EE223Electrical Circuits

Dr. Sarika Khushalani Solanki

Electric current

Describes charge in motion, the flow of charge

This phenomenon can result from moving electrons in a conductive material or moving ions in charged solutions

Charge

• Charge is a basic SI unit, measured in Coulombs (C)• Counts the number of electrons (or positive charges) present.• One Coulomb is quite large, 6.24*1018 electrons.

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Electric current

An ampere (A) is the number of electrons having a total charge of 1 C moving through a given cross section in 1 s.

As defined, current flows in direction of positive charge flow

Current

• i = dq/dt – the derivitive or slope of the charge when plotted against time in seconds

• Q = ∫ i dt – the integral or area under the current when plotted ∙against time in seconds

4321

Current amps

5 sec

Q delivered in 0-5 sec= 12.5 Coulombs

AC and DC Current•DC Current has a constant value

•AC Current has a value that changes sinusoidally

Notice that AC current changes in value anddirection

DC vs. AC• A current that remains constant with

time is called Direct Current (DC)• Such current is represented by the capital

I, time varying current uses the lowercase, i.

• A common source of DC is a battery.• A current that varies sinusoidally with

time is called Alternating Current (AC)• Mains power is an example of AC

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Electrical Circuits

Electric circuit

An electric circuit is an interconnection of electrical elements linked together in a closed path so that electric current may

flow continuously

Circuit diagrams are the standard for electrical engineers

Rate of flow of charge from node a to node b

Rate of flow of charge from node b to node a

(i = current)

Voltage

Driving “force” of electrical current between two points

Vab

Vba

Voltage at terminal a with respect to terminal b

Voltage at terminal b with respect to terminal a

Vab = -Vba

Note: In a circuit, voltage is often defined relative to “ground”

Direction of current• The sign of the current indicates the direction in

which the charge is moving with reference to the direction of interest we define.

• We need not use the direction that the charge moves in as our reference, and often have no choice in the matter.

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Direction of Current II

• A positive current through a component is the same as a negative current flowing in the opposite direction.

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Voltage

• Electrons move when there is a difference in charge between two locations.

• This difference is expressed at the potential difference, or voltage (V).• It is always expressed with reference to two locations

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Voltage II

• It is equal to the energy needed to move a unit charge between the locations.

• Positive charge moving from a higher potential to a lower yields energy.

• Moving from negative to positive requires energy.

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VoltageThe voltage across an element is the work (energy) required to move a

unit of positive charge from the “-” terminal to the “+” terminal

A volt is the potential difference (voltage) between two points when 1 joule of energy is used to move 1 coulomb of charge from one point to the other

Power

The rate at which energy is converted or work is performed

A watt results when 1 joule of energy is converted or used in 1 second

Power and Energy

• Voltage alone does not equal power.• It requires the movement of charge, i.e. a current.• Power is the product of voltage and current

• It is equal to the rate of energy provided or consumed per unit time.• It is measured in Watts (W)

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vip

Circuit schematic example

What is a circuit?• An electric circuit is an interconnection of electrical elements.• It may consist of only two elements or many more:

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Units• When taking measurements, we

must use units to quantify values• We use the International

Systems of Units (SI for short)• Prefixes on SI units allow for

easy relationships between large and small values

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Charge II

• In the lab, one typically sees (pC, nC, or μC)• Charge is always multiple of electron charge• Charge cannot be created or destroyed, only transferred.

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Passive Sign Convention• By convention, we say that an

element being supplied power has positive power.

• A power source, such as a battery has negative power.

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Conservation of Energy• In a circuit, energy cannot be created or destroyed.• Thus power also must be conserved• The sum of all power supplied must be absorbed by

the other elements.• Energy can be described as watts x time.• Power companies usually measure energy in watt-

hours

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Circuit Elements

• Two types:• Active• Passive

• Active elements can generate energy

• Generators• Batteries• Operational Amplifiers

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Circuit Elements II

• Passives absorb energy• Resistors• Capacitors• Inductors

• But it should be noted that only the resistor dissipates energy ideally.• The inductor and capacitor do not.

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Ideal Voltage Source

• An ideal voltage source has no internal resistance.• It also is capable of producing any amount of current needed to

establish the desired voltage at its terminals.• Thus we can know the voltage at its terminals, but we don’t know in

advance the current.

