Maths for electonics

There’s so many things to remember in the field of electronics and if you’re like me, you need a cheat sheet. This is my reference of the mathematical functions, along with a few reminders of the theory behind them. I add to it when I learn something new so let me know if you have any suggestions. I hope you find it useful in your electronic adventures!

Resistance

Fun fact - Resistivity is a standard metric that can be used to compare the resistance of materials. It’s defined as the resistance of 1 cubic meter of a material, measured between the two opposing faces of the cube.

Resistance of wire

$$R = {\rho L \over A}$$

  • R - Resistance in ohms
  • $\rho$ - Resistivity of the material in ohms per cubic meter (Rho)
  • L - length in meters
  • A - cross-sectional area of the wire in square meters

Ohms Law

$$E = I * R$$

  • E - Voltage or electromotive force in volts
  • I - Current in ampheres
  • R - Resistance in ohms

Resistors in parallel

Calculating the total resistance of any number of resistors in parallel $$Rt = {1 \over {1/R1 + 1/R2 + 1/Rn}}$$

  • Rt - total resistance in Ohms
  • R1 to Rn - each parallel resistance in Ohms

Calculating the total resistance of two resistors in parallel can be done with a simpler equation $$Rt = {{R1 * R2} \over {R1 + R2}}$$

  • Rt - total resistance in Ohms
  • R1 - first parallel resistance in Ohms
  • R2 - second parallel resistance in Ohms

Power

DC Power

$$P = IE$$

  • P - Power in watts
  • I - Current in ampheres
  • E - Voltage or electromotive force, in volts

AC Power (RMS)

Root Mean Square (RMS) is a measure of voltage for AC circuits that functions as an equivalent voltage compared with DC circuits. RMS voltage allows us use a single set of functions to work with both AC and DC circuits without needing to factor in the oscilating duty cycle of an alternating voltate. For example, we can calculate the power of an AC circuit using the same function we’d use for a DC circuit because the RMS voltage value caters for the differing electromotive force.

$$Vrms = {Vpeak * 0.707}$$

  • Vrms - RMS voltage
  • Vpeak - peak voltage
  • 0.707 - a constant

Capacitance

Capacitance is the property of an electric circuit that apposes changes in voltage.

Charge and voltage across a capacitor

Quantity of charge in a capacitor $$Q = CE$$

  • Q - charge in Coulombs
  • C - capacitance in Farads
  • E - Voltage or electromotive force in Volts

Energy stored in a capacitor

$$En = CE^2/2$$

  • En = energy in joules (watt seconds)
  • C = capacitance in farads
  • E = voltage

Capacitors in series

Calculating the total capacitance of any number of capacitors in series $$Ct = 1 / (1/C1 + 1/C2 + 1/Cn)$$

  • Ct - total capacitance in Farads
  • C1 to Cn - each capacitance in Farads

Capacitor time constant

The time taken for a capacitor to charge to 63.2% is called the Time Constant of the capacitor.

$$T = {C * R}$$

  • T - Time constant to reach 63.2% of total charge in seconds
  • C - Capacitance in Farads
  • R - Resistance in Ohms

A capacitor is considered fully charged after 5 time constants (T). Charge is not linear, its logarithmic. The same applies to discharging a capacitor, taking 5 time constants (T) to discharge fully.

Calculating the charge of a capacitor at any point in time $$Ec = {Ea(1 - {e^-t \over RC})}$$

  • Ec - voltage across the capacitor after time t
  • Ea - the applied voltage
  • t - time in seconds since the voltage was applied
  • R - resistance in ohms
  • C - capacitance in farads
  • e - 2.71828 Euler mathematical constant

Capacitive Reactance

$$Xc = {1 \over {2 \pi f C}}$$

  • Xc - capacitive reactance in ohms
  • $2 \pi$ - a numeric constant
  • f - frequency in hertz
  • C - capacitance in farads

Inductance

Inductance is the measure of back EMF in a circuit $$L(uH) = (r^2 N^ 2) / (24r + 25l)$$

  • L = Inductance in uH
  • N = number of turns
  • R = radius in cm
  • L = wire length in cm

In an inductive circuit, current lags voltage by 90 degrees unless there is also resistance: $$Angle = tan Xl / R.$$ In a capacitive circuit, current leads voltage by 90 degrees, unless there is also resistance: $$Angle = tan Xc / R.$$

