!!!UNDER CONSTRUCTION!!!
Bridging the Gap between Circuits and Electronics
HOW do you get to Carnegie Hall? Practice. Irwin and Nelms propose that learning circuit analysis is similar to learning a musical instrument. 1 During my undergraduate studies, a professor told me candidly that without hundreds of attempts at solving circuit problems many students may never obtain a passing grade. Practice. The same way that ’noodling’ on a guitar doesn’t often equip someone to play Purple Rain’s solo, unstructured or under-practiced circuit study may seldom result in fluency of the tools used to build sophisticated electronic systems. This is true for electrical engineers, computer engineers, mechancial engineers, aerospace engineers, physicists, mathematicians, artists, CEOs, hobbyists, professors - everybody. The tutorial-based practice outlined below builds circuit analysis fundamentals and reinforces several specific skills that must be combined to analyze or design refined electonic systems. Where possible, examples (and associated references) are given such that the difficulty related to a particular set of exercises progressively escalates. If at any point an example feels overwhelming, you may want to rewind your study to a specific ‘parachute point’ that will be outlined at the begining of each section. The organization of this tutorial consists of the following sections:
- Differences between a 'circuits course' and an 'electronics course'
- Basic engineering mathematics
- Ohm's law and I-V curves
- Kirchhoff's voltage law (KVL) and circuit 'loops'
- Kirchhoff's current law (KCL) and circuit 'nodes'
- Mesh analysis
- Nodal analysis
- Voltage dividers and current dividers
- Series and parallel elements
- Thévenin & Norton equivalent circuits
- Complex numbers, rational polynomials, sinusoids, magnitude & phase
- Energy storage elements
- Time-domain solutions for transient RC circuits
- Laplace transforms (notions for frequency-domain RC circuits)
- Exponential functions, log functions, decibels and their plots
- Simple RC filters
- Chemistry and materials
I. Differences between a ‘circuits course’ and an ’electronics course’
Circuits courses provide foundational concepts and tools needed to analyze and design sophisticated hardware related to electrical, electronic, power, computer, and electro-mechanical systems. A circuits course builds on basic physics, linear algebra and differential equations to analze various electrical component configurations as abstractions electrical/electronic systems. Electronics courses provide useful configurations and physical models with the aim of processing signals. An electronics course uses results and patterns from a circuits course (and other prereqs). Electronics as a discipline aims to create systems the reliably perform an engineered function (usually with inputs and outputs). Electronics courses involve more complex devices and mathematics which are enabled by semiconductor-based, solid-state-physics models (that are usually linearized) and engineering-oriented applied math techniques involving input-output relationships (that usually involve impulse responses, frequency responses and transfer functions).
A circuits course is similar to a collection of Sodoku puzzles. The puzzles are fun and require skills, practice and problem-solving. However, the completed puzzles are not valuable beyond entertainment or achieving proficiency towards the underlying problem solving mechanisms.
Both circuits and electronics courses aim to identify, model and predict the behavior of electrical/electronic circuits. Below is a table of some of the finer points that often differ between the two courses:
| Circuits Course | Electronics Course | |
|---|---|---|
| Goals: | Introduce concepts | Apply concepts towards engineered systems |
| Build problem solving skills | Process and store signals | |
| Texts: | Nilsson, James William, and Susan A. Riedel. Electric circuits. Pearson Education Limited, 2020. | Sedra, A., Smith, K. C., Carusone, T. C., & Gaudet, V. (2020). Microelectronic circuits 8th edition. |
| Irwin, J. D., & Nelms, R. M. (2020). Basic engineering circuit analysis (12th ed.). John Wiley & Sons. | Jaeger, R. C., Blalock, T. N., & Blalock, B. J. (1997). Microelectronic circuit design (pp. 1033-1036). New York: McGraw-Hill. | |
| Hayt, W. H. (2024). Engineering Circuit Analysis (10th ed.), McGraw-Hill Education | Razavi, B. (2021). Fundamentals of microelectronics. John Wiley & Sons. | |
| Concepts: | Elementary components | Complex & active components |
| Resistor, capacitors, inductors | Diodes, transistors, amplifiers | |
| I-V curves | Input/output relationships | |
| Linear (Ohmic) relationships | Linear & nonlinear relationships | |
| Systems of equations | Iterative/simulation solutions | |
| Exact solutions | Approximations | |
| Basic physics models | Solid-state physics models |
II. Basic engineering mathematics
These courses use mathematics. At a minimal, first glance, students should be comfortable with concepts like fractions, algebra, trigonometry, solving systems of equations, exponential functions, logarithmic functions, complex numbers, rational functions, and complex exponential-sinusoidal function relationship. As a parachute point consider a pre-calculus textbook 2. Alternatively, some signals & systems textbooks provide comprehensive reviews of basic engineering mathematics that are needed 3.
