Applied Circuit Analysis

Chapter 4 - Series Circuits

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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.

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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.

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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.

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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.

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Recognizing Nodes II

• Examine the two circuits shown here.• They are equivalent circuits.

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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.

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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.

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1 2TR R R

1 2 3T NR R R R R

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.

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1 1

2 2

N N

V IR

V IR

V IR

Series Resistors III

• We can examine the power dissipated in series resistors as well.

• The power through the individual resistors is:

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21 1

22 2

2N N

P I R

P I R

P I R

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

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1 2T NP P P P

21 2T NP I R R R

2T TP I R

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.

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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:

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1

0M

mm

v

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…

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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:

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1 2 3 4 5 0V V V V V

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

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2 3 5 1 4V V V V V

voltage drops voltage rises

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.

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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.

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Voltage Sources in Series II

• Take the series connected sources shown here.

• Applying KVL to the circuit:

• Or

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1 2 3 0abV V V V

1 2 3abV V V V

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:

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1 1 2 2 n nV IR V IR V IR

1 2eq nR R R R

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

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eq

VI

R

1 21 2

nn

eq eq eq

RR RV V V V V V

R R R

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.

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1 21 2

1 2 1 2

R RV V V V

R R R R

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.

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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.

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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.

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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.

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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.

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