Author: Connor

  • Boolean Algebra Circuits tool

    Boolean_Algebra_Simulator

    Numerically Applied – Boolean_Algebra_Simulator


    How it works: Select number of inputs (A–E). Add columns (layers) → add/edit/delete gates inside columns. Each gate wires to any previous-layer signal via dropdown. Gates in same column are parallel (no intra-column wiring). Select any signal as F. Truth table always evaluates correctly from gate types + connections. Expressions update on edit; use “Rebuild Expressions” after changing upstream gates. Final output shows both symbolic and traditional Boolean algebra form ( * = AND, + = OR, ! = NOT ).



    Build your circuit, edit gates as needed, select output F, then Generate Truth Table.

  • Circuits 1 Calculator

    circuits_equation_calculator.html

    Numerically Applied

    circuits_equation_calculator.html

    Enter values in consistent SI units. Calculations auto-log to CSV (persisted in browser).

    Basic Circuit Laws

    1. Ohm’s Law — V = IR

    Solves for any one variable when the other two are known. Fundamental starting equation.

    2. Power in Resistor — P = VI / I²R / V²R

    Computes power using any available pair. Derived from Ohm’s law substitutions.

    Inductors & Capacitors — Basic Relations

    3. Inductor Voltage — v = L di/dt

    Voltage induced by changing current. Provide numeric di/dt at the instant of interest.

    4. Capacitor Current — i = C dv/dt

    Current due to changing voltage across capacitor.

    5. Energy in Inductor — w_L = ½ L I²

    Magnetic energy stored. Often used with i_L(0) initial condition.

    6. Energy in Capacitor — w_C = ½ C V²

    Electric energy stored. Dual of inductor energy.

    First-Order RC & RL Circuits — Natural & Step Responses

    7. RC Time Constant — τ = RC

    Governs speed of exponential transients in RC circuits.

    8. RL Time Constant — τ = L/R

    Governs speed of exponential transients in RL circuits.

    9. RC Natural Response — v(t) = V₀ e^(-t/τ)

    Source-free capacitor voltage decay (t ≥ 0). Vss = 0. Find V₀ from DC analysis at t=0−. Matches source sheet algorithm.

    10. RC Step Response — v(t) = Vss + (V₀ − Vss) e^(-t/τ)

    Complete response when DC source is applied. Find Vss (t→∞, cap open) and V₀ (t=0−) first per PDF 5-step method.

    11. RL Natural Response — i(t) = I₀ e^(-(R/L)t)

    Source-free inductor current decay. I₀ from t=0− (inductor current continuous).

    12. RL Step Response — i(t) = Iss + (I₀ − Iss) e^(-(R/L)t)

    Complete response for RL circuit with DC source. Iss found with inductor as short-circuit (t→∞).

    Second-Order RLC Circuits — Parameters & Transient Response Forms

    13. RLC Parameters — α, ω₀, Damping Type, Roots / ωd

    First step for any RLC analysis. Computes Neper frequency α and resonant frequency ω₀. Classifies overdamped / critically damped / underdamped and gives s1,s2 or ωd. Use exact formulas from source sheet step 5.

    14. Overdamped RLC Response — r(t) = rss + A₁e^{s₁t} + A₂e^{s₂t}

    For α > ω₀. Supply A1/A2 (solved from initial conditions i_L(0), v_C(0) per PDF) and s1/s2 from Parameters calculator. rss usually 0 for natural response.

    15. Critically Damped RLC Response — r(t) = rss + (A₁ + A₂ t) e^{-α t}

    For α = ω₀ (repeated root). Form includes linear t multiplier. Obtain α from Parameters calculator.

    16. Underdamped RLC Response — r(t) = rss + e^{-αt} (A₁ cos(ωd t) + A₂ sin(ωd t))

    For α < ω₀ (ringing case). Most common. Supply ωd and α from Parameters calculator. A1/A2 from initial conditions.


    Calculation Log — CSV Data Store

    Every successful calculation is automatically recorded here and saved in browser localStorage (persists across refreshes). Data is stored internally as a CSV structure. Use Export to download as .txt file (CSV content).