III. Ohm’s law and I-V curves
$$ I_T = G V_T = \frac{V_T}{R}$$
IV. Kirchhoff’s voltage law (KVL) and circuit ’loops’
V. Kirchhoff’s current law (KCL) and circuit ’nodes'
VI. Mesh analysis
Level 1 Mesh Analysis: Consider the following circuit:
These equations are linear and may be scaled. If the resistors are scaled by 1/1000 and the currents are scaled by 1000, resulting system of equations becomes $$\begin{align} I_1 - I_2 = -20A \newline 2I_1 + 4I_2 = 5V \newline \end{align} $$
Level 2 Mesh Analysis: Consider the following circuit:
Note that two supermesh selections are possible to avoid $I_S$. Proceeding with the supermesh comprised by outermost loop gives the KVL equation:
$$-V_{S1}+V_{R1}+V_{R2}+V_{R3}+V_{R4}=0$$ $$-V_{S1}+I_1R_1+I_2R_2+I_3R_3+I_3R_4=0$$ $$2000I_1+500I_2+1000I_3+3000I_3=5V$$ $$2000I_1+500I_2+4000I_3=5V$$
The remaining equation may be obtained either by using the remaining supermesh or the left-most mesh. The left-most mesh gives the KVL equation:
$$-V_{S1}+I_1R_1+(I_1-I_2)R_5+V_{S2}=0$$ $$2000I_1+200I_1-200I_2=2V$$ $$2200I_1-200I_2=2V$$
These equations are linear and may be scaled. If the resistors are scaled by 1/1000 and the currents are scaled by 1000, resulting system of equations that can be written as the following matrix
$$ \begin{bmatrix} 0 & 1 & -1 \newline 2 & 0.5 & 4 \newline 2.2 & -0.2 & 0 \end{bmatrix} \begin{bmatrix} I_1 \newline I_2 \newline I_3 \end{bmatrix} = \begin{bmatrix} -20 \newline 5 \newline 2 \end{bmatrix} $$
This gives the solution $I_1=-0.583\mu A$, $I_2=-16.408mA$, and $I_3=3.592mA$.
Level 3 Mesh Analysis: Coming soon…
VII. Nodal analysis
Level 1 Nodal Analysis: Consider the following circuit:
The next equation can be obtained by applying Kirchhoff’s current law (KCL) at node defined by $V_2$. The approach is to sum the currents entering and equate them to the sum of currents leaving. The mechanics of this technique can be developed differently. The solution here uses the ‘currents leaving’ the node $V_2$ as the starting point and the ‘currents entering’ the node $V_2$ as current sources (like $I_S$). The resulting KCL equation gives
$$\begin{align}\sum I_\text{Leaving} &= \sum I_\text{Entering} \newline I_{R1} + I_{R2} &= I_S \newline \frac{V_2-V_1}{R_1} + \frac{V_2-V_3}{R_2} &= I_S \newline \frac{-V_1}{R_1} + V_2\Bigg(\frac{1}{R1}+\frac{1}{R2}\Bigg) + \frac{-V_3}{R_2} &= I_S \newline -0.5m\mho V_1 + 1.5m\mho V_2 - 1m\mho V_3&= 20mA\newline -0.5V_1 + 1.5V_2 - 1V_3&= 20.\end{align}$$.
The final node $V_3$ gives the third equation needed: $$\begin{align}\sum I_\text{Leaving} &= \sum I_\text{Entering} \newline I_{R2} + I_{R3} &= 0 \newline \frac{V_3-V_2}{R_2} + \frac{V_3}{R_3} &= 0 \newline \frac{-V_2}{R_2} + V_3\Bigg(\frac{1}{R_2}+\frac{1}{R_3}\Bigg) &= 0 \newline -1m\mho V_2 + 1.333m\mho V_3 &= 0 \newline -1 V_2 + 1.333 V_3 &= 0. \end{align}$$.
