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• EENG223: CIRCUIT THEORY I

DC Circuits:

Basic Laws Dr. Hasan Demirel

• EENG223: CIRCUIT THEORY I

Resistance and Ohms Law All materials resist the flow of current.. Resistance R of an element denotes its ability to resist the flow of electric

current, which is measured in ohms (). A cylindrical material of length l and cross-sectional area A has the

following resistance:

A

lR

R= resistance of the element in ohms ()

= resistivity of the material in ohm-meters (-m)

l = lenghth of the cylindrical material in meters (m)

A=Cross sectional area of the material in meters2 (m2)

• EENG223: CIRCUIT THEORY I

Resistance and Ohms Law Resistance: Basic Concepts and Assumptions:

Conductors (e.g. Wires) have very low resistance (50 M) that can be usually ignored ( omitted from circuit for analysis).

Resistors have a medium range of resistance and must be accounted for the circuit analysis.

Conceptually, a light bulb is similar to the resistor.

Properties of the bulb control how much current flows and how much power is

dissipated (absorbed then emitted as light and heat).

As with the circuit elements, we need to know the current through and voltage

across the device are ralated.

The relationship between the current and voltage can be linear or nonlinear.

• EENG223: CIRCUIT THEORY I

Resistance and Ohms Law Ohms Law states that the voltage v accross a resistor is directly

proportional to the current i flowing through the resistor.

Ri = voltage in volts (V),

i = current in (A),

R = resistance in ().

Sign is determined by passive sign convention (PSC). Materials with linear relationships between the current and the voltage

satisfy the Ohms Law.

Resistor symbol

Ohmlaw applies Ohms Law does Not apply

Resistance R is equal to the slope m in (a).

• EENG223: CIRCUIT THEORY I

Resistance and Ohms Law Resistors and Passive Sign Convention (PSC)

Note that the the relationship between current and voltage are sign sensitive. PSC is satisfied if the current enters the positive terminal of an element:

f PSC is satisfied : =iR f PSC is not satisfied : =iR

PSC satisfied PSC not satisfied PSC not satisfied PSC satisfied

( =iR) ( =iR) ( =iR) ( =iR)

• EENG223: CIRCUIT THEORY I

Equations derived from Ohms Law

(1 = 1 V/A)

Resistors cannot produce power , so the power absorbed by a resistors will always be positive.

1 = 1 V/A

iR i

R Ohms Law:

R

Riip

22 Recall:

R

i

• EENG223: CIRCUIT THEORY I

Short Circuit and Open Circuit Short Circuit as Zero resistance:

An element (or wire) with R =0 is called a short circuit. Short circuit is just drawn as a wire (line).

• EENG223: CIRCUIT THEORY I

Short Circuit and Open Circuit Short Circuit as Voltage Source (0V):

An ideal voltage sorce with Vs=0 V is equivalent to a short circuit. Since =iR and R =0, =0 regardless of i. You could draw a source with Vs=0 V, but it is not done in practice. You cannot connect a voltage source to a short circuit. f connected, usually wire wins and the voltage source melts (smoke

comes out) if not protected.

• EENG223: CIRCUIT THEORY I

Short Circuit and Open Circuit

An element (or wire) with R= is called an open circuit. Such an element is just omitted.

• EENG223: CIRCUIT THEORY I

Short Circuit and Open Circuit Open Circuit as Current Source (0 A):

An ideal current source with I=0 A is equivalent to a open circuit. Since =iR and if I=0 A, then R = . You could draw a source with I=0 A, but it is not done in practice. You cannot connect a current source to an open circuit. f connected, usually you blow the current source (smoke comes

out) if not protected. The insulator (air) wins. Else, sparks fly.

• EENG223: CIRCUIT THEORY I

Conductance: inverse of resistance

Conductance is the ability of an element to conduct electric current . Conductance is the inverse of resistance.

i

RG

1

Units: siemens (S) or mho ( )

Ri

Gi

R

Riip

22

G

iGip

22

• EENG223: CIRCUIT THEORY I

Circuit Building Blocks: Nodes, Branches and Loops:

In circuit analysis, we need a common language and

framework for describing circuits.

In this course, circuits are modelled to be the same as networks.

Networks are composed of nodes, braches and loops.

• EENG223: CIRCUIT THEORY I

Circuit Building Blocks: Nodes, Branches and Loops:

A branch represents a single element such as voltage source, resistor or current source.

How many branches? Branch: a single two-terminal circuit element. Wire segments are not counted as branches. Examples: voltage source/current source/resistors.

There are 5 branches

• EENG223: CIRCUIT THEORY I

Circuit Building Blocks: Nodes, Branches and Loops:

A node is a point of connection between two or more branches.

How many nodes? Node: a connection point between two or more branches. May include a portion of circuit (more than a single point). Essential Node: the point of connection between three or

more brances.

There are 3 nodes (a, b and c)

2 essential nodes (b and c)

• EENG223: CIRCUIT THEORY I

Circuit Building Blocks: Nodes, Branches and Loops:

A loop is a closed path in a circuit.

How loops? Loop: a closed path in a circuit. Independent Loop: A loop is independent if it contains at least

one branch which is not a part of any other independent loop.

There are 6 loops.

There are 3 independent loops.

• EENG223: CIRCUIT THEORY I

Kirchhoffs Laws: Overview

Ohms Law is not sufficient to analyze circuits alone.

