Sample Works

The usage of complex numbers in analyzing Alternating Current (AC) Resistor-Inductor-Capacitor (RLC) circuits to find impedance and phase angle

Session: May 2023

Introduction and Aim

The exploration aim is to explore mathematical methodologies used to analyze AC circuits, and to demonstrate that mathematical methods using complex numbers may be more efficient than those using only real numbers, even for real-life applications of mathematics. Findings shall be verified experimentally to determine whether the theoretical methods used are accurate. The focus is on impedance and phase angle for resistors, inductors, and capacitors connected to an AC power supply. This analysis explores a multitude of topics from the IB AA HL curriculum, including Euler’s form of complex numbers, Maclaurin series, Argand diagrams, vectors and phasors, trigonometric functions because of AC’s periodic motion, differentiation, integration, as well as differential equations to highlight the need for complex numbers in AC circuit analysis. This investigation will first introduce the mathematical physics required, then why complex numbers should be used, the impedances of the three individual circuit components, their combined impedance represented visually, a worked example problem using our findings, and finally, an experimental verification.

I chose this exploration because I have a passion for Physics, especially for the electromagnetism topic. I have a keen interest in circuit building and have always wanted to know more about the electronic devices we use in our daily lives. Personally, I find it fascinating how imaginary numbers may make real life applications of mathematics easier, especially in circuits which is arguably more focused on the experimental as opposed to the theoretical part of physics. To see whether this seemingly contradictory idea of using imaginary numbers in real-life applications is valid, I will build my own circuit to compare the experimental value to my manually-calculated value.

Contextualizing Physical Aspects

To thoroughly understand the mathematics applied in circuits, the physics behind it must briefly be established in the context of mathematics. Alternating current (AC) is current, or moving charges, that changes direction periodically (Tsokos). AC is sinusoidal (Figure 1) as electrons oscillate with a frequency, which means instantaneous voltage

V

and current

i

can be defined with trigonometric equations using angular frequency. Angular frequency is defined by

\omega=2\pi f

, where

\omega

has the unit rads⎺¹, and

f

is the frequency in Hertz (Tsokos). Voltage, also known as potential difference, is defined as the electrical potential energy transferred from an electron during its movement from one point in the circuit to another, and is measured in Volts (Tsokos). Current is defined as the rate of change of charge q, is measured in Amperes, and can be algebraically represented through Equation 1:

i=\frac{\Delta q}{\Delta t}=\frac{dq}{dt}

where

t

is the time interval in which charge flows (Tsokos). Chosen values of

i

and

f

in Figure 1 are examples. We have established that voltage and current are sinusoidal as AC is sinusoidal. In the standard sine function form

f(x)=a\sin(bx-d)+K

,

a

is the amplitude, hence we can say the following where

V_0

and

i_0

are the maximum/peak voltage and current, respectively:

V(t)=V_0\sin(\omega t)

and

i(t)=i_0\sin(\omega t)

Figure 1: Sinusoidal current in AC-powered circuits. Graphed on Desmos

Resistors are circuit components which resist, or oppose, the flow of charges, and resistance is measured in ohms and denoted by

R

(Tsokos). Ohm’s Law relates resistance, voltage, and current:

R=\frac Vi

. Inductors are circuit components which oppose change in charge through storing inductance, which is measured in Henry and denoted by

L

(Electronics Tutorials Editors. “The”). Inductance is defined by the following equation with

V_L

being the induced voltage of the inductor, Equation 2:

V_L=L\frac{\Delta i}{\Delta t}=L\frac{di}{dt}

. A negative sign sometimes accompanies this equation to represent the opposition of motion. This explains why induced current lags voltage by a phase difference of

\frac \pi 2

, because the derivative of a negative sinusoidal

i

is a negative cosine graph, meaning a cosine graph reflected over the

x

-axis (Figure 2). Capacitors are circuit components which can store charge across parallel plates, where capacitance is measured in Farad and denoted by

C

(Tsokos). Ohm’s Law for a capacitor states the following, with

V_C

being the induced voltage of the capacitor,

i=C\frac{dV_c}{dt}

(All About Circuits Editors), leading to Equation 3:

V_C=\frac1C\int i~dt

, as capacitance is constant in a given circuit. This relation explains that the induced current leads voltage by

\frac \pi 2

because the derivative of the sinusoidal

i

is cosine (Figure 3). The chosen values of current and voltage are examples. These components’ circuit symbols are depicted below in a series RLC circuit diagram (Figure 4).

Figure 2: current (green) lags

V_L

( (red) by

\frac \pi 2

. Graphed on Desmos

Figure 3: current (blue) leads

V_C

(red) by

\frac \pi 2

. Graphed on Desmos

Complex Number

Mathematically Modelling Household Metal Containers: An Investigation into the Most Suitable Container(s) to Store a Hot Beverage

Session: May 2022

Page count: 16

Modeling and Optimization into different factors of the safety nets to minimize the casualties from a falling brick from a construction site.

1. Introduction

These days, skyscrapers are mostly built in cities for various purposes. However, there are possible dangers of freefalling bricks or metal materials that cause serious damage to workers and even passers-by. A normal force that does the slightest damage in the human head is 84 grams, but considering the weight a normal construction brick,2.04 kg(CivilSir, 2023), the brick itself is a deadly weapon for the passengers below. Therefore, to prevent casualties, safety nets are implemented in building walls in countries such as Singapore, England and Japan. Yet in Vietnam safety nets are not mandatory, domestic construction firms purposely exclude the safety nets to save money. This fails to assure the safety of the peasants and workers themselves.

After secondary research, I discovered how vulnerable the construction fields in Vietnam are, since it could not assure the safety of the workers and the passengers compared to other nations that implement nets, even sold in the internet widely, while this is not a common item that is found explicitly in Vietnam. Therefore, my IA focuses on dealing with the issue of safety. I will think of possible dimensions, angles, thickness and material for safety nets so that this could prevent hazardous issues in the future. In order to understand where the brick fell, I found a construction site with an unfinished building, which could increase the credibility of my result rather than scaling down the experiment. After the experiment, my plan is to find the dimension that has high efficiency by plotting a reference point on the position where the brick fell and draw a normal distribution model to find out the range of the distance where it has the highest percentage.

After acquiring the dimension, my IA will extend by taking into consideration conditional situations such as the brick penetrating through the barrier or bouncing off. Inorder to prevent this, I will use an theoretical model by calculating the impulse of the brick when first hitting the barrier and apply the “V” Safety net to check if the net could absorb the energy.

My aim for this math IA experiment is to model and optimize different factors of the safety nets to minimize the casualties from a falling brick from a construction site.

A+ aim and rational given. Introduction given.

Criterion C: 1 out of 3 version

IA Work

Sample Works

The usage of complex numbers in analyzing Alternating Current (AC) Resistor-Inductor-Capacitor (RLC) circuits to find impedance and phase angle

Introduction and Aim

Contextualizing Physical Aspects

Mathematically Modelling Household Metal Containers: An Investigation into the Most Suitable Container(s) to Store a Hot Beverage

Introduction and Rationale

Aim and Methodology

Variables and Significance

Key Formulae Used

1. Introduction