The resulting system of equations is
$$ \begin{bmatrix} 1 & 0 & 0 \newline -0.5 & 1.5 & -1 \newline 0 & -1 & 1.333 \end{bmatrix} \begin{bmatrix} V_1 \newline V_2 \newline V_3 \end{bmatrix} = \begin{bmatrix} 5 \newline 20 \newline 0 \end{bmatrix} $$
which gives the solution $V_1=5V$, $V_2=30V$, and $V_3=22.5V$.
Level 2 Nodal Analysis: Consider the following circuit:
The next equation can be obtained by applying Kirchhoff’s current law (KCL) at node defined by $V_1$. The approach is to sum the currents entering and equate them to the sum of currents leaving. The resulting KCL equation gives
$$\begin{align}\sum I_\text{Leaving} &= \sum I_\text{Entering} \newline I_{R1} + I_{R2} + I_{R5} &= 0 \newline \frac{V_1-V_{S1}}{R_1} + \frac{V_1-V_2}{R_2} + \frac{V_1-V_{S2}}{R_5} &= 0 \newline V_1\Bigg(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_5} \Bigg) - \frac{V_2}{R_2} - \frac{V_{S1}}{R_1} - \frac{V_{S2}}{R_5} &= 0 \newline \end{align}$$.
Considering the intent of expressing this equation later in a system, it may be rearranged to move the constants to one side.
$$\begin{align} V_1\Bigg(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_5} \Bigg) - \frac{V_2}{R_2} &= \frac{V_{S1}}{R_1} + \frac{V_{S2}}{R_5} \newline 7.5m\mho V_1 - 2m\mho V_2 &= 2.5mA + 15mA \newline 7.5 V_1 - 2 V_2 &= 17.5 \newline \end{align}$$.
The next equation can be obtained by applying Kirchhoff’s current law (KCL) at node defined by $V_2$. $$\begin{align}\sum I_\text{Leaving} &= \sum I_\text{Entering} \newline I_{R2} + I_{R3} &= 20mA \newline \frac{V_2-V_{1}}{R_2} + \frac{V_2-V_3}{R_3} &= 20mA \newline -\frac{V_1}{R_2} + V_2\Bigg( \frac{1}{R_2}+\frac{1}{R_3}\Bigg) - \frac{V_3}{R_3} &= 20mA \newline -2m\mho V_1 + 3m\mho V_2 -1m\mho V_3 &= 20mA \newline -2V_1 + 3V_2 - V_3 &= 20 \end{align}$$.
The final equation can be obtained by applying Kirchhoff’s current law (KCL) at node defined by $V_3$. $$\begin{align}\sum I_\text{Leaving} &= \sum I_\text{Entering} \newline I_{R3} + I_{R4} &= 0A \newline \frac{V_3-V_2}{R_3} + \frac{V_3}{R_4} &= 0A \newline -\frac{V_2}{R_3} + V_3\Bigg( \frac{1}{R_3} + \frac{1}{R_4} \Bigg) &= 0A \newline -1m\mho V_2 + 1.333m\mho V_3 &= 0A \newline -V_2 + 1.333V_3 &= 0 \newline \end{align}$$.
The resulting system of equations is
$$ \begin{bmatrix} 7.5 & -2 & 0 \newline -2 & 3 & -1 \newline 0 & -1 & 1.333 \end{bmatrix} \begin{bmatrix} V_1 \newline V_2 \newline V_3 \end{bmatrix} = \begin{bmatrix} 17.5 \newline 20 \newline 0 \end{bmatrix} $$
which gives the solution $V_1=6.165V$, $V_2=14.369V$, and $V_3=10.777V$.