    Timestamp Equation Inputs Result
    Standalone HTML • Dark theme #212121 • Green accents #22c55e • No rounded corners • Data export .txt (CSV)

  • Common CNC Mill G-Codes: List

    Common CNC Mill G-Codes: List

    CodeUseParameters
    G00Rapid MovementG00X_Y_Z_
    G01Feed MovementG01X_Y_Z_F_
    M6Tool ChangeT_M6
    M3Spindle On ClockwiseS____M3
    M4Spindle On CCWS____M4
    M5Stop SpindleM5
    M0Force Pause MachineM0
    M1Optional Pause MachineM1
    M8Coolant OnM8
    M9Coolant OffM9
    G41Cutter Comp LeftG41D__
    G42Cutter Comp RightG42D__
    G43Toll Height CompG43H__
    G40Cancels Cutter and Height CompG40
    G02Clockwise CircleG02X_Y_Z_I_J_K_
    G03CCW CircleG03X_Y_Z_I_J_K_
    M30Stops and Rewinds ProgramM30
    G28HomeG28
    G81Straight DrillG81X_Y_Z_R_F_
    G82Spot DrillG83X_Y_Z_R_P_F_
    G83Peck DrillG83X_Y_Z_R_Q_P_F_
    G80Stop Canned CycleG80
    G54-G59Work offsetsG54, G55, G56…
    G53Cancels Work offsets and uses Machine CoordinatesG53

    Safety Codes

    CodeUseParameters
    G17Sets plane to XY PlaneG17
    G20Set to InchesG20
    G40Cancels Height/Diameter Cutter CompG40
    G80Cancels Canned CyclesG80

  • 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…=11+1+1…=1

    As with regular algebra 1+0=1

    1+0=11+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(A+B)=C

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

    AB=CA*B=C

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

    AB+CD=EA*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.

    AA’

    It is distributive

    (A+B)=A+B(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+B)’=C

    A NAND Gate is similar.

    (AB)=C(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.

    (AB)=A+B(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)=AB(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

    ABC
    000
    010
    100
    111

    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

    ABCDF

    Karnaugh Map

    AB/CD00011110
    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

  • Circuit Analysis

    Section 1: Basic Circuit Analysis Groundwork

    NameSymbolUseUnits
    AmpACurrent FlowA
    ColumbCElectric ChargeA · s
    VoltVElectric Potentialkg · m² · s⁻³ · A⁻¹
    OhmΩResistancekg · m² · s⁻³ · A⁻²
    WattWPowerkg · m² · s⁻³
    JouleJEnergykg · m² · s⁻²
    FaradFCapacitancekg⁻¹ · m⁻² · s⁴ · A²
    HenryHInductancekg · m² · s⁻² · A⁻²
    HertzHzFrequencys⁻¹

    Basic Equations

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

    V=IRV=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/VI=R/V
    R=V/IR=V/I

    Example:

    Section 2: Parallel and Series Calculations

    The equation for Parallel Resistance is

    1/R1+1/R2+1/Rn...=1/R1/R_1+1/R_2+1/R_n…=1/R

    The Equation for Series Resistance is

    R1+R2+Rn...=RR_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. This means that when you add the Voltage Source to the Voltage drop of the Resistors, the total voltage will equal 0.

    This is best shown with a example.

    The Voltage source provides 10 Volts, the Resistors R1 and R2 provide a Voltage drop that together will equal 10 Volts. To calculate this we need one additional equation and concept.

    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.

    VR1=VSourceR1R1+R2V_{R1}=V_{Source}*\frac{R1}{R1+R2}

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

    Using this equation and the previous example we can solve for the Current in R1.

    VR1=VSource100200+100=1013=3.33…V_{R1} = V_{Source}*\frac{100}{200+100} = 10*\frac{1}{3}=3.33…

    Using the same Method for R2.

    VR2=VSource200200+100=1023=6.66…V_{R2} = V_{Source}*\frac{200}{200+100} = 10*\frac{2}{3}=6.66…

    Remember, this is a voltage drop, when we sum the Currents in this equation 10VSource -3.33VR1-6.67VR2 = 0 we can see that Kirchhoff’s Voltage law is true. !!!!!

