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Applied Circuit Analysis Chapter 4 Series Circuits Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multi-Element Circuits • So far we have considered circuits limited to one resistor. • From now on we will consider circuits with more than one resistor. • We will begin by looking at circuit topology. 2 Nodes Branches and Loops • Circuit elements can be interconnected in multiple ways. • To understand this, we need to be familiar with some network topology concepts. • A branch represents a single element such as a voltage source or a resistor. • A node is the point of connection between two or more branches. • A loop is any closed path in a circuit. 3 Nodes • A node is usually indicated by a dot in a circuit, although we do not follow this convention in this book. • If a short circuit (wire) connects two nodes, the nodes are considered as one. • The circuit shown has three nodes. 4 Recognizing Nodes • It is important to keep track of the topology of a circuit. • Any single circuit can be drawn a multitude of ways that are functionally equivalent. • Keeping track of nodes is an important part of this. 5 Recognizing Nodes II • Examine the two circuits shown here. • They are equivalent circuits. 6 Network Topology • A loop is independent if it contains at least one branch not shared by any other independent loops. • Two or more elements are in series if they share a single node and thus carry the same current. • Two or more elements are in parallel if they are connected to the same two nodes and thus have the same voltage. 7 Series Resistors • Two resistors are considered in series if the same current pass through them • Take the circuit shown: • The total resistance is: RT R1 R2 • More generally, the total resistance equals the sum of the resistances. RT R1 R2 R3 RN 8 Series Resistors II • Because the same current I passes through each resistor, we can calculate the voltage across each resistor: V1 IR1 V2 IR2 VN IRN • This indicates the voltage drop across each resistor depends on its resistance. 9 Series Resistors III • We can examine the power dissipated in series resistors as well. • The power through the individual resistors is: P1 I 2 R1 P2 I 2 R2 PN I 2 RN 10 Power in Series Resistors • The total power delivered to the series circuit is: PT P1 P2 PN • Because the current through each resistor is the same, the power can be expressed as: PT I 2 R1 R2 RN • Or PT I 2 RT 11 Kirchoff’s Laws • Ohm’s law is not sufficient for circuit analysis. • Kirchoff’s laws complete the needed tools. • There are two laws: – Current law (KCL) – Voltage law (KVL) • KCL will be covered in the next chapter. 12 KVL • Kirchoff’s voltage law is based on conservation of energy. • It states that the algebraic sum of currents around a closed path (or loop) is zero. • It can be expressed as: M v m 1 m 0 13 KVL II • As an example, consider the circuit shown. • Starting at any branch and go around the loop in either direction. • If we start at the voltage source and go around clockwise… 14 KVL III • The voltages we would see are – V1,+V2,+V3,-V4, and +V5 in that order. • For example, as we reach branch 3, the positive terminal is met first, so the voltage is written as positive. • KVL will yield: V1 V2 V3 V4 V5 0 15 Alternate KVL • From the last example, one can see an alternative way to express KVL. • If we separate the negative and positive voltages from the path we took, we have: V2 V3 V5 V1 V4 • Or voltage drops voltage rises 16 Drops vs. Rises • Voltage rises occur when we travel across through an element going from – to +. • Voltage drops occur when we go from + to -. • A voltage rise is said to take place in an active element. • A voltage drop occurs in a passive one. 17 Voltage Sources in Series • One application of KVL is dealing with multiple voltage sources. • A number of applications require multiple voltages to be supplied to a circuit. • KVL helps us to understand how this can be accomplished easily. 18 Voltage Sources in Series II • Take the series connected sources shown here. • Applying KVL to the circuit: • Or Vab V1 V2 V3 0 Vab V1 V2 V3 19 Voltage Division • Series resistors are often used to provide voltage division. • If we apply Ohm’s law to each resistor, the voltage drops are: V1 IR1 V2 IR2 Vn IRn • Because the resistors are in series, the equivalent resistance is: Req R1 R2 Rn 20 Voltage Division II • If a voltage V is applied across the resistors, the current through them is: I V Req • We can thus express the voltage across the resistors as R1 V1 Req R2 V V2 V Req Rn Vn Req V 21 Voltage Division III • The most common application is with two resistors. • Applying the formula that was just presented, the voltages are: R1 R2 V1 V V2 V R1 R2 R1 R2 • One can see that two resistors may be used to create any voltage between 0 and V. 22 Ground Connections • Like measuring distance, voltage must be measured from a reference point. • The most common reference point used is the earth. • Or more specifically, the ground on which the building you are in sits. • This is why this reference point is referred to as ground. 23 Grounding • Electrical equipment that is connected to ground is said to be grounded or earthed. • Part of the wiring of any building is a wire that is connected to a large metal rod driven deep into the ground. • This ensures a good connection to ground. 24 Grounding II • Proper grounding is vital to making electrical equipment safe. • Imagine an electrical device sitting on a wooden table. • If the device is damaged, a charge might accumulate on the frame of the device, since the table will not conduct electricity. 25 Grounding III • Any person touching, or possibly even just going near the device may get a serious shock. • In older homes, the incoming water pipe was used a grounding as it was galvanized steel. • However, with the rise of plastic piping, this is no longer the case. 26 Ground Symbols • A ground is a point of reference. • We attach the value of 0V to ground. • The three symbols shown below all represent ground. • Earth ground is shown in a and b • Chassis ground is shown in c. 27