EE-111

LINEAR CIRCUIT
ANALYSIS
Chapter 2: Basic Laws

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 Chapter Overview
› Objectives
– Introduction to fundamental laws for analyzing electric circuits.
– Foundation for understanding how circuits operate.
› Topics Covered:
– Ohm’s Law
– Nodes and Branches
– Kirchhoff’s Current Law (KCL)
– Kirchhoff’s Voltage Law (KVL)
– Series Resistors
– Parallel Resistors
– Voltage and Current Dividers

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 Ohm's Law
› Defines the relationship between Voltage
(𝑣), Current (𝑖), and Resistance (𝑅):
– 𝑣 = 𝑖𝑅
› Resistance (R) depends on material
properties: length (𝑙), area (𝐴), and
resistivity (𝜌):
𝑙
– 𝑅=𝜌
𝐴

› The resistor is the simplest passive

element in circuits.
› Unit of resistor is ohm (Ω)
› 1Ω = 1 𝑉Τ𝐴
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 Resistance
› Follows the passive sign convention.
› Value of R can range from zero to infinity.
› A short circuit is a circuit element with
resistance approaching zero. 𝑣 = 𝑖𝑅,
voltage is zero, current can be any value.
› An open circuit is a circuit element with
𝑣
resistance approaching infinity. 𝑖 = ,
𝑅
current is zero, voltage can be any value.
› Variable resistors have adjustable
resistance.
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 Resistance
› Reciprocal of resistance R is known as conductance and
denoted by G.
› The unit of conductance is the mho (ohm spelled backward)
or reciprocal ohm, with symbol Type equation here..
› A short circuit is a circuit element with resistance
approaching zero. 𝑣 = 𝑖𝑅, voltage is zero, current can be
any value.
› An open circuit is a circuit element with resistance
𝑣
approaching infinity. 𝑖 = , current is zero, voltage can be
𝑅
any value.
› Variable resistors have adjustable resistance.
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Circuit Elements
Distinguishing feature is i-v relationship,
also called element law
𝑖 = 𝑖(𝑣)
Where i is the dependent variable and v is
independent variable or effect and cause
Slope of i-v curve is always reciprocal of
some resistance called dynamic resistance

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Circuit Elements
1 𝑑𝑖
=𝑔=
𝑟 𝑑𝑣
V-I characteristic
◼ Test current
1 𝑑𝑣
◼ 𝑟= =
𝑔 𝑑𝑖
◼ Slope varies from
point to point
◼ Nonlinear curve

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 Straight Line Characteristics
Straight line has i-v as linearly proportional, will have
constant dynamic resistance throughout
1 ∆𝑖
=𝑔=
𝑅 ∆𝑣
1 𝑖
For straight line through origin =
𝑅 𝑣

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Sources
Voltage sources maintains a prescribed
voltage across its terminals regardless of
the current through it v = vs

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Sources
Current sources maintains a prescribed
current regardless of voltage across its
terminals i = is

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Basic Laws

Element laws relate terminal voltages and

currents of individual elements regardless
of their interconnections
Connection laws (Kirchhoff's laws) relate
voltages and currents shared at the
interconnections regardless of type of
elements

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Branches

Circuit is a network with each element as

a branch
Each branch has its branch current &
branch voltage
Label voltages and currents with correct
polarities and directions

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Branches

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Nodes

Leads of two or more elements join

together to form a node
All leads converging on a node have same
potential

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Nodes

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Series / Parallel Connectivity

Two or more elements are in series if they

are cascaded or connected sequentially
and consequently carry the same current
Two or more elements are in parallel if
they are connected to the same two
nodes and consequently have the same
voltage across them.

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How many branches and nodes does the
circuit has? Identify the elements that are
in series and in parallel.

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How many branches and nodes does the
circuit has? Identify the elements that are
in series and in parallel.

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How many branches and nodes does the
circuit has? Identify the elements that are
in series and in parallel.

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Reference Node
Potential difference only has significance in
a circuit
Refer potential of all nodes to a common
node called reference or datum
Bottom node is the most likely choice
It is most convenient to designate the
node having largest number of connections

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Branch voltages vs Node voltages

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Loops and Meshes
Loop is a closed path such that no node
is traversed more than once
Mesh is a loop that contains no other
loop

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Loops

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Overview of Kirchhoff’s Laws
Establish relationship between branch
currents associated with a node and
branch voltages associated with a loop
These laws stem from charge conservation
and energy conservation principles
respectively

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KCL
At any instant sum of all currents entering
a node must equal sum of all currents
leaving that node
෍ 𝑖𝐼𝑁 = ෍ 𝑖𝑂𝑈𝑇
𝑛 𝑛

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Kirchhoff’s Current Law

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Kirchhoff’s Current Law for Boundaries

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KCL - Example

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KVL
At any instant sum of all voltage rises
along a loop must equal sum of all
voltage drops around that loop
෍ 𝑣𝑅𝐼𝑆𝐸 = ෍ 𝑣𝐷𝑅𝑂𝑃
𝑙 𝑙
The algebraic sum of the voltages equals
zero for any closed path (loop) in an
electrical circuit

