Category: Electrical Engineering

Digital Logic
Section 1: Intro, Notation and basic operations

Boolean Algebra is very similar to regular algebra with a couple key differences. The only possible outputs are 1 or 0. 1+1+1… = 1.

$1+1+1…=1$

As with regular algebra 1+0=1

$1+0=1$

Addition is equivalent to a Or Gate. The mathematic representation of a OR Gate, with Inputs A,B and Output C is.

$(A+B)=C$

Multiplication is equivalent to Multiplication. The mathematic representation of a AND Gate, with Inputs A,B and Output C is.

$A*B=C$

Multiple Gates can be added and multiplied. Two And Gates with Inputs (A,B), (C,D) and Output E is.

$A*B+C*D=E$

A NOT Gate is another fundamental component. The simplest explanation is it inverts whatever it is give 1=0 and 0=1. It will give the complement of whatever it is given. The representation of NOT A with Boolean Algebra is.

$A’$

It is distributive

$(A+B)’=A’+B’$

This can be built into equations. A NOR Gate is on as long as Both inputs are 0.

$(A+B)’=C$

A NAND Gate is similar.

$(A*B)’=C$

Demorgan’s theorem is a way to convert between AND and OR Gates. This will be a useful feature later.

Converting a NAND Gate to a OR gate is as follows. A NAND Gate is equivalent to Two NOT Inputs and a OR Gate.

$(A*B)’=A’+B’$

Converting a NOR Gate to a AND Gate is as follows. A NOR Gate is equivalent to Two NOT Inputs and a AND Gate.

$(A+B)’=A’B’$

Note: From this point further, i will be dropping the *, it would otherwise make the larger equations unnecessarily cumbersome.

XOR Gates

Section 2: Truth Tables

Truth tables provide every input and output, there are various useful things we can do with this, including create equations from truth tables.

A simple example for a AND Gate

A B C
0 0 0
0 1 0
1 0 0
1 1 1

This takes every possible input, and gives every possible output. Every Equation has a truth table and this can be used to convert equations and circuits into different gates. This is done occasionally for power efficiency, some types of gates are more efficient than others.

Section #: Minterm and Maxterms

You can derive various equations from a truth table. These equations are as a default in the MinTerm or Maxterm form. I will explain that in the next section

Section #: SR Latch

Section #: Karnaugh Maps

Karnaugh Maps are a type of Truth Table, which can be used to relatively easily simplify Equations into fewer Gates. In the given map, you draw various squares and rectangles in order to create various equations. With Karnaugh Maps, there are various rules and conventions for creating resulting equations/circuits.

Rules
- All selected outputs must be squares and rectangles.
- There must be a 2ⁿ number of ones, 2, 4, 8 are all valid while 3, 5, 6 and 7 are never valid.
- All cells must be horizontally or vertically, but not horizontally adjacent.
- There should be the fewest number of equations.
SAMPLE EQUATION

Truth Table

A B C D F

Karnaugh Map

^AB/_CD 00 01 11 10
00
01
11
10

Section #: Finite State Machines

Or Gates

And Gates

D Latches

Multiplexers

Section 3: Circuit Design

Truth Tables

Time Graphs

Minterms and max terms

SOP AND POS

Karnaugh Maps

Don’t care

Half Adders

Full adders

Half Subtractor

Full Subtractor

Finite State Machines
June 7, 2026

A	B	C
0	0	0
0	1	0
1	0	0
1	1	1

Circuit Analysis

Section 1: Basic Circuit Analysis Groundwork

Name	Symbol	Use	Units
Amp	A	Current Flow	A
Columb	C	Electric Charge	A · s
Volt	V	Electric Potential	kg · m² · s⁻³ · A⁻¹
Ohm	Ω	Resistance	kg · m² · s⁻³ · A⁻²
Watt	W	Power	kg · m² · s⁻³
Joule	J	Energy	kg · m² · s⁻²
Farad	F	Capacitance	kg⁻¹ · m⁻² · s⁴ · A²
Henry	H	Inductance	kg · m² · s⁻² · A⁻²
Hertz	Hz	Frequency	s⁻¹

Basic Equations

The Fundamental Equations we will use often throughout this course are.

V=I*R

V=IR is the simplest and most fundamental equation. Memorize it, we will be using it alot. Voltage(V) is Equal to Current(I) Multiplied by Resistance(R). You can rearrange this equation as needed. To solve for Current and Resistance.

I=R/V

R=V/I

Symbols

Section 2: Parallel and Series Calculations

The equation for Parallel Resistance is

1/R_1+1/R_2+1/R_n…=1/R

The Equation for Series Resistance is

R_1+R_2+R_n… =R

Exercise 1:

Section 3: KVL and KCL

Kirchhoff’s Voltage Law: All voltages in a Loop must add up to Zero.

EXAMPLE IMAGE

MATH

Kirchhoff’s Current Law: All Currents entering a node must add up to Zero

EXAMPLE IMAGE

MATH

Exercise 1: Solve for the Voltage in the Resistor R2

Exercise 2: Solve for the Current in Resistor R2

Voltage Drop

A necessary component of using KVL Effectively is Voltage Drop. Given two Resistors in Series, the Voltage drop across each of them will be proportional to the resistance. As a example, a 12v source, that runs through two resistors R1=1ohm and R2=2ohm. The Voltage Drop will be equal to.

