor, and the power supplied by the current source 0 5x P 5 W or 0 8 G G G 0 5 P 5 W 5A 05 Dr Louie Dr Louie 7

8 /5/0 Parallel Circuits t may be easier to combine parallel resistors two at a time (easy to compute productoversum) : DeltaWye Circuits t is possible for elements to be arranged such that they are not in series or parallel s c b a Tee connection connection Dr Louie Dr Louie DeltaWye Circuits Often interested in converting a Delta ( ) to a Wye (Tee) or vice versa nterested in terminal characteristics Convert from Delta to Wye (or vice versa) so that same resistance is seen from terminals (,) and (,) c DeltaWye Circuits esistance between terminals, in Wye is: = n Delta it is: = b ( a b ) c b a b a Dr Louie 5 Dr Louie DeltaWye Circuits Setting them ual and solving b ( a b ) = epeating for terminals, and, gives c ( c b ) = a ( c b ) = b c a DeltaWye Circuits Solving, and in terms of a, b, c by combining uations b c a b c a c a b c a b a b c c b a Product of adjacent divided by sum or resistors (see Fig 9) Dr Louie 7 Dr Louie 8 8

9 /5/0 DeltaWye Circuits Solving a, b and c in terms of,, by combining uations a b c b c a DeltaWye Circuits How are the uations reduced when = = or a = b = c? Y Y This occurs in balanced three phase networks Sum of products of all combinations of resistors taken two at a time divided by the opposite Wye resistor (see Fig 9) Dr Louie 9 Dr Louie 50 9