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: More generally, the total resistance equals the sum of the resistances. 8

Series Resistors II Because the same current I passes through each resistor, we can calculate the voltage across each resistor: 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: 10

Power in Series Resistors The total power delivered to the series circuit is: Because the current through each resistor is the same, the power can be expressed as: Or 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: 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 – V 1,+V 2,+V 3,-V 4, and +V 5 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: 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: Or 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 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: Because the resistors are in series, the equivalent resistance is: 20

Voltage Division II If a voltage V is applied across the resistors, the current through them is: We can thus express the voltage across the resistors as 21

Voltage Division III The most common application is with two resistors. Applying the formula that was just presented, the voltages are: 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