Electric Circuit Theory

Electric Circuit Theory

Nam Ki Min

010-9419-2320

nkmin@korea.ac.kr

Circuit Elements Chapter 2

Nam Ki Min

Review 3

Chapter One Objectives

1. Understand and be able to use SI units and the standard prefixes for powers of 10.

2. Know and be able to use the definitions of voltage and current.

3. Know and be able to use the definitions of power and energy.

4. Be able to use the passive sign convection to calculate the power for an ideal basic circuit element given its voltage and current.

Introduction 4

• Passive elements

• Active elements

 Circuit Elements

 There are two types of elements found in electric circuits:

 Passive Elements

• An active element is capable of generating electrical energy while a passive element is not.

• Examples of passive elements are resistors, capacitors, and inductors.

 Active Elements

• Typical active elements include generators, batteries, and operational amplifiers.

• The most important active elements are voltage or current sources that generally deliver power to the circuit connected to them.

• Voltage sources

• Current sources

• Resistors

• Capacitors

• Inductors (coils)

 Five Ideal Basic Circuit Elements

Introduction 5

Voltage sources

Resistors

Capacitors

Inductors (coils)

Voltage & current sources

2.1 Voltage and Current Sources 6

 Electrical Sources

• A discharging battery converts chemical energy to electric energy, whereas a battery being charged converts electric energy to chemical energy.

• A dynamo is a machine that converts mechanical energy to electrical energy and vice versa.

 A electrical source is a device that is capable of converting nonelectrical energy to electrical energy and vice versa.

- Generator: mechanical energy → electrical energy - Motor: electrical energy → mechanical energy

Discharging battery Battery charging

2.1 Voltage and Current Sources 7

 Ideal Voltage or Current Sources

There are two kinds of sources: independent and dependent sources.

Symbols for ideal independent voltage sources

• An ideal independent current source is an active element that provides a specified current completely independent of the voltage across the source.

• In other words, an ideal independent voltage source delivers to the circuit whatever current is necessary to maintain its terminal voltage.

- That is, the current source delivers to the circuit whatever voltage is necessary to maintain the designated current

Symbol for ideal independent currents source.

 Ideal Independent Sources

• An ideal independent source is an active element that provides a specified voltage or current that is completely independent of other circuit variables.

- Physical sources such as batteries and generators may be regarded as approximations to ideal voltage sources.

𝑣

𝑖

Ideal independent voltage source Practical

𝑣

𝑖

current source Practical

2.1 Voltage and Current Sources 8

 Ideal Dependent Sources

• An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current.

• Dependent sources are useful in modeling elements such as transistors, operational amplifiers and integrated circuits.

• Dependent sources are usually designated by diamond-shaped symbols.

9

 Resistor

2.2 Electrical Resistance

Electrical resistance

• The capacity of materials to impede the flow of current(the flow of electric charge).

Resistor

• A circuit element used to model this behavior.

• A component that is specifically designed to have a certain amount of resistance.

Color bands

Resistance material (carbon composition)

Insulation coating Leads

Physical resistor with resistance R Circuit symbol

• The principal applications of resistors are to limit current in a circuit, to divide voltage, and, in certain cases, to generate heat.

10

 Ohm’s Law

2.2 Electrical Resistance

 The most important fundamental law in electronics is Ohm’s law, which relates voltage, current, and resistance.

 Georg Simon Ohm (1787-1854) studied the relationship between voltage, current, and resistance and formulated the equation that bears his name.

 In terms of current, Ohm’s law states.

𝑖 = 𝑣

𝑅 𝐼 = 𝑉

𝑅

Voltage (V)

Current (mA)

0 10 20 30

0 2.0 4.0 6.0 8.0 10

• One ohm (1 Ω) is the resistance if one ampere (1 A) is in a material when one volt (1 V) is applied.

𝑅 = 𝑉 𝐼

1 V

1 A = 1 Ω

• Conductance is the reciprocal of resistance.

𝐼 = 𝑉 𝑅

𝑅 = 1

𝑅 S(siemens) :Ω mho(inverted omega)

2.2 Electrical Resistance 11

 Two Possible Reference Choices at the Terminals of a Resistor

 Ideal Resistors

 The resistance of ideal resistor is constant and its value does not vary over time.

 We use ideal resistors in circuit analysis.

Color bands

Resistance material (carbon composition)

Insulation coating Leads

 For purpose of circuit analysis, we must reference the current in the resistor to the terminal voltage.

