Electronics Math Simplified
Electronics math is often seen as hard because it is presented in its complex algebra form and often assumes the reader understands all the symbols used.
I hope to simplify the math for my readers by using the unit names and simpler equations

Basic Math:

Volts = Amps * Ohms, Otherwise known as V=IR
Amps = Volts / Ohms
Resistance = Voltage / Amps
Watts = Volts * Amps
Watts = Volts ^ 2 / Ohms
Watts = Current ^ 2 * Ohms
Resistors in series: R Total = R1+R2+...
Resistors in parallel: R Total = 1/((1/R1) + (1/R2) + ...)

Capacitors:

Current = Voltage Change per second * Farads
Voltage change per second = Current / Farads
Farads = Current / Voltage Change per second
Joules = (Farads / 2) * (Voltage ^ 2)
Capacitors in parallel: C Total = C1+C2+...
Capacitors in series: C Total = 1/((1/C1) + (1/C2) + ...)
Impedance = 1 / (2 * PI * Frequency * Farads)

Equations to spec a capacitor to result in a known ripple:
Voltage Change per second = 120 * Desired ripple voltage.
(use 60 for a half bridge rectifier, assuming 60hz)
Farads = Load Current / Voltage Change per second

Inductors:

Voltage = Current Change Per Second * Henries
Current Change Per Second = Voltage / Henries
Henries = Voltage / Current Change Per Second
Joules = (Henries / 2) * (Amps ^ 2)
Inductors in series: L Total = L1+L2+...
Inductors in parallel: L Total = 1/((1/L1) + (1/L2) + ...)
Impedance = 2 * PI * Frequency * Henries

Leds:

Leds are devices with a very sharp IV curve, As such they expect to be run off a constant current supply. A resistor in series with the led and a constant voltage power supply works as well. The following equation tells you the resistance needed:
Series Resistor = (Supply Voltage - LED Voltage) / LED Current
Led current is typically 10mA, LED Voltage is typically about 2v for red, yellow and green LEDs and 3v for blue and white LEDs. If using multiple LEDs in series, add there voltages together. Do not use LEDs in parallel without each having their own resistor
When reading LED data sheets, LED voltage is Vf (Voltage Forward) and current is If (Current Forward)

AC math:

RMS = Root Mean Square. Meaning they take the square root of the mean (Average) of the square of many samples. It is not the same as 'average' and compensates for the fact that power increases to the square of the voltage or current in a circuit.
AC is considered to have two types of power, Apparent power and Real power. Apparent power is measured by RMS voltage * RMS current Real power is measured by RMS (Voltage * Current), Where they measure voltage and current simultaneously, multiply, and sample the result many times to result in an RMS value.
Power Factor = Apparent Power / Real power
Low power factors are a result of current draw that is not in phase with voltage, and results in excess current on power distribution wires, As it is the RMS current that dictates dissipation in wires.

Thermal Math:

Heat sinks are rated in C/W, this is how many C they rise above ambient to dissipate 1W. Thermal resistance (Rth) is also measured in C/W. They can be added together to find the total thermal resistance from die to ambient, and hence the die temperature.
Temperature Rise = (Rth Die to Case + Rth Case to Heat sink + Rth Heat sink to Ambient) * Wattage
Die Temperature = Ambient + Temperature Rise
Rule of thumb is to design for 30c ambient for heat sinks in free air, 40c for a well ventilated case with fans, 50c for a well ventilated case without fans, And 60c+ for poorly ventilated cases, Its best to measure the case in question if ventilation is poor.
To figure out the required heat sink thermal rejection for a given die temperature and dissipation, use:
Case To Heat sink Temp Differential = Wattage * (Rth Die to Case + Rth Case to Heat sink)
Temperature Budget = Desired Die Temperature - Ambient - Case To Heat sink Temp Differential
Desired Heat sink C/W = Temperature Budget / Wattage

Transistor Math:

Gain of a transistor is known as HFE. This is the ratio of collector current that will be allowed to flow vs base current. Transistor gain varies wildly depending on collector current and temperature. If you wish to ensure a transistor switches on fully (Goes into saturation mode), You should always consult the HFE vs Ice (Current Collector Emitter) Graph of the transistor in question for your expected load current.
Your base current should be at least:
Base Current = Load * Overhead Current / HFE
Where Overhead Current should at least be 1.3, Recommended 1.5+ to ensure good saturation.

Filter Math:

LC Filters:
Inductor Impedance = 2 * PI * Frequency * Henries
Inductor Admittance = 1 / (2 * PI * Frequency * Henries)
RL Filter Gain:
Filter Current = Voltage/(Impedance+Ohms)
High pass Gain = Impedance / Sqrt(Ohms ^ 2 + Impedance ^ 2)
Low pass Gain = Ohms / Sqrt(Ohms^2 + Impedance^2)
LR Filter Phase:
High pass Phase = arctan(Ohms / Impedance)
Low pass Phase = arctan(-(Impedance / Ohms))

RC filters:
Capacitor Impedance = 1 / (2 * PI * Frequency * Farads)
Capacitor Admittance = 2 * PI * Frequency * Farads
3db Frequency for RC filter = 1 / Admittance
Resonant frequency of LC = 1 / (2 * PI * sqrt(Ohms * Farads))
RC filter Gain:
Filter Current = Voltage/(Impedance+Ohms)
Low pass gain = 1/(Sqrt(1+(Admittance * Ohms)^2))
High pass gain = Admittance * Ohms / (Sqrt(1+(Admittance * Ohms)^2))
RC filter Phase:
Low Pass phase = arctan(-(Admittance * Ohms)^2)
High Pass phase = arctan(1/(Admittance * Ohms)^2)

SMPS Math:

Feedback compensation: The goal of feedback compensation is to reduce the gain of the feedback loop to below 1 at the 180 degree phase frequency of your output filter. The easiest way to calculate this is to make a bode plot of your LC output filter and feedback loop via simulation, And then reduce your feedback loop frequency until you have less then 1 gain at the -180 degree point. (Both bode plots phase delay should be added together to arrive at the total phase delay of the smps)

Input Power = Output Power / Efficiency
Input Current = Output Voltage * Output Current / (Input Voltage * Efficiency)
Output Power = Input Power * Efficiency
Output Current = Input Voltage * Input Current * Efficiency / Output Voltage

Conversion to other units:

Watt Hours = Amp Hours * Average Voltage
1 Joule = 1 Watt Second
745 Watts = 1 Horse Power
1 Watt Hour = 3.412 British Thermal Unit (BTU)
ω = Radians per second = 2 * Pi * Frequency

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