Level 3 Nodal Analysis: Coming soon…
VIII. Voltage dividers and current dividers
IX. Series and parallel elements
X. Thévenin & Norton equivalent circuits
XI. Complex numbers, rational polynomials, sinusoids, magnitude & phase
Imaginary numbers enable analysis of oscillations by linking a simple exponential function to sine and cosine functions. This is important because when designing electronic systems, arbitrary signals will be modeled as a summation of sine and cosine functions that can be expressed as exponential functions. When manipulating these signals the equivalent exponential expression will offer a compact and convienent form that can carry insight in both time and frequency domain analysis. $j=\sqrt{-1}$, $j^2 = -1$, $-j = \frac{1}{j}$ $$\frac{1}{j} = \frac{1}{j}\frac{j}{j} = \frac{j}{-1} = -j$$ $$a+jb = Re^{j\theta} = R\angle \theta$$ $$ \theta = \text{tan}^{-1}\bigg(\frac{b}{a}\bigg)$$ $$ \theta = -j\text{ln}\bigg(\frac{a+jb}{R}\bigg)$$ $$H = \frac{a+jb}{c+jd} = \frac{R_1e^{j\theta_1}}{R_2e^{j\theta_2}} $$ $$|H| = \Bigg|\frac{a+jb}{c+jd}\Bigg|=\frac{|a+jb|}{|c+jd|} = \frac{R_1}{R_2} $$ $$\angle(H) =\angle\Bigg(\frac{R_1e^{j\theta_1}}{R_2e^{j\theta_2}}\Bigg) = \angle\Bigg(\frac{R_1e^{j\theta_1}e^{-j\theta_2}}{R_2}\Bigg) = \angle\Bigg(\frac{R_1}{R_2}e^{j(\theta_1-\theta_2)}\Bigg) =\theta_1 - \theta_2$$
Changing angles as a function of time leads to concepts like angular velocity, frequency and period. Complex exponentials provide a compact notation for sinusoidal waveforms. This can be proven by several approaches. Valuable intuition can be outlined through the development of $e^t$ as an ’eigenfunction’ for differential operators and Taylor’s series expansions.
$$e^{\pm j \pi} = e^{\pm j 180^o} = -1$$
$$e^{j\omega t} = \text{cos}(\omega t) + j \text{sin}(\omega t)$$
$$e^{j\omega \pi} = \text{cos}(\omega \pi) + j \text{sin}(\omega \pi)$$
$$e^{j\omega \pi} - 1 = 0$$
$$e^{j\omega t + \phi} = \text{cos}(\omega t + \phi) + j \text{sin}(\omega t + \phi)$$
A Taylor series is an expansion of a function that results in infinite terms defined by the function’s derivative. The result gives a representation of a function at and around a point. Considering a finite amount of terms, the abbreviated expansion approximates the original function a and around a point. This approximation becomes more accurate as more terms in the series are considered. The definition of the Taylor series is
$$\begin{align} f(t)|_{t=a}&=f(a)+\frac{f’(a)}{1!}(t-a)+\frac{f’’(a)}{2!}(t-a)^2+\frac{f’’’(a)}{3!}(t-a)^3+\dots \newline &=\sum_0^\infty\frac{f^{(n)}(a)}{n!}(t-a)^n. \end{align}$$
As a result, the function $f(t)$ can be approximated at (and around) point $a$ to an accuracy related to the amount of summation terms. Applying this expansion at the point $a=0$ is called the Maclaurin series. By keeping a the partial sum (as opposed to infinite summation terms) results in an approximation to a polynomical degree $k$ where the $k^\text{th}$ term is the highest term retained in the summation. This treatment is often referred to as approximating a function to the $k^{th}$ order. For example, approximating $f(t)$ to the ‘zeroth order’ implies keeping $k=0$ summation terms to represents $f(t)$ as a constant at point $t=a$ and gives a poor approximation of the function as points ‘away’ from $a$ are considered. Likewise, approximating $f(t)$ to the ‘first order’ implies keeping $k=1$ to represent $f(t)$ as a line that passes through point $t=a$ and gives a slightly better approximation of the function as points ‘away’ from $a$ are considered. Similarly, approximating $f(t)$ to the ‘second order’ can improve approximations by using quadradtic functions, and so on.
