Notes › Fundamentals of Electric Circuits (Sadiku) Lecture 1

Basic Concepts

calendar_today June 12, 2025 article 849 words access_time 7 min replay Modified June 26, 2025

Table of Contents

What is Circuit Analysis?

Definition 01.1 (Electric Circuit).

An interconnection of electrical elements

Analyzing circuits is important to understand:
- How does it respond to a given input?
- How do the interconnected elements and devices interact?

Charge and Current

Definition 01.2 (Electric Charge).

An electrical property of the atomic particles of which matter consists, measured in coulombs (C)

The fundamental charge $e$ is equal to -1.602e-19 C
In 1 C of charge there are 1/(1.602e-19) = 6.24e18 electrons
Charge is quantized and conserved

Definition 01.3 (Electric Current).

The time rate of change of charge, measures in amperes (A) = 1 C/s

$$i = \frac{\mathrm{d}q}{\mathrm{d}t}$$$$Q = \int_{t_{0}}^{t} i \, \mathrm{d}t$$

Current that never reverses direction is direct current (DC)
Current that changes direction is alternating current (AC)
By convention, we take current to flow in the direction of positive charges

Example 01.4.

(Example 1.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 34])

1In [1]: -1.602e-19*4.6e3
2Out[1]: -7.3692e-16

Example 01.5.

(Practice Problem 1.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 34])

1In [2]: 1.602e-19*10e9
2Out[2]: 1.602e-09

Example 01.6.

(Example 1.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 34])

1In [5]: diff(5*x*sin(4*pi*x))
2Out[5]: 20⋅π⋅x⋅cos(4⋅π⋅x) + 5⋅sin(4⋅π⋅x)
3In [18]: _.subs(x, 0.5)
4Out[18]: 10.0⋅π

Example 01.7.

(Practice Problem 1.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 34])

1In [19]: diff(20-15*x-10*exp(-3*x)).subs(x, 1)
2Out[19]:-15 + 30*exp(-3)
3In [3]: N(_)
4Out[3]: -13.5063879489641

Example 01.8.

(Example 1.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 34])

1In [4]: Q = Piecewise((8, (t<1)&(t>0)),(8*t**2, (t>1)))
2In [5]: integrate(Q, (t, 0, 2))
3Out[5]: 80/3

Voltage

Moving a point charge requires work
- This work requires an external electromotive force (emf), typically a battery
- This emf is also known as voltage or potential difference
The voltage $v_{ab}$ between two points $a$ and $b$ in a circuit is the work, in joules (J), needed to move a unit charge, in coulombs, from $b$ to $a$
- Measured in volts (V) = 1 J/C = 1 Nm/C

$$v_{ab} = \frac{\mathrm{d}w}{\mathrm{d}q}$$

Definition 01.9 (Voltage/Potential Difference).

The energy required to move a unit charge from GND (-) to VCC (+), measured in volts (V).

$v_{ab}$ can be interpreted in two ways:
1. Point $a$ is at a potential $v_{ab}$ higher than point $b$ (the voltage drops from $a$ to $b$)
2. The electric potential at point $a$ with respect to point $b$ is $v_{ab}$
It follows that $v_{ab} = -v_{ba}$

Power and Energy

Definition 01.10 (Power).

The time rate of expending/absorbing energy, measured in watts (W)

$$p = \frac{\mathrm{d}w}{\mathrm{d}t}$$

It follows that $p = \frac{\mathrm{d}w}{\mathrm{d}t} = \frac{\mathrm{d}w}{\mathrm{d}q} \frac{\mathrm{d}q}{\mathrm{d}t} = vi$
- If $p > 0$, power is delivered/absorbed by the element
- If $p < 0$, power is being delivered by the element
The voltage polarity and current direction must conform to the passive sign convention to be positive

Definition 01.11 (Passive Sign Convention).

Satisfied when the current enters through the positive terminal of an element and $p = +vi$. If the current enters through the negative terminal, $p = -vi$.

By the law of conservation of energy, the sum of power in a circuit must always be zero

$$\begin{align} \sum p &= 0 \\ \text{Power Absorbed - Power Supplied} &= 0 \\ \text{Power Absorbed} &= \text{Power Supplied} \end{align} $$

Definition 01.12 (Energy).

The capacity to do work, measured in joules (J).

The energy absorbed/supplied by an element from time $t_0$ to time $t$ is

$$\begin{align} w &= \int_{t_{0}}^{t} p \, \mathrm{d}t \\ &= \int_{t_{0}}^{t} vi \, \mathrm{d}t \end{align}$$

Power utility companies measure energy in watt-hours (Wh):

$$1 \text{Wh} = 3600 \text{J}$$

Example 01.13.

(Example 1.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 38])

 1In [6]: i_t = 5*cos(60*pi*t)
 2
 3In [7]: v1 = 3*i_t
 4
 5In [8]: v2 = 3*diff(i_t)
 6
 7In [9]: (i_t * v1).subs(t, 3e-3)
 8Out[9]: 75*cos(0.18*pi)**2
 9
10In [10]: N(_)
11Out[10]: 53.4667234336902
12
13In [11]: (i_t * v2).subs(t, 3e-3)
14Out[11]: -4500*pi*sin(0.18*pi)*cos(0.18*pi)
15
16In [12]: N(_)
17Out[12]: -6395.84554679224

Example 01.14.

(Practice Problem 1.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 39])

 1In [6]: i_t = 5*cos(60*pi*t)
 2
 3In [13]: v1 = 6*i_t
 4
 5In [14]: v2 = 6 + 10 * integrate(i_t, (t, 0, t))
 6
 7In [15]: N((i_t * v1).subs(t, 5e-3))
 8Out[15]: 51.8237254218789
 9
10In [16]: N((i_t * v2).subs(t, 5e-3))
11Out[16]: 18.2642465093081

Example 01.15.

(Practice Problem 1.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 40]) Answer is interpreted as 33.12 kilowatt-hours

1In [18]: p = 12 * 115
2
3In [19]: p * 24
4Out[19]: 33120

Circuit Elements

Active elements generate energy, do not
There are two kinds of voltage/current sources: independent and dependent sources

Definition 01.16 (Ideal Independent Source).

An active element that provides a specified voltage or current that is completely independent of other circuit elements.

Ideal independent voltage source delivers whatever current is necessary to maintain its terminal voltage
- And vice versa for ideal independent current source

Definition 01.17 (Ideal Dependent Source).

An active element in which the source quantity is controlled by another voltage or current.

Example 01.18.

(Practice Problem 1.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 42])

Sources

Alexander & Sadiku, Fundamentals of Electric Circuits

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