Series and Parallel Resistors


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1 Series and Parallel Resistors 1
2 Objectives To calculate the equivalent resistance of series and parallel resistors. 2
3 Examples for resistors in parallel and series R 4 R 5 Series R 6 R 7 // R 8 R 4 // R 5 3
4 Elements in Series Suppose two elements are connected with nothing coming off in between KCL says that the elements carry the same current. We say these elements are in series. i 1 i 2 = 0 i 1 = i 2 4
5 Resistors in Series i R 1 R 2 R 3 + i R i R i R V TOTAL Each resistor has the same current (labeled i). Each resistor has voltage ir, given by Ohm s law. The total voltage drop across all 3 resistors is V TOTAL = i R 1 + i R 2 + i R 3 = i (R 1 + R 2 + R 3 ) 5
6 i R 1 R 2 R 3 + v  When we look at all three resistors together as one unit, we see that they have the same IV relationship as one resistor, whose value is the sum of the resistances: So we can treat these resistors as just one equivalent resistance, as long as we are not interested in the individual voltages. Their effect on the rest of the circuit is the same, whether lumped together or not. i R 1 + R 2 + R 3 + v  6
7 Example Find R eq Solution v 1 = ir 1, v 2 = ir 2 If we apply KVL to the loop, we have v + v 1 + v 2 = 0 Then v = v 1 + v 2 v = i(r 1 + R 2 ), R eq = R 1 + R 2 v = ir eq 7
8 The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances. For N resistors in series then, 8
9 Parallel Elements KVL tells us that any set of elements which are directly connected by wire at both ends carry the same voltage. We say these elements are in parallel. KVL clockwise, start at top: Vb Va = 0 Va = Vb 9
10 Parallel Resistors Resistors in parallel carry the same voltage. All of the resistors below have voltage V R. The current flowing through each resistor could definitely be different. Even though they have the same voltage, the resistances could be different. R 1 R 2 R 3 + V R i 1 i 2 i 3 _ i 1 = V R / R 1 i 2 = V R / R 2 i 3 = V R / R 3 10
11 Example : Find R equ From Ohm s law, v = i 1 R 1 = i 2 R 2 Applying KCL at node a gives the total current i as i  i 1  i 2 = 0 i = i 1 + i 2 where 11
12 If we view the three resistors as one unit, with a current i TOTAL going in, and a voltage V R, this unit has the following IV relationship: i TOTAL = i 1 + i 2 + i 3 = V R (1/R 1 + 1/R 2 + 1/R 3 ) in other words, V R = (1/R 1 + 1/R 2 + 1/R 3 ) 1 i TOTAl So to the outside world, the parallel resistors look like one: i TOTAL i TOTAL + V R R 1 R 2 R 3 _ i 1 i 2 i 3 + V R _ R EQ = + + R eq R R R 3 12
13 In case of two equal resistance in parallel The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum. R 1 = R 2, then R eq = R 1 / 2. R 1 = R 2 = = R N = R, then 13
14 Example: Resistance in series and Parallel Find total power expended in the circuit R = R 1 + R R n V p = = = = = 0. 7mW R R ( ) k 145k R = ( )k 14
15 Example Find R AB Combining resistors, The circuit is now reduced to: and further to: R AB = 8K 15
16 Example Find R eq of the circuit 16
17 Example Find equivalent resistance 17
18 18
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