Chapter 2Chapter 2

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An ideal voltage source provides a prescribed voltage across its terminals irrespective of the current flowing through it. The amount of current supplied by the source is determined by the circuit connected to it.

Ideal voltage sources

Various representations of an electrical system

An ideal current source provides a prescribed current to any circuit connected to it. The voltage generated by the source is determined by the circuit connected to it.

Symbol for ideal current source

The sources described so far have the capability of generating a prescribed voltage or current independent of any other element within the circuit. Thus, they are termed independent sources.

There exists another category of sources, however, whose output (current or voltage) is a function of some other voltage or current in a circuit. Theseare called dependent (or controlled) sources. A different symbol, in the shape of a diamond, is used to represent dependent sources and to distinguish them from independent sources.

The symbols typically used to represent dependent sources are depicted below. The table illustrates the relationship between the source voltage or current and the voltage or current it depends on—vx or ix, respectively—which can be any voltage or current in the circuit.

Symbols for dependent sources

Branch

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.

Definition of a branch

Node

A node is the junction of two or more branches (one often refers to the junction of only two branches as a trivial node). The figure below illustrates the concept.

Definitions of node and supernode

Loop

A loop is any closed connection of branches. Various loop configurations are illustrated below.

Definition of a loop

Mesh

A mesh is a loop that does not contain other loops.

Definition of a mesh

Network Analysis

The analysis of an electrical network consists of determining each of the unknown branch currents and node voltages.

Electric current is defined as the time rate of change of charge passing through a predetermined area.

Current flow in an electric conductor

In order for the current to flow, there must exist a closed circuit.In order for the current to flow, there must exist a closed circuit.

The figure below depicts a simple circuit, composed of a battery (e.g., a The figure below depicts a simple circuit, composed of a battery (e.g., a dry-cell or alkaline 1.5-V battery) and a lightbulb.dry-cell or alkaline 1.5-V battery) and a lightbulb.

Note that in the circuit of this figure, the current Note that in the circuit of this figure, the current ii flowing from the battery flowing from the battery to the lightbulb is equal to the current flowing from the lightbulb to the to the lightbulb is equal to the current flowing from the lightbulb to the battery. In other works, no current (and therefore no charge) is “lost” battery. In other works, no current (and therefore no charge) is “lost” around the closed circuit. This principle was observed by the German around the closed circuit. This principle was observed by the German scientist G.R. Kirchhoff and is now known as scientist G.R. Kirchhoff and is now known as Kirchhoff’s current law Kirchhoff’s current law (KCL). (KCL). Kirchhoff’s current law states that because charge cannot be Kirchhoff’s current law states that because charge cannot be created but must be conserved, created but must be conserved, the sum of the currents at a node must the sum of the currents at a node must equal zero.equal zero. Formally, Formally,

A simple electric circuit

The significance of Kirchhoff’s current law is The significance of Kirchhoff’s current law is illustrated below.illustrated below.

Illustration of Kirchhoff’s current law

The total work per unit charge associated with the motion of charge between two points is called voltage. Thus, the units of voltage are those of energy per unit charge; they have been called volts in honor of Alessandro Volta.

Experimental observations led Kirchhoff to the formulation of the second of his laws, Kirchhoff’s voltage law, or KVL. The principle underlying KVL is that no energy is lost or created in an electric circuit; in circuit terms, the sum of all voltages associated with source must equal the sum of the load voltages, so that the net voltage around a closed circuit is zero.

Voltages around a circuit

In general, refer to elements that provide energy In general, refer to elements that provide energy as as sourcessources and to elements that dissipate energy and to elements that dissipate energy as as loads. loads. Standard symbols for a generalized Standard symbols for a generalized source-and-load circuit are shown below.source-and-load circuit are shown below.

Sources and loads in an electric circuit

The concept of reference voltage finds a practical use in the The concept of reference voltage finds a practical use in the ground voltageground voltage of a circuit. Ground represents a specific of a circuit. Ground represents a specific reference voltage that is usually a clearly identified point in a reference voltage that is usually a clearly identified point in a circuit.circuit.

Analogy between electrical and earth ground

The electric power generated by an active element, or that dissipated or storedby a passive element, is equal to the product of the voltage across the element and the current flowing through it.

The passive sign convention

FOCUS ON METHODOLOGYTHE PASSIVE SIGN CONVENTION

1. Choose an arbitrary direction of current flow.

2. Label polarities of all active elements (voltage and current sources).

3. Assign polarities to all passive elements (resistors and other loads); for passive elements, current always flows into the positive terminal.

4. Compute the power dissipated by each element according to the following rule: If positive current flows into the positive terminal of an element, then the power dissipated is positive (i.e., the element absorbs power); if the current leaves the positive terminal of an element, then the power dissipated is negative (i.e.,the element delivers power).

The relationship between current and voltage at the terminals of a circuit element defines the behavior of that element within the circuit. The figure below depicts the representation that is employed to denote a generalized circuit element.

Generalized representation of circuit elements

An ideal resistor is a device that exhibits linear resistance properties according to Ohm’s law, which states that

The resistance of a material depends on a property called resistivity. The inverse of resistivity is called conductivity. For a cylindrical resistence element (shown below), the resistance is proportional to the length of the sample l and inversely proportional to its cross sectional area A and conductivity σ .

The resistance element

In addition to the resistance in ohms, the maximum allowable power dissipation (or power rating) is typically specified for commercial resistors. Exceeding this power rating leads to overheating and can cause the resistor to literally burn up. For a resistor R, the power dissipated can be expressed by

A circuit element with resistance approaching A circuit element with resistance approaching zero is called a zero is called a short circuit.short circuit.

The short circuit

A circuit element whose resistance approaches A circuit element whose resistance approaches infinity is called an infinity is called an open circuit.open circuit.

The open circuit

Definition

Two or more circuit elements are said to be in series if the current from one element exclusively flows into the next element. From KCL, it then follows that all series elements have the same current.

The voltage across each resistor in a series circuit divides in direct proportion to the individual series resistances.

Two or more circuit elements are said to be in parallel if the elements share the same terminals. From KVL, it follows that the elements will have the same voltage.

Current Divider

The ohmmeter is a device that, when connected across a circuit element, can measure the resistance of the element.

The resistance of an element can be measured only when the element is disconnected from any other circuit.

Ohmmeter and measurement of resistance

1. The ammeter must be placed in series with the element whose current is to be measured (e.g., resistor R2).

2. The ammeter should not restrict the flow of current (i.e., cause a voltage drop), or else it will not be measuring the true current flowing in the circuit. An ideal ammeter has zero internal resistance.

The ammeter is a device that, when connected in series with a circuit element, can measure the current flowing through the element.

Measurement of current

The voltmeter is a device that can measure the voltage across a circuit element.

1. The voltmeter must be placed in parallel with the element whose voltage it is measuring.

2. The voltmeter should draw no current away from the element whose voltage it is measuring, or else it will not be measuring the true voltage across that element. Thus, an ideal voltmeter has infinite internal resistance.

Measurement of voltage