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

    EXAMPLE IMAGE

    MATHMATH

    Exercise 1: Solve for the Voltage in the Resistor R2

    Exercise 2: Solve for the Current in Resistor R2

    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=IVP=I*V

    Power is equal to Current(I) Squared times Resistance

    P=I2RP=I^2*R

    Power is equal to Voltage(V) Squared times Resistance

    P=V2/RP=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, Vin+
      Must Equal Vin-
      Vin- = Vin+
    • Input(-)
      Inverting Input, Vin-
      Must Equal Vin+
      Vin+ = Vin-
    • Vout
      Output
      Vout =A(Vin+ – Vin-)
    • Supply(+)
      Provides power to the Op Amp
      Vout Cannot exceed Supply(+)
    • Supply(-)
      Provides power to the Op Amp
      Vout 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:ResistanceCapacitanceInductance
    Series:R1+R2+Rn=R1/C1+1/C2+1/Cn=1/CL1+L2+Ln=L
    Parallel:1/R1+1/R2+1/Rn=1/RC1+C2+Cn=C1/L1+1/L2+1/Cn=1/L
    EquationR=V/IC=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.

    IR=V/RI_R=V/R
    IL=1/L0tvdtI_L=1/L*\int_0^t{v}dt
    IC=CdvdtI_C=C\frac{dv}{dt}

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

    VR=IRV_R=IR
    VL=LdidtV_L=L\frac{di}{dt}
    VC=1C0tidtV_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.

    RC

    RL

    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

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

    Example:

    Norton Equivalent Steps

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

    Example:

    Section #: AC Current

    Sinusoidal Sources

    VRMS

    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

  • CNC Hand Programming

    CNC Hand programming is handwriting programs. While as a fact no-one handwrites programs anymore, for beginners looking for in depth knowledge this is an extremely useful skillset.

    Vertical Mill Hand Programming.

    G and M Codes: Numerical Control

    To start, with some of the simplest and most basic codes, we have G01 and G00.

    G00
    G00 is for Bulk and Fast movement, do not use this for cutting. It is solely intended to move quickly from your current position closer to your material. Typically you will move to your X0.0 and Y0.0 then to a few inches above the part.

    Task: Move your tool to one inch above your part.

    Example:
    (X0.0Y0.0 Center of Material)
    (Z0.0 Top of the Material)
    
    G00X0.0Y0.0; (moves to center of the part)
    G00Z1.0; (Moves to one inch above your part)

    G01
    G01 is for cutting, you need a feed rate that is typically in Inches Per Minute (IPM). The Program will not run without a feedrate(F__).

    Task: From your previous position, move to the bottom left corner, one inch away in the X and Y. Then move down and cut 0.020″ deep up to the top left corner one inch away in the X and Y. Use a Feedrate of Five Inches Per Minute.

    Example:
    
    G01X-1.0Y-1.0F5.0; (Moves to bottom left corner)
    Z-0.010; (The Program will stay in G01 and reuse the feedrate of 5.0IPM)
    X-1.0Y1.0; (Moves directly up and travels two inches)

    As you can see here, the machine will remember the last G00 or G01 used. You do not need to write it each time, you can simply write it once then input the rest of your coordinates. Ensure that you are in the correct Cutting(G01) or Rapid(G00) mode otherwise you can break a tool, damage the part or crash the machine.

    Comments (I am a comment)

    At the beginning, at tool changed, the beginning of blocks and occasionally in the middle, there will very often be Comments in parentheses. The program will ignore anything within the parentheses.

    (Zero is on top of block)
    (XY is in the top left corner)
    (Use tool 5, .125R 3/4 End Mill)

    These lines provide useful information, the program will ignore them but the operator can see them at the machine.

    M3 and M4
    M3 will turn on your spindle Clockwise, M4 will turn your spindle on CounterClockWise. You will almost always use M3, nearly every CNC Machine tool is designed to turn Clockwise. S____ will set the speed in Rotations Per Minute (RPM). Be very careful that you did not mistype the speed value or it can severely damage something.

    Task: Turn the Spindle to 2500 RPM.

    S2500M3; (Sets the spindle speed to 2500RPM, Then turns the spindle on Clockwise)

    Work Offsets: G54
    Your work offset tells the Machine where your workpiece is. There is a separate Article about how to set your work offset here (LINK). For now we will assume it is already set.

    For Mills, your Work Offset can be any number between G54-G59. It will include three numbers, your X, Y and Z values. Ensure these are set correctly, otherwise you can scrap a part or crash the machine.

    Task: Call your work offset.

    Example:
    G54; (Sets your work offset to G54)
    
    (It is that simple, just don't forget it)

    Height Offsets: G43 and Height
    Another set of Data you need to pay attention to is your Tool Height. All of your tools are different lengths, they can wear down, and when you replace them it’s nearly impossible to get them in the exact same place.