V_{R1}=V_{Source}*\frac{R1}{R1+R2}

This is true for any number of resistors in Series. The denominator represents Total Resistance in Series.

Rember this, it will be important later Voltage Division can be used to calculate the Voltage Drop across any Resistor. The Drop may not be relative to your ground(I will explain this with the Node Current method), but it will tell you the drop of the voltage across a resistor which can be used to calculate current.

Section 4: Power Equations

You can solve many equations using Conservation of Energy. Fundamentally this involved Power equations, the sum of all the power entering or leaving the system must be Zero.

The fundamental equations are.

Power is Equal to Current(I) times Voltage(V)

P=I*V

Power is equal to Current(I) Squared times Resistance

P=I^2*R

Power is equal to Voltage(V) Squared times Resistance

P=V^2/R

Exercise 1:

Section 5: Mesh Voltage and Node Current Methods

Two methods, that fundamentally are extensions of KVL/KCL are the Mesh Voltage and Node Current Methods. I will describe them together, then go into them separately. The Mesh Voltage and Node Current Methods, involve getting a series of Linear Equations, then solving them simultaneously. You can solve these linear equations manually, but i highly suggest using a calculator for this.

Mesh Voltage

Mesh Voltage fundamentally relies on Kirchoff’s Voltage Law, the Voltage in a Loop must equal Zero.

Node Current

Section 6: Source Conversion and Voltage Dividers

When evaluating a circuit, it may be convenient or necessary to convert between a voltage source and a Current source.

Section 7: Op Amps

Previously, before Digital Electronics Opamps were used to do analog Math operations.

Op Amps have a couple features and properties that make them very useful. Understanding these features and properties makes solving simple circuits with them extremely straight forward.

The Op Amp will try to make the Voltage of both Inputs equal. A Ideal Op Amp will not allow any current to pass through either input. The Voltage provided by the Output cannot exceed the Supply(+/-).

Features and Equations.

Input(+)
Non-Inverting Input, V_in+
Must Equal V_in-
V_in- = V_in+
Input(-)
Inverting Input, V_in-
Must Equal V_in+
V_in+ = V_in-
V_out
Output
V_out =A(V_in+ – V_in-)
Supply(+)
Provides power to the Op Amp
V_out Cannot exceed Supply(+)
Supply(-)
Provides power to the Op Amp
V_out Cannot exceed Supply(-)

Amplitude

Integrator Circuit

Sumer Circuit

Example 1:

Section 8: Applications of Calculus and Inductors/Capacitors

Capacitors and Inductors store energy in various ways. Capacitors store energy in Electric Potential, Inductors store Energy in a magnetic field. In DC circuits, when you apply a voltage and current to one of them, they eventually become Saturated.

The Natural Response, is when a Saturated Capacitor or Inductor is suddenly disconnected from the power source. What they do following that is called the Natural response, which is defined by First Order Differential Equations.

“n” here means for any quantity of resistors, capacitors or inductors that are all in either series of parallel.

Component:	Resistance	Capacitance	Inductance
Series:	R1+R2+Rn=R	1/C1+1/C2+1/Cn=1/C	L1+L2+Ln=L
Parallel:	1/R1+1/R2+1/Rn=1/R	C1+C2+Cn=C	1/L1+1/L2+1/Cn=1/L
Equation	R=V/I	C=i(t)/(dv/dt)	L=v(t)/(di/dt)

Types of circuits RLC, RL, RC circuits

Natural response vs step response

There are various current equations for a Resistor, Inductor and Capacitor.

I_R=V/R

I_L=1/L*\int_0^t{v}dt

I_C=C\frac{dv}{dt}

There are various voltage equations for a Resistor, Inductor and Capacitor.

V_R=IR

V_L=L\frac{di}{dt}

V_C=\frac1C\int_0^t{idt}

RL First Order Differential Equation

RC First Order Differential Equation

RL Power Equation

RC Power Equation

Time Constant

RC Example:

RL Example

From these we can get two differential equations which are straight forward to solve.

Section 9: Thevenin and Norton Equivalent Circuits.

Thevenin and Norton Equivalent circuits are ways to compress a more complicated circuit into a single equivalent Resistor and Voltage/Current Source. If you need to test various different components this can greatly simplify the system.

Thevenin Equivalent Steps

Open the Load Resistor (Remove it from the circuit)
Find the equivalent voltage for the given components Terminals
Short all the Voltage Sources and Open all the Current Sources.
Find the equivalent Resistance.

Example:

Norton Equivalent Steps

Open the Load Resistor (Remove it from the circuit)
Find the equivalent current for the given wire left by the Short.
Short all the Voltage Sources and Open all the Current Sources.
Find the equivalent Resistance.

Example:

Section #: AC Current

Sinusoidal Sources

V_RMS

Steady Stage Response

Section #: Phasors

Capacitors and Inductors

Super Nodes and

Voltage and Current Source conversion

Thevenin’s Theorem

Thevenin Equivalent

OpAmps

Section 2: Capacitors, Inductors and Step Response

Section 3: AC Circuits and Phasors

Filter Circuits

Capacitors and Inductors

Response

June 7, 2026

^AB/_CD	00	01	11	10
00
01
11
10

Category: Electrical Engineering

Digital Logic

Circuit Analysis