Circuit Analysis I

(ENGR 2405)

Chapter 1

Review: Charge, Current,

Voltage, Power

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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Electric Charge

When an amber rod is

rubbed with fur, some of

the electrons on the

atoms in the fur are

transferred to the amber:

Electric Charge:

Water (H2O) molecule can

be polarized by electrostatic

Induction For example, the

water molecule has more

positive charges on one side

of the molecule and negative

charges on the other side.

Thus, water can be slightly

attracted to a static electric

charge. A demonstration of

that can be seen in bending a

stream of water with a

charged plastic comb.

Electric Charge:

Conductors and Insulators

Electric Charge

Some materials can become

polarized – this means that their

atoms rotate in response to an

external charge. This is how a

charged object can attract a

neutral one.

Electric Charge

The electrons in an atom are in a cloud

surrounding the nucleus, and can be separated

from the atom with relative ease.

Electric Charge

We find that the total electric charge of the

universe is a constant:

Electric charge is conserved.

Also, electric charge is quantized in units of e.

The atom that has lost an electron is now

positively charged – it is a positive ion

The atom that has gained an electron is now

negatively charged – it is a negative ion

Many descriptions of electric charge use

terms that might lead you to the conclusion

that charge is a substance. Phrases like:

“Charge on a sphere”

“Charge transferred”

“Charge carried on the electron”

However, charge is a property of particles,

one of many properties, such as mass.

21.5 Charge is Quantized

Charge is Quantized

. Since the days of Benjamin Franklin, our understanding of

of the nature of electricity has changed from being a type of

‘continuous fluid’ to a collection of smaller charged particles.

The total charge was found to always be a multiple of a certain

elementary charge, “e”:

The value of this elementary charge is one of the fundamental

constants of nature, and it is the magnitude of the charge

of both the proton and the electron. The value of “e” is:

Charge is Quantized

Elementary particles either carry no charge, or carry a single

elementary charge. When a physical quantity such as charge

can have only discrete values, rather than any value, we say

the quantity is quantized. It is possible, For example, to find

a particle that has no charge at all, or a charge of +10e, or -6e,

but not a particle with a charge of, say, 3.57e.

Conductors and Insulators

Conductors are materials through which charge can move freely; examples include metals (such as

copper in common lamp wire), the human body, and tap water.

Nonconductors—also called insulators—are materials through which charge cannot move freely;

examples include rubber, plastic, glass, and chemically pure water.

Semiconductors are materials that are intermediate between conductors and insulators; examples

include silicon and germanium in computer chips.

Superconductors are materials that are perfect conductors, allowing charge to move without any

hindrance.

The properties of conductors and insulators are due to the structure and electrical nature of atoms.

Atoms consist of positively charged protons, negatively charged electrons, and electrically neutral

neutrons. The protons and neutrons are packed tightly together in a central nucleus.

When atoms of a conductor come together to form the solid, some of their outermost (and so most

loosely held) electrons become free to wander about within the solid, leaving behind positively

charged atoms ( positive ions).We call the mobile electrons conduction electrons.

There are few (if any) free electrons in a nonconductor.

Charge

• Charge is a basic SI unit, measured in

Coulombs (C)

• Counts the number of electrons (or positive

charges) present.

• Charge of single electron is 1.602*10-19 C

• One Coulomb is quite large, 6.24*1018

electrons.

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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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Electric Current:

Although an electric current is a stream of moving charges, not all moving

charges constitute an electric current. If there is to be an electric current through

a given surface, there must be a net flow of charge through that surface. Two

examples are given.

1. The free electrons (conduction electrons) in an isolated length of copper wire

are in random motion at speeds of the order of 106 m/s. If you pass a

hypothetical plane through such a wire, conduction electrons pass through it in

both directions at the rate of many billions per second—but there is no net

transport of charge and thus no current through the wire. However, if you

connect the ends of the wire to a battery, you slightly bias the flow in one

direction, with the result that there now is a net transport of charge and thus an

electric current through the wire.

Electric Current:

The figure shows a section of a conductor, part of a conducting loop in which current has been

established. If charge dq passes through a hypothetical plane (such as aa’) in time dt, then the current i

through that plane is defined as:

The charge that passes through the plane in a time interval extending from 0 to t is:

Under steady-state conditions, the current is the same for planes aa’, bb’, and cc’ and for all planes that

pass completely through the conductor, no matter what their location or orientation.

The SI unit for current is the coulomb per second, or the ampere (A):

Electric Current:

Electric Current, Conservation of Charge, and Direction of Current:

Current

• The movement of charge is called a current

• Historically the moving charges were

thought to be positive

• Thus we always note the direction of the

equivalent positive charges, even if the

moving charges are negative.

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Current II

• Current, i, is measured as charge

moved per unit time through an

element.

• Unit is Ampere (A), is one

Coulomb/second

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dt

dqi

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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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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Electric Potential:

The potential energy per unit charge at a point in an electric field is called the electric

potential V (or simply the potential) at that point. This is a scalar quantity. Thus,

The electric potential difference V between any two points i and f in an electric field is equal

to the difference in potential energy per unit charge between the two points. Thus,

The potential difference between two points is thus the negative of the work done by the

electrostatic force to move a unit charge from one point to the other.

If we set Ui =0 at infinity as our reference potential energy, then the electric potential V must

also be zero there. Therefore, the electric potential at any point in an electric field can be

defined to be

Here W∞ is the work done by the electric field on a charged particle as that particle

moves in from infinity to point f.

The SI unit for potential is the joule per coulomb. This combination is called the volt

(abbreviated V).

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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Power in Electric Circuits:

In the figure, there is an external conducting path between

the two terminals of the battery. A steady current i is

produced in the circuit, directed from terminal a to

terminal b. The amount of charge dq that moves between

those terminals in time interval dt is equal to i dt.

This charge dq moves through a decrease in potential of

magnitude V, and thus its electric potential energy

decreases in magnitude by the amount

The power P associated with that transfer is the rate of

transfer dU/dt, given by

The unit of power is the volt-ampere (V A).

Energy and Power in Electric Circuits

When the electric company sends you a bill,

your usage is quoted in kilowatt-hours (kWh).

They are charging you for energy use, and kWh

are a measure of energy.

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

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.

• Passive sign convention is

satisfied if the direction of

current is selected such that

current enters through the

terminal that is more

positively biased.

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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. 38

Dependent Sources

• A dependent source has its output

controlled by an input value.

• Symbolically represented as a

diamond

• Four types:

– A voltage-controlled voltage source

(VCVS).

– A current-controlled voltage source

(CCVS).

– A voltage-controlled current source

(VCCS).

– A current-controlled current source

(CCCS).

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Dependent Source example

• The circuit shown below is an example

of using a dependent source.

• The source on the right is controlled by

the current passing through element C.

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Circuit Applications of

Dependent Sources • Dependent sources are good models

for some common circuit elements: – Transistors: In certain modes of operation,

transistors take either a voltage or current input

to one terminal and cause a current that is

somehow proportional to the input to appear at

two other terminals.

– Operational Amplifiers: Not covered yet, but the

basic concept is they take an input voltage and

generate an output voltage that is proportional to

that.

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TV Picture Tube

• Old style cathode Ray Tubes (CRT) are a good example of the

flow of electrons

• A hot filament is the source of electrons

• Charged plates accelerate and steer a thin stream (beam) of

electrons

• The beam strikes a phosphor coated screen causing light

emission.

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