12

 Power in a Resistor: Watt’s Law

2.2 Electrical Resistance

 Three equations for power in circuits

𝑝 = 𝑣𝑖

𝑝 = 𝑣𝑖 = 𝑖

𝑅 𝑝 = 𝑣

𝑅

in terms of current in terms of voltage

𝑝 = 𝑖

𝐺

𝑝 = 𝑣𝑖 = 𝑣

𝐺

 One watt(W) is the amount of power when one joule of energy is used in one second.

13

 Branches, Nodes, and Loops

2.4 Kirchhoff’s Laws

 Network or Circuit

• Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concepts of network topology.

• To differentiate between a circuit and a network, we may regard a network as an interconnection of elements or devices, whereas a circuit is a network providing one or more closed paths.

 Branches

• A branch is any portion of a circuit with two terminals connected to it.

• A branch may consist of one or more circuit elements.

• In practice, any circuit element with two terminals connected to it is a branch.

 Nodes

• A node is the junction of two or more branches

• A node is usually indicated by a dot in a circuit.

• If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node.

14

Loop

 Branches, Nodes, and Loops

2.4 Kirchhoff’s Laws

 Loops(closed paths)

• A loop is any closed path in a circuit.

• A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once.

• A loop is said to be independent if it contains a branch which is not in any other loop.

Independent loops or paths result in independent sets of equations.

Loop

A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology:

𝑏 = 𝑙 + 𝑛 − 1

15

 Kirchhoff’s Laws

2.4 Kirchhoff’s Laws

 Kirchhoff’s Current Law (KCL)

• Ohm’s law by itself is not sufficient to analyze circuits. However, when it is coupled with Kirchhoff’s two laws, we have a sufficient, powerful set of tools for analyzing a large variety of electric circuits.

• Kirchhoff’s laws were first introduced in 1847 by the German physicist Gustav Robert Kirchhoff (1824–1887).

• These laws are formally known as Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL).

• The algebraic sum of all the currents at any node equals zero or the algebraic sum of currents entering a node (or a closed boundary) is zero (because charge cannot be created but must be conserved)

𝑖 + −𝑖

+ −𝑖

+ −𝑖

= 0 For node 1

For node 2

𝑖

+𝑖

+𝑖

+ −𝑖 =0

• The sum of the currents entering a node is equal to the sum of the currents leaving the node.

𝑖 = 𝑖

+𝑖

+𝑖

𝑖

+𝑖

+𝑖

= 𝑖

2.4 Kirchhoff’s Laws 16

 Kirchhoff’s Current Law (KCL)

• Mathematically, KCL implies that

𝑖

= 0

𝑁 𝑛=𝑖

𝑖

Note that KCL also applies to a closed boundary. This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point.

2.4 Kirchhoff’s Laws 17

 Kirchhoff’s Voltage Law (KVL)

• Kirchhoff’s second law is based on the principle of conservation of energy.

No energy is lost or created in an electric circuit; in circuit terms, the sum of all voltages associated with sources must equal the sum of the load voltages, so that the net voltage around a closed circuit is zero.

• The algebraic sum of all voltages around any closed path(or loop) in a circuit equals zero.

• Mathematically, KVL states that

𝑣

= 𝑣

𝑣

𝑣

𝑣

𝑣

+ 𝑣

= 𝑣

1.5 + 1.5 = 3 V = 𝑣

𝑣

= 0

𝑀 𝑚=𝑖

𝑣

+ 𝑣

+ (−𝑣

) = 0

Original circuit Equivalent circuit

−𝑉

+ 𝑉

+ 𝑉

− 𝑉

= 0

𝑉

= 𝑉

+ 𝑉

− 𝑉

2.5 Analysis of a Circuit Containing Dependent Sources 18

 Analysis with Dependent Sources

 A dependent source is a source that generates a voltage or current that depends on the value of another voltage or current in the circuit.

 When a dependent source is present in a circuit to be analyzed, one can initially treat it as an ideal source and write the node or mesh equations accordingly.

 In addition to the equation obtained in this fashion, there will also be an equation relating the dependent source to one of the circuit voltages or currents.

 This constraint equation can then be substituted in the set of equations obtained by the techniques of nodal and mesh analysis, and the equations can subsequently be solved for the unknowns.

19 Fundamental Laws of Circuit Analysis

Summary

 Ohm’s Law

 Kirchhoff’s Current Law (KCL)

 Kirchhoff’s Voltage Law (KVL)

Ohm’s Law establishes the proportionality of voltage and current in a resistor.

Specifically,

𝑣 = 𝑖𝑅

If the current flow in the resistor is in the direction of the voltage drop across it,

or

𝑣 = −𝑖𝑅

Kirchhoff’s current law states that the algebraic sum of all the currents at any node in a circuit equals zero.

Kirchhoff’s voltage law states that the algebraic sum of all the voltages around any closed path in a circuit equals zero.