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Chap 3_Simple Resistive Circuits_v2

# Chap 3_Simple Resistive Circuits_v2 - Chapter 3 Simple...

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Chapter 3: Simple Resistive Circuits This chapter presents several techniques that are useful in solving and thinking about circuits. You should think of these things as “tools” in a toolbox; keep them handy and take them out when you need them. Be very careful, however, that you apply them correctly. It is a very common problem that students apply rules where they are not valid. To avoid that problem, make sure you understand the derivation of each of these rules; if you don‟t, you run the risk of misusing them. 3.1 Resistors in Series Circuit elements are in series if they carry the same current. What the means is that at the node where they are connected, there are only two elements; there can be no “T” connecting a third circuit element or else they are not in series. We can often simplify circuits containing resistors in series, as follows. The resistors in the circuit to the right are in series (they are also in series with the voltage source), so the current in them is the same (we could apply KCL to prove this). In that case, we have, by KVL: 4 3 2 1 R R R R i v s s . This equation suggests that, given v s , the current will stay the same if we replace the four resistors with an “equivalent” resistor 4 3 2 1 R R R R R eq . In other words eq s s R i v v s R 1 R 2 R 4 R 3 i s

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We can think of “equivalence” as follows. If the resistors were placed in a box, anything connected to the box at terminals a, b would not know the difference between the four original resistors and the equivalent resistor. This is an important idea that will come up in other contexts throughout the course. More generally, for k resistors in series, we can define an equivalent given by k k eq R R R R R 1 3 2 1 . 3.2 Resistors in Parallel Circuit elements are in parallel if the same voltage is across them. We can often simplify a circuit containing resistors in parallel. The four resistors in the figure to the right are in parallel with each other, and with the voltage source. a) b) R 1 R 2 R 4 R 3 v - + i a) b) R eq v - + i v s R 1 R 2 R 3 i s R 4 i 1 i 2 i 3 i 4
Since the same voltage across each resistor is the same, we have 1 1 R v i s 2 2 R v i s 3 3 R v i s 4 4 R v i s But these currents add to the source current, so 4 3 2 1 i i i i i s . Then 4 3 2 1 4 3 2 1 1 1 1 1 R R R R v R v R v R v R v i s s s s s s This last equation suggests that, given v s , the current will stay the same if we replace the four resistors with an equivalent R eq ; that is, eq s s R v i provided we define 4 3 2 1 1 1 1 1 1 R R R R R eq This equivalence is illustrated in the figure below, in analogy with the equivalence for series resistors.

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Chap 3_Simple Resistive Circuits_v2 - Chapter 3 Simple...

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