Notes › Fundamentals of Electric Circuits (Sadiku) Lecture 2

Basic Laws

calendar_today June 12, 2025 article 1558 words access_time 12 min replay Modified June 27, 2025

Table of Contents

Resistance and Ohm’s Law

Resistance

Definition 02.1.

Resistance is the behavior of a material to resist the flow of electric charge, represented by

$$R = \rho \frac{\ell}{A},$$

where $\rho$ is the resistivity in $\Omega m$, $\ell$ is length, and $A$ is cross-sectional area.

Resistivities of Common Materials

Material	Resistivity (Ω·m)	Usage
Silver	1.64 × 10⁻⁸	Conductor
Copper	1.72 × 10⁻⁸	Conductor
Aluminum	2.8 × 10⁻⁸	Conductor
Gold	2.45 × 10⁻⁸	Conductor
Carbon	4 × 10⁻⁵	Semiconductor
Germanium	47 × 10⁻²	Semiconductor
Silicon	6.4 × 10²	Semiconductor
Paper	10¹⁰	Insulator
Mica	5 × 10¹¹	Insulator
Glass	10¹²	Insulator
Teflon	3 × 10¹²	Insulator

The Resistor

The simplest passive element
Used to model the current-resisting behavior of a material
$R$ is the resistance of the resistor

Types of Resistors

Fixed resistors have constant $R$
Variable resistors, such as potentiometers, have adjustable $R$

Ohm’s Law

Definition 02.2 (Ohm's Law).

The voltage $v$ across a resistor is directly proportional to the current $i$ through the resistor.

$$v \propto i$$

The proportionality constant for this relationship is the resistance, $R$. So, the equation becomes

$$v = iR.$$

To obey the passive convention, the current direction and voltage polarity must flow from a high potential (VCC) to low potential (GND)
Since $R \in [0, \infty]$, we must consider two extremal values of $R$:
- When $R = 0$, there is a short circuit; the voltage is zero but the current can be anything
- When $R = \infty$, there is an open circuit; the current is zero but the voltage can be anything

Definition 02.3 (Short Circuit).

Any circuit element with $R \rightarrow 0$. In practice, this is a connecting wire between two points in the circuit, in which the charge carriers take the path of least resistance.

Definition 02.4 (Open Circuit).

Any circuit element with $R \rightarrow \infty$. In practice, this is a literal gap in the circuit, since air is a perfect insulator.

Conductance

Definition 02.5 (Conductance).

The ability of an element to conduct electric current; it is measured in mhos ($\mho$)/siemens (S).

Ohm’s Law under conductance may be expressed as:

$$ i = Gv$$

Power Dissipated by a Resistor

Nonlinear in terms of current or voltage
Can be expressed in terms of $R$:

$$p = i^2 R = \frac{v^2}{R}$$

Or in terms of $G$:

$$p = v^2 G = \frac{i^2}{G}$$

Since $R$ and $G$ are always positive, power dissipated is always positive; therefore, resistors always absorb power (adheres to the passive convention)

Example 02.6.

(Practice Problem 2.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 60])

1In [1]: eq = Eq(110**2 / 15, x * 110)
2
3In [2]: solve(eq, x)
4Out[2]: [7.33333333333334]

Example 02.7.

(Practice Problem 2.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 61])

1In [1]: v = 3e-3 * 10e3
2
3In [2]: G = 1/10e3
4
5In [3]: p = 3e-3**2 * 10e3
6
7In [4]: v, G, p
8Out[4]: (30.0, 0.0001, 0.09)

Example 02.8.

(Practice Problem 2.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 61])

 1In [2]: p = 30*cos(t)**2
 2
 3In [3]: v = 15*cos(t)
 4
 5In [4]: i = p / v
 6
 7In [5]: R = v**2 / p
 8
 9In [6]: i, R
10Out[6]: (2⋅cos(t), 15/2)

Nodes, Branches, and Loops

Definition 02.9 (Branch).

Any two-terminal element in a circuit, such as a power source or resistor. The number of branches equals the number of circuit elements.

Definition 02.10 (Node).

The point of connection between two or more branches.

Typically indicated by a dot in a circuit
If a short circuit connects two nodes, the two nodes may be treated as a single node

Definition 02.11 (Loop).

Any closed path in a circuit.

A “closed path” starts at a node, passes through a set of nodes, and returns to the starting node without passing through any node more than once.
Is independent if it contains a branch that is not part of any other independent loop
- Independent loops result in independent sets of equations

A network with $b$ branches, $n$ nodes, and $l$ independent loops will satisfy the following relation (The Fundamental Theorem of Network Topology):

$$b = l + n - 1$$

Two elements are in series if they exclusively share a single node
- Consequently, they share the same current
Two elements are in parallel if they are connected to the same two nodes
- Consequently, they share the same voltage

Example 02.12.

(Practice Problem 2.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 63])

5 elements, so 5 branches
3 nodes
4 $\Omega$ is in parallel with 10V
1 $\Omega$ is in parallel with 2 $\Omega$

Kirchoff’s Laws

Kirchoff’s Current Law

Based on the principle of conservation of charge

Definition 02.13 (Kirchoff's Current Law).