    G43 is the code to turn on height offset compensation. If you include a height offset value without it, the code will do nothing. Then you could scrap a part or crash the machine.

    There is another Article on how to set your height. Your Height will be stored in your Tool Offsets, under the H#. # is whatever tool you are using. T5 is H5, when setting up, it is very common when setting up a machine to leave a tool in it’s existing position. When you do so make sure you change your Height Value as well.

    Task: Set your height value to H2

    Example:
    G43H2;

    Tool Changes: M6
    In order to run a program effectively you may need multiple tools, in order to toolchange, you use the code M6.

    Task: Switch to tool 19.

    Example:
    T19M6; (Calls up T19 then switches tools with M6.)

    Remember to turn the spindle back on before cutting! M3S____

    Exercise Section:
    Create a program, that switches to T5, a half inch endmill. Then cuts a 1 inch square in the center of a 2 inch block using a spindle speed of 2,000 RPM and a Feedrate of 5IPM.

    Answer

    O4001;
    G90 G80 G40 G20 G17; (Safety line, explained in the next section)
    T5M6; (Selects Tool 5 and switches using M6)
    G43H5; (Enables Length based compensation with G43 and Selects Height 5)
    G54: (Selects work offset G54)

    (Rapid Moves)
    G00X0.0Y0.0;(Moves to X0Y0)
    Z1.0;(One inch about the Part)
    <INCOMPLETE>

    Radial offsets: G41 and G42
    Often, your tool will not be the exact right size for the task. It may be 0.001″ too big or “-0.003 too small. We can compensate for this using G41 and G42, this is often handled by the CAM Software, but here we will do this manually.

    The Offset D number will have two numbers a Diameter which is typically set to 0.000, and a Wear offset. The Wear offset is used to make small adjustments. You can program you toolpath to match a specific tool, such as a 1/4″, 3/8″ or 1/2″, or more rarely you can program the toolpath to follow the edge of your part. Then set the Diameter to the size of your tool. This is uncommon and will typically only be done when learning. We will do it in the next exercise.

    G41: Offsets the Tool to the Left of the Toolpath.

    G42: Offsets the Tool to the Right of the Toolpath.

    If your tool is travelling on the near side of a block, left to right. You would use G42, this offsets the tool to the Right, which will compensate for your wear or tool size. If travelling from right to left, you would use G41. This offsets the tool to the left of the tool path.

    Task: Draw a 1 inch square Offset clockwise, then the same square Offset counter-clockwise.

    Canned Cycles

    Canned Cycles are prepackaged functions that are useful to simplify programs and improve readability. The most common ones are for drilling, listed here:

    G81 Straight Drill Cycle

    G82 Spot Drill Cycle

    G83 Peck Drill Cycle

    G80 Cancel Canned Cycle.

    Every Canned Cycle will start with a G8_ Code and end with a G80 code in a self contained block. If you do not include a G80 at the end, the program will interpret any additional coordinates as additional places to drill. This could result in a undesirable amount of holes in your work piece.

    Example:
    G01 X1.0Y1.0F5.0; (Feed towards first hole)
    G83 X1.0Y1.0R0.1Z-1.0Q.200F2.0; (At X1.0Y1.0, a Peck drilling cycle is initiated with a Retard of R0.1 a peck depth of Q0.200 and a Feedrate of F2.0)
    X1.0Y2.0; (Drills a second hole at X1.0 and Y2.0)
    G80; (Cancels Canned Cycle)

    Feeds and Speeds

    Exercise, Facing program

    Exercise 123 Block

    Lathe Hand Programming

    G00

    G00 works exactly the same way as on a mill, but in the X and Z Axis. It is Best Practice and often necessary to move each Axis in a specific order, otherwise you could crash the machine. When approaching a work piece, you should almost always approach with the Z First then the X. If you move the X fire you can crash into the Tail Stock.

    When returning to home, you should always make sure you are clear of the part, home the X(G28U0.0) first and then the Z(G28W0.0). This is done for the same reason as approaching the part, if you do not you can crash into the tail stock. If you are using a Drill, ID(Internal Diameter) Threader or Boring bar make sure you retract it First before sending X Home.

    G01 and G00

    Work offset

    Tools

    Tool Changes

    Constant Surface Speed

    G50

    Arcs