DC Circuits: Basic Laws - Eastern Mediterranean...

EENG223: CIRCUIT THEORY I

DC Circuits:

Basic Laws Dr. Hasan Demirel


Resistance and Ohm’s 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 (Ω). • A cylindrical material of length l and cross-sectional area A has the

following resistance:

A

lR

R= resistance of the element in ohms (Ω)

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

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

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


Resistance and Ohm’s Law • Resistance: Basic Concepts and Assumptions:

Conductors (e.g. Wires) have very low resistance (<0.1 Ω), which is usually be neglected (i.e. We will assume that wires have zero resistance).

Insulators (e.g. air) have very large resistance (>50 MΩ) 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.


Resistance and Ohm’s Law • Ohm’s Law states that the voltage v accross a resistor is directly

proportional to the current i flowing through the resistor.

Riυ υ = voltage in volts (V),

i = current in (A),

R = resistance in (Ω).

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

satisfy the Ohm’s Law.

Resistor symbol

Ohm’law applies Ohm’s Law does Not apply

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


Resistance and Ohm’s 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:

İf PSC is satisfied : υ =iR

İf PSC is not satisfied : υ =−iR

PSC satisfied PSC not satisfied PSC not satisfied PSC satisfied

(υ =iR) (υ =−iR) (υ =−iR) (υ =iR)


Equations derived from Ohm’s Law

(1 Ω = 1 V/A)

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

• 1 Ω = 1 V/A

iRυ i

υR • Ohm’s Law:

R

υRiυip

22 • Recall:

R

υi


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).


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 υ =iR and R =0, υ =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. • İf connected, usually wire wins and the voltage source melts (smoke

comes out) if not protected.


Short Circuit and Open Circuit

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


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 υ =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. • İf connected, usually you blow the current source (smoke comes

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


Conductance: inverse of resistance

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

υ

i

RG

1

• Units: siemens (S) or mho ( )

Riυ

Gi

R

υRiυip

22

G

iGυυip

22


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.



• 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



• 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)



• 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.


Kirchhoff’s Laws: Overview

• Ohm’s Law is not sufficient to analyze circuits alone.

• Kirchhoff’s Laws help Ohm’s law to form the foundation for circuit analysis: The defining equations for circuit elements (Ohm’s Law). Kirchhoff’s Current Law (KCL). Kirchhoff’s Voltage Law (KVL).

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

• Kirchhoff’s laws tell us how the voltages and currents in different branches are related.


Kirchhoff’s Current Law

• Kirchhoff’s 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:


Kirchhoff’s Current Law

• Applying KCL to node a:

312

321 0

IIII

IIII

T

T

321 IIIIT

• Equivalent circuit can be generated as follows:


Kirchhoff’s Current Law: for Closed Boundaries

4231

4321 0

IIII

IIII

• KCL applies to a closed boundaries:

Closed boundary


Kirchhoff’s Current Law: Example

• Apply KCL to the each essential node in the circuit.

• Essential node 1:



i2

i1 i3 i4

231321 0 iiiiii

mA50mA5 4343 iiii

mA50mA5 142142 iiiiii


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.


Kirchhoff’s Voltage Law: KVL

• Kirchhoff’s 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


Kirchhoff’s Voltage Law: KVL

• Apply KVL to each loop in the following circuit:

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


Kirchhoff’s 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


Applying Basic Laws: Example • Calculate vo and i.

V48A8162

06412412

60642412

0

00

vii

iii

ivivi

• Apply KVL around the loop:


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


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

(Resistors in series add)


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)


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


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


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)


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

following circuit.


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

following circuit.


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

circuit.


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

in series.

• More tools are needed.


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(Δ) Networks

Two equivalent forms of (Y) Network

Two equivalent forms of (Δ) Network


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

network (vice versa).


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


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


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


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

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