Energy stored in an inductor

Energy stored in the magnetic field of an inductor $$En = LI^2 / 2$$

  • En = Energy in joules (watt-seconds)
  • L = Inductance in henries (H)
  • I = Current in amps

Inductors in series and parallel

Inductance in series are added, while inductors in parallel are calculated using the inverse of the inverse equation. Both assume the magnetic lines of the individual inductors are not coupled: $$Lt = 1 / (1/L1 + 1/L2 + 1/Ln)$$

Inductor time constant

Time constant of an inductance to reach 63.2% of the maximum current. Like capacitance, it takes 5 time constants to reach 100% $$T = L/R$$

  • T = time in seconds
  • L = inductance in Henry’s (H)
  • R = resistance in ohms of the wire

Inductive Reactance

Inductive Reactance in an AC circuit $$Xl = {2 \pi f L}$$ $$L = {Xl \over {2 \pi f}}$$

Inductive coupling

Mutual inductance of two coils is reached with all the magnetic lines of the two coils are linked, and the mutual inductance is: $$M = Sqrt(L1 + L2)$$

If the two coils are not 100% coupled, then $$M = k * Sqrt(L1 + L2)$$ Where k is the percentage of coupling between the two coils.

You can work out the efficiency of coupling by transposing that equation: $$k = M / Sqrt(L1 + L2)$$


Impedance transformers

$$Np / Ns = Sqrt(Zp / Zs)$$

  • Np = primary turns
  • Ns = secondary turns
  • Zp = primary impedance from power source
  • Zs = secondary impedance of the load

Impedance is the ratio of voltage to current $$Z = E/I$$

  • Z = Impedance in ohms
  • E = Electromotive force in volts
  • I = Current in amps

Wavelength

The wavelength of a signal can be calculated using the frequency of the signal and the speed through which it propogates a material or free space, described by:

$$λ = {C \over f}$$

  • λ - Wavelength
  • C - Speed of the wave
  • f - frequency in Hz

A wave traveling through free space moves at the speed of light or 299,792,458m/s. Howeve, it’s common to use a rounded value 300 for the wave speed along with a frequency in MHz.

An example

What is the wavelength of a signal at 146.450MHz?

$${300 \over 146.450} = 2.048m$$


Resonant circuits

The frequency of a series or parallel LC circuit is calculated with: $$Fr = {1 \over {2(pi) \sqrt{LC}}}$$

  • Fr = Frequency in hertx
  • L - inductance in Henry’s
  • C - Capacitance in Farads

Impedance of a series LC circuit: $$Zlcs = { jL{ {w^2 - w0^2} \over w} }$$

  • Where $w = 2(pi)f$ and $w0 = {1 \over \sqrt{LC}}$
  • Zlcs - Series LC impedance in ohms
  • f - Frequency in Hertz
  • L - inductance in Henry’s
  • C - Capacitance in Farads

Impedance of a parallel LC circuit: $$Zlcp = { j{1 \over C} {w \over {w^2 - w0^2}} }$$

  • Where $w = 2(pi)f$ and $w0 = {1 \over \sqrt{LC}}$
  • Zlcp - Parallel LC impedance in ohms
  • f - Frequency in Hertz
  • L - inductance in Henry’s
  • C - Capacitance in Farads

Bandwidth

Calculating the bandwidth of an FM signal $$BW = 2(fd + fs)$$

  • fd = Frequency deviation = 5kHz (allowed deviation)
  • fs = Modulation frequency = 3kHz (voice / audio deviation)

For example, a frequency deviation of 5kHz and a modulation deviation of 3kHz: $$BW = 2(5 + 3)$$ The bandwidth is 16kHz


Critical Resistance

Critical resistance is used to stop an oscillating signal from traversing a circuit. For example, parasitic oscillation can be prevented using a parasitic stopper in the form of an RF choke which increases the resistance of the parasitic circuit to a value greater than or higher than the critical resistance of the frequency of oscillating circuit:

$$Cr = 2Sqrt(L/C)$$

  • Cr = Critical Resistance in ohms
  • L = Inductance in Henrys
  • C = Capacitance in Farads
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