$$\begin{align} e^{j\omega t}|_{a=0}&=f(a)+\frac{f’(a)}{1!}(t-a)+\frac{f’’(a)}{2!}(t-a)^2+\frac{f’’’(a)}{3!}(t-a)^3+\dots \newline &=e^{0}+j\omega e^0 t+\frac{(j\omega)^2 e^0}{2} t^2+\frac{(j\omega)^3 e^0}{6} t^3+\frac{(j\omega)^4 e^0}{24} t^4+\dots \newline &=1+j\omega t+\frac{(j\omega)^2}{2} t^2+\frac{(j\omega)^3}{6} t^3+\frac{(j\omega)^4}{24} t^4+\dots \end{align}$$
$$\begin{align} \text{cos}(\omega t)|_{a=0}&=f(a)+\frac{f’(a)}{1!}(t-a)+\frac{f’’(a)}{2!}(t-a)^2+\frac{f’’’(a)}{3!}(t-a)^3+\dots \newline &= \text{cos}(0) - \omega\text{sin}(0)t - \frac{\omega^2}{2}\text{cos}(0)t^2 + \frac{\omega^3}{6}\text{sin}(0)t^3 + \frac{\omega^4}{24}\text{cos}(0)t^4 + \dots \newline &= 1 - \frac{\omega^2}{2}t^2 + \frac{\omega^4}{24}t^4 - \frac{\omega^6}{720}t^6 + \dots \end{align}$$
$$\begin{align} \text{sin}(\omega t)|_{a=0}&=f(a)+\frac{f’(a)}{1!}(t-a)+\frac{f’’(a)}{2!}(t-a)^2+\frac{f’’’(a)}{3!}(t-a)^3+\dots \newline &=\text{sin}(0) + \omega\text{cos}(0)t + \frac{\omega^2}{2}\text{sin}(0)t^2 - \frac{\omega^3}{6}\text{cos}(0)t^3 - \frac{\omega^4}{24}\text{cos}(0)t^4 + \dots \newline &= 0 + \omega t - \frac{\omega^3}{6}t^3 + \frac{\omega^5}{120}t^5 + \dots \end{align}$$
Considering the point $a=0$, the Taylor series expressions for $\text{cos}(\omega t)$ and $\text{sin}(\omega t)$ can be combined and rearranged to form the Taylor series expansion for $e^{j\omega t}$ as follows:
$$\begin{align} e^{j\omega t} &= \text{cos}(\omega t) + j\text{sin}(\omega t) \newline &= \bigg( 1 - \frac{\omega^2}{2}t^2 + \frac{\omega^4}{24}t^4 - \frac{\omega^6}{720}t^6 + \dots \bigg) + j\bigg( \omega t - \frac{\omega^3}{6}t^3 + \frac{\omega^5}{120}t^5 + \dots \bigg) \newline &= 1 - \frac{\omega^2}{2}t^2 + \frac{\omega^4}{24}t^4 - \frac{\omega^6}{720}t^6 + j\omega t - j\frac{\omega^3}{6}t^3 + j\frac{\omega^5}{120}t^5 + \dots \newline &= 1 + j\omega t - \frac{\omega^2}{2}t^2 - j\frac{\omega^3}{6}t^3 + \frac{\omega^4}{24}t^4 + j\frac{\omega^5}{120}t^5 - \frac{\omega^6}{720}t^6 + \dots \newline &= 1 + j\omega t + \frac{(j\omega)^2}{2}t^2 + \frac{(j\omega)^3}{6}t^3 + \frac{(j\omega)^4}{24}t^4 + \frac{(j\omega)^5}{120}t^5 + \frac{(j\omega)^6}{720}t^6 + \dots \newline &= \frac{1}{0!} + \frac{j\omega}{1!}t + \frac{(j\omega)^2}{2!}t^2 + \frac{(j\omega)^3}{3!}t^3 + \frac{(j\omega)^4}{4!}t^4 + \frac{(j\omega)^5}{5!}t^5 + \frac{(j\omega)^6}{6!}t^6 + \dots \newline &=\sum_0^\infty\frac{(j\omega)^n}{n!}(t)^n \newline &=\sum_0^\infty\frac{f^{(n)}(a)}{n!}(t-a)^n. \end{align}$$
This proof properly shows the validity of Euler’s formula using only tools available to students up to a second university calculus course. Interestingly, this fact serves as primary foundation of Complex Analysis as well as a cornerstone for signals & systems design and analysis.