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KIRCHHOFF’S VOLTAGE LAW

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KVL - Example

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Example – Applying the Basic Laws

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Example – Applying the Basic Laws

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Example – Applying the Basic Laws

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Interconnection of Sources
Voltage sources are connected in series
◼ vs = vs1 + vs2
◼ Never in parallel, violates KVL
Current sources are connected in parallel
◼ is = is1 + is2
◼ Never in series, violates KCL

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Ideal Current Sources: Series

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Resistors in Series

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Resistors in Parallel

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Resistors in Parallel

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Voltage and Current Dividers

Voltages are attenuated, while nature of

signal is preserved
𝑅2
◼ 𝑣𝑜 = 𝑣𝑖
𝑅1+𝑅2
Gain gives the output / input ratio
𝑣𝑜 1
◼ =
𝑣𝑖 1+𝑅1ൗ𝑅2
Variable gain attenuator
𝐾𝑅
𝑣𝑜 = 𝑣𝑖
𝑅

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Voltage Divider

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Current Dividers

Current division is inversely proportional to

the resistances
𝑅1
◼ 𝑖𝑜 = 𝑖
𝑅1+𝑅2 𝐼
Current gain gives the output / input ratio
𝑖𝑜 1
◼ =
𝑖𝐼 1+𝑅2ൗ𝑅1

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Current Divider

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Applying Dividers to Circuit Analysis

Voltage dividers are in series and carry

same current and share a simple node

Current dividers are in parallel and share

the same voltage and share same node
pair

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Resistive Bridges & Ladders

Voltage dividers are combined to

implement more complex networks
Resistive Bridge
Resistive Ladder

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Resistive Bridges
1 1
𝑣𝑜 = − 𝑣𝑠
1+𝑅1ൗ𝑅2 1+𝑅3ൗ𝑅4

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Resistive Bridges

Balanced bridge
𝑅1 𝑅3
=
𝑅2 𝑅4
Wheatstone bridge
𝑅2
𝑅𝑥 = 𝑅𝑣
𝑅1

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n

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Resistive Ladder

Comprises n resistor pairs

Continued fraction computation

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R2R Ladder

N identical series resistors of value R

N+1 identical shunt resistors of value 2R
At each node the equivalent resistance is
R
It halves the voltage and current in each
shunt arm to the right

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Resistor Network - Comments

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Practical Sources and Loading
› A voltage source is designed to supply a prescribed
voltage regardless of the current drawn by the load.
› In a practical source, however, it is found that the
voltage decreases somewhat as the load current is
increased.

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Practical Voltage Source Model
› Drop in voltage across the terminals of a voltage source is linearly
proportional to the load current.
› 𝑣 = 𝑣𝑠 − 𝛼𝑖 where α is a suitable proportionality constant.
› α must have the dimensions of resistance, 𝑣 = 𝑣𝑠 − 𝑅𝑠 𝑖
› 𝑣 is the voltage at the terminals of the source, 𝑖 is the current drawn
by the load, and 𝑅𝑠 is called the internal resistance of the source.
› 𝑣 = 𝑣𝑠 only when 𝑖 = 0 , 𝑣𝑠 is called the open-circuit voltage of the
practical source. This is the voltage that we would measure across the
terminals of the practical source in the absence of any load."

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Practical Voltage Source Model

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› The voltage available at the terminals equals the voltage
produced by the ideal source less the voltage drop across
the internal resistance.
› This resistance 𝑅𝑠 appears at the output, it is also called the
output resistance of the source.
› Thus, internal resistance, series resistance, and output
resistance are equivalent designations for 𝑅𝑠 .

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Exercise 2.15 A certain battery is powering three 50-mA bulbs
in parallel. If disconnecting one of the bulbs causes the
battery voltage to change from 8.910 V to 8.930 V, what are
the values of 𝑣𝑠 and 𝑅𝑠 for this particular battery? (Hint: Use
Equation (2.37) twice.)

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 Practical Current Source Model
› The current supplied by a practical current source is not strictly
independent of the load voltage but usually decreases as the latter is
increased.
› When this decrease is linearly proportional to the load voltage, the
𝑖 − 𝑣 characteristic of the source can be written as 𝑖 = 𝑖𝑠 − 𝛽𝑣𝑖
› Since the proportionality constant 𝛽 must have the dimensions of the
reciprocal of resistance, the 𝑖 − 𝑣 characteristic can be expressed as
1
𝑖 = 𝑖𝑠 − 𝑣
𝑅𝑠

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The Loading Effect
› The fact that connecting a load to a source causes its
output to drop in magnitude is referred to as the
loading effect.

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 Credits
The images and material used in the preparation of these
lecture slides are taken from:
“Fundamentals of Electric Circuits”, Charles K. Alexander,
Matthew N. O. Sadiku. 5th ed. Copyright © 2013 by The
McGraw-Hill Companies, Inc.

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