Kirchhoffs Laws help Ohms law to form the foundation for circuit analysis: The defining equations for circuit elements (Ohms Law). Kirchhoffs Current Law (KCL). Kirchhoffs Voltage Law (KVL).

The defining equations (from Ohms law) tell us how the voltage and current within a circuit element are related.

Kirchhoffs laws tell us how the voltages and currents in different branches are related.

• EENG223: CIRCUIT THEORY I

Kirchhoffs Current Law

Kirchhoffs current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.

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

KCL also applies to a closed boundary:

• EENG223: CIRCUIT THEORY I

Kirchhoffs Current Law

Applying KCL to node a:

312

321 0

IIII

IIII

T

T

321 IIIIT

Equivalent circuit can be generated as follows:

• EENG223: CIRCUIT THEORY I

Kirchhoffs Current Law: for Closed Boundaries

4231

4321 0

IIII

IIII

KCL applies to a closed boundaries:

Closed boundary

• EENG223: CIRCUIT THEORY I

Kirchhoffs Current Law: Example

Apply KCL to the each essential node in the circuit.

Essential node 1:

Essential node 2:

Essential node 3:

i2

i1 i3 i4

231321 0 iiiiii

mA50mA5 4343 iiii

mA50mA5 142142 iiiiii

• EENG223: CIRCUIT THEORY I

Ideal Current Sources: Series

Ideal currtent sources cannot be connected in series.

Recall: ideal current sources guarantee the current flowing through source is at specified value.

Recall: the current entering a circuit element must be equal to the current leaving the circuit element: Iin = Iout.

Ideal current sources do not exist. Tecnically allowed if : I1 = I2 , but it is a bad idea.

• EENG223: CIRCUIT THEORY I

Kirchhoffs Voltage Law: KVL

Kirchhoffs voltage law (KVL) states that the algebraic sum of voltages around a closed path (or loop) is zero.

sum of voltage drops = sum of voltage rises

or

• EENG223: CIRCUIT THEORY I

Kirchhoffs Voltage Law: KVL

Apply KVL to each loop in the following circuit:

Loop 1: Loop 2: Loop 3: Loop 4: Loop 5: Loop 6:

• EENG223: CIRCUIT THEORY I

Kirchhoffs Voltage Law (KVL) Example:

Calculate V2, V6 and VI.

V50301010

V30)106).(105(

V10)102).(105(

10010

33

6

33

2

6262

I

II

V

V

V

VVVVVV

• EENG223: CIRCUIT THEORY I

Applying Basic Laws: Example Calculate vo and i.

V48A8162

06412412

60642412

0

00

vii

iii

ivivi

Apply KVL around the loop:

• EENG223: CIRCUIT THEORY I

Applying Basic Laws: Example Calculate I2, I3, I7, V3, and VI .

• EENG223: CIRCUIT THEORY I

Series Resistors and Voltage Division The equivalent resistance of any number of resistors

connected in series is the sum of individual resistance.

Consider the following circuit:

Applying KVL

4321

4321

4321

4321

11111

)(

GGGGG

RRRRR

RIRRRRI

IRIRIRIRV

eq

eq

eqss

sssss

• EENG223: CIRCUIT THEORY I

Series Resistors and Voltage Division

(Applying KVL)

voltage accross each resistor, is proportional to its resistance. Larger the resistance, larger the voltage drop on that resistor:

voltage division principle

(voltage divider circuit)

• EENG223: CIRCUIT THEORY I

Parallel Resistors and Current Division The equivalent resistance of N resistors number of resistors

can be calculated by:

Consider the following circuit:

4321

4321

4321

4321

1111

111111

)1111

(

RRRR

RRRRRR

R

V

RRRRV

IIIII

eq

eq

eq

s

s

s

• EENG223: CIRCUIT THEORY I

Parallel Resistors and Current Division

4321

4321

11111

GGGGG

RRRRR

eq

eq

Resistors in parallel have more complicated relationship It is easier to express in conductance

• EENG223: CIRCUIT THEORY I

Parallel Resistors and Current Division

(Applying KCL at node a)

Given the total current i entering to node a the current is shared by the resistors by inverse proportion to their resistance:

Current division principle

(current divider circuit)

• EENG223: CIRCUIT THEORY I

Resistor Networks and Equivalent Resistance Example: Calculate the Req for the

following circuit.

• EENG223: CIRCUIT THEORY I

Resistor Networks and Equivalent Resistance Example: Calculate the Req for the

following circuit.

• EENG223: CIRCUIT THEORY I

Resistor Networks and Equivalent Resistance Example: Calculate the Rab for the

following circuit.

• EENG223: CIRCUIT THEORY I

Resistor Networks and Equivalent Resistance Example: Calculate the Geq for the following

circuit.

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations There are cases where the resistors are neither in parallel nor

in series.

More tools are needed.

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Three-terminal equivalent networks can be used to simplify the

analysis of the circuits. There are two types of three-terminal networks:

Wye (Y) Networks

Delta() Networks

Two equivalent forms of (Y) Network

Two equivalent forms of () Network

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Every Wye (Y) network is functionally equivalent to a Delta ()

network (vice versa).

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Example: Convert the following D network to a Y network.

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Example: Convert the following Ynetwork to a D network.

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Example: Calculate Req and Power delivered by the source.

• EENG223: CIRCUIT THEORY I

Resistor Networks: Wye-Delta Transformations Example: Calculate Rab and and use it to calculate i.