Two other important expressions for sinusoids come from Euler’s formula. Note that the cosine funciton is even (e.g. $\text{cos}(-x) = \text{cos}(x)$) and the sine function is odd (e.g. $\text{sin}(-x)=-\text{sin}(x)$). First, an expression for the cosine funciton maybe obtained.
$$\begin{align} e^{j\omega t} + e^{-j\omega t} &= \text{cos}(\omega t) + j\text{sin}(\omega t) + \text{cos}(-\omega t) + j\text{sin}(-\omega t) \newline e^{j\omega t} + e^{-j\omega t} &= \text{cos}(\omega t) + j\text{sin}(\omega t) + \text{cos}(\omega t) - j\text{sin}(\omega t) \newline e^{j\omega t} + e^{-j\omega t} &= 2\text{cos}(\omega t) \newline \end{align}$$
This gives
$$\begin{align} \text{cos}(\omega t) = \frac{e^{j\omega t}+e^{-j\omega t}}{2}. \newline \end{align}$$
Next, an expression for the sine function may be obtained.
$$\begin{align} e^{j\omega t} - e^{-j\omega t} &= \text{cos}(\omega t) + j\text{sin}(\omega t) - \text{cos}(\omega t) + j\text{sin}(\omega t) \newline e^{j\omega t} - e^{-j\omega t} &= j2\text{sin}(\omega t) \newline \end{align}$$
This gives
$$\begin{align} \text{sin}(\omega t) = \frac{e^{j\omega t}-e^{-j\omega t}}{2j}. \newline \end{align}$$
Also, an insight about how derivatives of sinusoids relate to $90^\text{o}$ phase shifts can be shown by using Euler’s formula.
$$\begin{align} z &= \text{cos}(\omega t) + j\text{sin}(\omega t) \newline \frac{dz}{dt} &= -\omega\text{sin}(\omega t) + j\omega\text{cos}(\omega t) \newline &= j^2\omega\text{sin}(\omega t) + j\omega\text{cos}(\omega t) \newline &= j\omega\big(\text{cos}(\omega t)+j\text{sin}(\omega t)\big) \newline &= j\omega z \newline \end{align}$$
Note that differentiating the complex function $z$ in respect to $t$ has the effect of multipling the function by $j$ which implies a $\frac{\pi}{2}=90^\text{o}$ phase shift. This fact is related to how inductors and capacitors impose $90^\text{o}$ phase shifts due to differential relationships with respective currents and voltages. Also note that the result is scaled in amplitude by $\omega$. Both of these facts are shown in a more efficent and compact form by using the exponential function instead of the sum of cosine and sine functions. This compactness is used as an advantage in many analyses and results in various forms of Fourier series, Fourier transform and the analysis of sinusoids in optics, communication systems, radar, sonar, etc. Consider the compact exponential form
$$\begin{align} z &= e^{j\omega t} \newline \frac{dz}{dt} &= j\omega e^{j\omega t} \newline &= j\omega z \end{align}$$
which has many less terms and steps in its analysis.
XII. Energy storage elements
|
|
||
|---|---|---|---|
| Capacitance - Farads [F] | Inductor - Henries [H] | ||
| Field | Electric | Magnetic | |
| Current | $i_C(t)=\frac{dq}{dt}=C\frac{dv_C(t)}{dt}$ | $i_L(t)= i_L(t_0)+\frac{1}{L}\int_{t_0}^t v_L(\tau)d\tau$ | |
| Voltage | $v_C(t)= v_C(t_0)+\frac{1}{C}\int_{t_0}^t i_C(\tau)d\tau$ | $v_L=L\frac{di_L(t)}{dt}$ | |
| Energy | $E_C=\frac{1}{2}Cv_C^2(t)$ | $E_L=\frac{1}{2}Li_L^2(t)$ | |
| Impedance | $Z_C=\frac{1}{j\omega C}=\frac{1}{sC}$ | $Z_L=j\omega L=sL$ | |
| Series | $C_\text{Eq}=\big( \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}\big)^{-1}$ | $L_\text{Eq} = L_1+L_2+\dots+L_n$ | |
| Parallel | $C_\text{Eq} = C_1+C_2+\dots+C_n$ | $L_\text{Eq}=\big( \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_n}\big)^{-1}$ | |
| Symobls | 
|
|
XIII. Time-domain solutions for transient RC circuits
$$f(t)=\frac{d}{dt}f(t)$$
where $f(0) = 1$ (note: if $f(0)=0$ nothing interesting happens).