The algebraic sum of currents entering/exiting a node is zero.

$$ \sum_{n=1}^{N} i_{n} = 0 $$

where $N$ is the number of branches connected to the node and $i_{n}$ is the $n$th current entering/exiting the node.

Currents entering a node is positive, while leaving current is negative
Alternative interpretation: the sum of currents entering a node is equal to the sum of the currents leaving the node
KCL may also be applied to any closed boundary, since a node is a closed surface shrunk to a point
A consequence of KCL is that a circuit cannot contain two different currents in series
Another consequence is that current sources in parallel may be added together

Kirchoff’s Voltage Law

Based on the principle of conservation of energy

Definition 02.14 (Kirchoff's Voltage Law).

The algebraic sum of all voltages around a loop is zero.

$$ \sum_{m = 1}^{M} v_{m} = 0 $$

where $M$ is the number of branches in the loop and $v_{m}$ is the $m$th voltage.

Can go clockwise or counterclockwise around the loop
The sign on each voltage is the polarity of the terminal encountered first
Alternative interpretation: the sum of voltage drops is equal to the sum of voltage rises
A consequence of KVL is that a circuit cannot contain two different voltages in parallel
Another consequence is that voltage sources in series may be added together

Example 02.15.

(Practice Problem 2.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 67])

Example 02.16.

(Example 2.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 67])

Example 02.17.

(Practice Problem 2.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 67])

Example 02.18.

(Example 2.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 68])

Example 02.19.

(Practice Problem 2.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 68])

Example 02.20.

(Practice Problem 2.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 69])

Series Resistance

Definition 02.21 (Equivalent Resistance of Series Resistors).

The equivalent resistance $R_{\text{eq}}$ of any number of resistors connected in series is the sum of the individual resistances.

$$R_{\text{eq}} = \sum_{n=1}^{N} R_{n}$$

Series resistors share the same current, but not the same voltage
However, the total voltage across all the resistors (the equivalent resistor) can be calculated as $R_{\text{eq}} i$

Voltage Division

A voltage source connected to a network of resistors in series effectively divides up that voltage, known as a voltage divider
The resistance of the $n$th resistor in the voltage divider is calculated as:

$$v_{n} = \frac{V_{s}R_{n}}{R_{\text{eq}}}$$

Parallel Resistances

Definition 02.22 (Equivalent Resistance of Parallel Resistors).

The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.

$$R_{\text{eq}} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}$$

For $n$ resistors in parallel, the equivalent resistance is

$$R_{\text{eq}} = \left(\sum_{n=1}^{N} \frac{1}{R_{n}}\right)^{-1}.$$

$R_{\text{eq}}$ is always less than the smallest resistor in parallel
If all the resistances in the parallel network are the same, then:

$$R_{\text{eq}} = \frac{R}{N}$$

Equivalent Conductance

Since $G = \frac{1}{R}$ by definition, it is possible to express parallel resistance in terms of conductance:

$$G_{\text{eq}} = \sum_{n=1}^{N}G_{n}$$

where $G_{\text{eq}}$ = $\frac{1}{R_{\text{eq}}}$

The same may be done for series resistance:

$$G_{\text{eq}} = \left(\sum_{n=1}^{N}\frac{1}{G_{n}}\right)^{-1}$$

Current Division

The total current in a network of parallel elements is effectively divided across those elements, known as current division
The current of the $n$th resistor in the current divider is calculated as:

$$i_{n} = i_{s}\left(\frac{R_{\text{eq}}}{R_{n}}\right),$$

or using the conductance definition:

$$i_{n} = \frac{i_{s}G_{n}}{G_{\text{eq}}}$$

Example 02.23.

(Practice Problem 2.12 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 76])

Example 02.24.

(Practice Problem 2.13 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 77])

Wye-Delta Transformations

Resistive networks that aren’t in series or parallel may be simplified to a three-terminal equivalent network
- Either the Wye/Tee network or the Delta/Pi network

Delta-Wye Conversion

Superimpose Y onto $\Delta$ by adding an extra node $n$. Each resistor in Y is the product of the resistors in the two adjacent $\Delta$ branches divided by the sum of the three $\Delta$ resistors
For the network below:

$$\begin{align} R_{1} &= \frac{R_{b}R_{c}}{R_{a}+R_{b}+R_{c}} \\ R_{2} &= \frac{R_{c}R_{a}}{R_{a}+R_{b}+R_{c}} \\ R_{3} &= \frac{R_{a}R_{b}}{R_{a}+R_{b}+R_{c}} \end{align} $$

Wye-Delta Conversion

Each resistor in the $\Delta$ network is the sum of all products of Y resistors divided by the opposite Y resistor
For the superimposed network:

$$\begin{align} R_{a} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}} \\ R_{b} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}} \\ R_{c} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}} \end{align}$$

Balanced Condition

If all the Y resistances are equal AND all the $\Delta$ resistances are equal, then:

$$R_{Y} = \frac{R_{\Delta}}{3} \ \text{or} \ R_{\Delta}= 3R_{Y}$$

Example 02.25.

(Practice Problem 2.14 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 80])

Example 02.26.

(Practice Problem 2.15 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 83])

Sources

Alexander & Sadiku, Fundamentals of Electric Circuits

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