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Ideal Current Source

• Current sources are the opposite of the voltage source:• They have infinite resistance• They will generate any voltage to establish the desired current

through them.• We can know the current through them in advance, but not the

voltage.

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Ideal sources• Both the voltage and current source ideally can

generate infinite power.• They are also capable of absorbing power from the

circuit.• It is important to remember that these sources do

have limits in reality:• Voltage sources have an upper current limit.• Current sources have an upper voltage limit.

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Resistivity

• Materials tend to resist the flow of electricity through them.• This property is called “resistance”• The resistance of an object is a function of its length, l, and cross

sectional area, A, and the material’s resistivity:

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lR

A

Ohm’s Law

• In a resistor, the voltage across a resistor is directly proportional to the current flowing through it.

• The resistance of an element is measured in units of Ohms, Ω, (V/A)• The higher the resistance, the less current will flow through for a

given voltage.

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V IR

Resistivity of Common Materials

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Short and Open Circuits• A connection with almost zero resistance is called a

short circuit.• Ideally, any current may flow through the short.• In practice this is a connecting wire.• A connection with infinite resistance is called an open

circuit.• Here no matter the voltage, no current flows.

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Linearity• Not all materials obey Ohm’s Law.• Resistors that do are called linear

resistors because their current voltage relationship is always linearly proportional.

• Diodes and light bulbs are examples of non-linear elements

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Power Dissipation

• Running current through a resistor dissipates power.

• The power dissipated is a non-linear function of current or voltage• Power dissipated is always positive• A resistor can never generate power

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22 v

p vi i RR

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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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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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• Voltage law

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KCL

• Kirchoff’s current law is based on conservation of charge• It states that the algebraic sum of currents entering a node (or a

closed boundary) is zero.• It can be expressed as:

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1

0N

nn

i

KVL

• Kirchoff’s voltage law is based on conservation of energy• It states that the algebraic sum of voltages around a closed path (or

loop) is zero.• It can be expressed as:

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1

0M

mm

v

Series Resistors• Two resistors are considered in series

if the same current pass through them• Take the circuit shown:• Applying Ohm’s law to both resistors

• If we apply KVL to the loop we have:

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1 1 2 2v iR v iR

1 2 0v v v

Series Resistors II

• Combining the two equations:

• From this we can see there is an equivalent resistance of the two resistors:

• For N resistors in series:

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1 2 1 2v v v i R R

1 2eqR R R

1

N

eq nn

R R

Voltage Division

• The voltage drop across any one resistor can be known.• The current through all the resistors is the same, so using Ohm’s law:

• This is the principle of voltage division

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1 21 2

1 2 1 2

R Rv v v v

R R R R

Multiple elements in a series circuit

Example: Resistors in seriesThe resistors in a series circuit are 680 Ω, 1.5 kΩ, and 2.2 kΩ. What is

the total resistance?

The current through each resistor?

Example: Voltage sources in series

What happens if you reverse a battery?

Find the total voltage of the sources shown

Parallel circuits

A parallel circuit has more than one current path branching from the energy source

Voltage across each pathway is the same

In a parallel circuit, separate current paths function independently of one another

Multiple elements in a parallel circuit

For parallel voltage sources, the voltage is the same across all batteries, but the current supplied by each element isa fraction of the total current

Example: Resistors in parallel

The resistors in a parallel circuit are 680 Ω, 1.5 kΩ, and 2.2 kΩ. What is the total resistance?

Example: Resistors in parallel

The resistors in a parallel circuit are 680 Ω, 1.5 kΩ, and 2.2 kΩ. What is the total resistance?

Voltage across each resistor?

Current through each resistor?

Parallel Resistors• When resistors are in parallel, the

voltage drop across them is the same

• By KCL, the current at node a is

• The equivalent resistance is:

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1 1 2 2v i R i R

1 2i i i

1 2

1 2eq

R RR

R R

Current Division

• Given the current entering the node, the voltage drop across the equivalent resistance will be the same as that for the individual resistors

• This can be used in combination with Ohm’s law to get the current through each resistor:

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1 2

1 2eq

iR Rv iR

R R

2 11 2

1 2 1 2

iR iRi i

R R R R

References

• An Introduction to Electrical Engineering - Aaron Glieberman• Fundamental Electrical Concepts – Northern Arizona University