$$\begin{align} f(t) &=&1 &&+ &&t &&+ &&\frac{t^2}{2}&&+ &&\frac{t^3}{3\cdot2}&&+ &&\frac{t^4}{4\cdot3\cdot2\cdot1}&&+ \dots + \frac{t^n}{n!} \newline & &\downarrow&&\nearrow&&\downarrow&&\nearrow&&\downarrow &&\nearrow&&\downarrow &&\nearrow&&\downarrow &&\newline f’(t) &=&1 &&+ &&t &&+ &&\frac{t^2}{2}&&+ &&\frac{t^3}{3\cdot2}&&+ &&\frac{t^4}{4\cdot3\cdot2\cdot1}&&+ \dots + \frac{t^n}{n!} \end{align}$$
$$f(t) = \sum_0^\infty \frac{t^n}{n!} = e^t$$
XIV. Laplace transforms (notions for frequency-domain RC circuits)
XV. Exponential functions, log functions, decibels and their plots
The compounding nature of exponential growth allows exponential functions can magnify errors, perturbations or changes. This means points that are nearby may be ‘stretched’ or expanded away from one another by the action of the function $e^t$. The inverse is true for logarithmic functions. Logarithmic functions ‘squish’ or compress distant points by an action that tends to normalize vastly differing scales onto similar scales. This inverse relationship describes that these two functions are inverse functions. Qualitatively, inverse functions tend to undo or reverse the action some other function. For beginning engineers, sometimes mathematical formality may seem overwhelming. Although a high degree of formalization is not needed for engineers to use logarithms, consider the definition of a function and its inverse for the sake of translating mathematical results into engineering practice. Formally, if a function is invertible then the function $f: X \rightarrow Y$ will hav an inverse function $f^{-1}:Y \rightarrow X$ that mapping each element $y \in Y$ to a unique element $x \in X$ to satisfy $f(x)=y$. For logarithmic functions this relationship can be described geometrically via symmetry and reflection about axes.
Quantative confirmation of the stretching properties of $e^t$ and the compressive properties of $\text{ln}(t)$ can be examined through the table below.
| t | $e^t$ | $\text{ln}(t)$ |
|---|---|---|
| 0.1 | 1.105 | -2.303 |
| 0.2 | 1.221 | -1.609 |
| 0.3 | 1.350 | -1.204 |
| $\vdots$ | $\vdots$ | $\vdots$ |
| 1 | 2.718 | 0 |
| 2 | 7.389 | 0.693 |
| 3 | 20.086 | 1.099 |
| $\vdots$ | $\vdots$ | $\vdots$ |
| 10 | 22,026 | 2.303 |
| 11 | 59,874 | 2.400 |
| 12 | 162,754 | 2.485 |
| $\vdots$ | $\vdots$ | $\vdots$ |
This table illustrates that as the variable $t$ is varied over an of $0.1 \leq t \leq 12$ the function $e^t$ varied over 160,000 (stretching small changes in $t$) units while the function $\text{ln}(t)$ varied only about 5 units (compressing all changes in $t$ big and small onto a similar scale). This compressive characteristic of logarithmic functions can be used to capture behaviors that encompass vastly different scales. For example, small changes in a logarithmic function between $t=0.1$ and $t=0.2$ appear at a similar magnitude scale for changes between $t=1$ and $t=2$ as well as even larger changes in $t$.
Some rules for manipulating exponential and logarithmic functions are given in the table below.
| Exponentials | Logarithms | |
|---|---|---|
| $a^0 = 1$ | $\text{log}_a(1)=0$ | |
| $a^1 = a$ | $\text{log}_a(a)=1$ | |
| $(ab)^m = a^m\cdot a^n = a^{m+n}$ | $\text{log}_a(m\cdot n)=\text{log}_a(m)+\text{log}_a(n)$ | |
| $(\frac{a}{b})^m=\frac{a^m}{b^m}$ | ||
| $\frac{a^m}{a^n} = a^{m-n}$, $(a \neq 0)$ | $\text{log}_a(\frac{m}{n})=\text{log}_a(m)-\text{log}_a(n)$ | |
| $\frac{1}{a}=a^{-1}$ | $\text{log}_a(\frac{1}{m})=-\text{log}_a(m)$ | |
| $(a^m)^n=a^{mn}$ | $a^{\text{log}_a(k)}=k$ | |
| $a^{\frac{m}{n}}=\sqrt[n]{a^m}$ | $\text{log}_a(a^k)=k$ |
XVI. Simple RC filters
XVII. Chemistry and materials
