Bonner County Amateur Radio Club – K7JEP K7JEP: In memory of Neal Fredstrum   # Formulas/Laws/Rules

(Assembled from various sources)

I am attempting to create a useful reference for most of the formulas/laws/theorems used by the amateur radio operator.  Most references that I have found are catch-all’s, such as the CRC manuals that include physics, chemistry, etc., or are designed for engineers with formulas hams will never need.

This reference is to be used as a “memory jogger” not how the entries were derived or specifically how to use them.  (Many times you just need to remember whether to use straight addition or the reciprocal formula.)   A few shortcuts and notes are included, as well as examples for the harder to understand formulas, and I feel confident they are correct but I have not verified them. Notes and shortcuts will be denoted in italics if they are from other sources or GHW if they are mine (as below).

I am having difficulty figuring out the best way to organize the information.  So far the best I have come up with is to present them in the order you probably learned them.

1. DC Circuits
2. AC Circuits
3. Antennas
4. DC Network Analysis

I’m violating my own preferences by requiring you to scroll numerous times but, presently, I don’t see a way around it.  If you decide you want a printed copy in PDF format, it can be downloaded here.

If I have left out a formula you feel should be included or you have any suggestion for re-organizing the reference, send me an email (kj7ziz-g@proton.me), your help is appreciated.

GHW- The hardest part from my perspective.

Most laws and theorems are easy to state and understand.  Such as Kirchhoff’s Current Law (KCL): “The algebraic sum of all currents entering and exiting a node must equal zero.”  But, in actual usage are much more complicated.  Most of these laws/theorems require you to analyze a circuit and formulate multiple equations that are solved by the use of simultaneous equations.

For me, the hardest part is in analyzing the circuit to “break” it down into its proper applicable parts, being consistent in my labeling, keeping track of what part I just solved (re-drawing the circuit and re-labeling if necessary), and writing equations that are correct.  For instance you may have Ct (current total) and several branch currents (C1, C2, C3…Cn).  All of which you must keep track of which you are currently solving and which you have combined to solve for another.

Once the equations are written, the simultaneous solving becomes easy.

## DC circuit equations and laws

### Ohm’s and Joule’s Laws

##### Ohm’s Law:

E = IR              I = E/R                       R = E/I

##### Joule’s Law:

P = IE                    P = E2/R                      P = I2R

Where:

E = Voltage in volts

I = Current in amperes (amps)

R = Resistance in ohms

P = Power in watts

NOTE: the symbol “V” (”U” in Europe) is sometimes used to represent voltage instead of “E”.   In some cases, an author or circuit designer may choose to exclusively use “V” for voltage, never using the symbol “E.” Other times the two symbols are used interchangeably, or “E” is used to represent voltage from a power source while “V” is used to represent voltage across a load (voltage “drop”).

##### Kirchhoff’s Laws:

The algebraic sum of all voltages in a loop must equal zero.”   Kirchhoff’s Voltage Law (KVL)

“The algebraic sum of all currents entering and exiting a node must equal zero.”   Kirchhoff’s Current Law (KCL)

##### Series circuit rules:

Components in a series circuit share the same current. Itotal = I1 = I2 = . . . In

Total resistance in a series circuit is equal to the sum of the individual resistances, making it greater than any of the individual resistances. Rtotal = R1 + R2 + . . . Rn

Total voltage in a series circuit is equal to the sum of the individual voltage drops.  Etotal = E1 + E2 + . . . En

##### Parallel circuit rules:

Components in a parallel circuit share the same voltage. Etotal = E1 = E2 = . . . En

Total resistance in a parallel circuit is less than any of the individual resistances. Rtotal = 1 / (1/R1 + 1/R2 + . . . 1/Rn)

Total current in a parallel circuit is equal to the sum of the individual branch currents. Itotal = I1 + I2 + . . . In

### Series and parallel component equivalent values

##### Series and parallel resistances:

Rseries = R1 + R2 + . . . Rn

Rparallel = 1/(1/ R1 +1/R2 + . . . 1/Rn )

Where: R = Resistance in Ohms

##### Series and parallel inductances:

Lseries = L1 + L2 + … Ln

Lparallel = 1/(1/L1+1/L2+… 1/Ln)

Where: L = Inductance in henrys

### Series and parallel capacitances

Cseries = 1/(1/C1+1/C2 + …1/Cn)

Cparallel = C1 + C2 + . . . Cn

Where: C = Capacitance in farads

##### Capacitor sizing equation

C = εA/d

Where:

ε = Permittivity of dielectric (absolute, not relative)

A = Area of plate overlap in square meters

d = Distance between plates in meters

ε = ε0 K

Where:

ε0 = Permittivity of free space

ε0 = 8.8562 x 10-12 F/m

K = Dielectric constant of material between plates (see table below)

##### Dielectric Constants
Dielectric                 K Dielectric                     K
Vacuum               1.0000 Quartz, fused            3.8
Air                       1.0006 Wood, maple             4.4
PTFE, Teflon       2.0 Glass                          4.9-7.5
Mineral oil           2.0 Castor oil                   5.0
Polypropylene     2.20-2.28 Wood, birch              5.2
ABS resin            2.4 – 3.2 Mica, muscovite        5.0-8.7
Polystyrene          2.45-4.0 Glass-bonded mica    6.3-9.3
Waxed paper       2.5 Porcelain, steatite      6.5
Transformer oil    2.5-4 Alumina Al2O3         8-10.0
Wood, oak           3.3 Water, distilled          80
Hard Rubber       2.5-4.8 Ta2O5                       27.6
Silicones              3.4-4.3 Ba2TiO3                    1200-1500
Bakelite               3.5-6.0 BaSrTiO3                  7500
##### A formula for capacitance in picofarads using practical dimensions:

C = 0.0885K(n-1) A /d = 0.225K(n-1)A’/ d’

Where:

K = Dielectric constant

A = Area of one plate in square centimeters

A’ = Area of one plate in square inches

d = Thickness in centimeters

d’ = Thickness in inches

n = Number of plates

### Inductors

Faraday’s laws: Voltage induced depends on rate of change of magnetic flux. For an EMF caused by a moving magnet, change of flux is proportional to strength of magnet, speed of magnet or coil, number of turns of coil and area of cross-section of coil.

Lenz’s law: The direction of induced EMF is such that it always opposes the change (movement in this example, or change of current through a coil) that causes it.

The voltage of the back-EMF can be calculated from the rate of change of current through the conductor and the details of construction of the conductor such as straight wire or coil, number of turns of coil, use of a magnetic core and so on (called inductance or, more correctly, self-inductance , symbol L).

L is defined in the equation:

E = L (dI/dt)

Where:

E is the amount of back-EMF

L is inductance

dI/dt is the rate of change of current

*The symbol‘d’ is used here to mean a small change of the quantity that follows – the notation of calculus. If E is measured in volts and dI /dt is the rate of change of current in amps per second, then L is in units of  henries (H),

Example: What back-EMF is developed when a current of 3A through a 0.5 H coil is reduced to zero in 20 ms?

The amount of back-EMF is found from:

E = L (dI/dt) = 0. 5 × (3/20 x 10 – 3) = 75 V

*The large EMF (equal to L(dI/dt)) which is generated when current is suddenly switched off in an inductive circuit can have destructive effects, causing sparking at contacts or breakdown of semiconductor junctions.

##### Inductor sizing equation

L = N2μA/ l

μ = μrμ0

Where:

L = Inductance of coil in Henrys

N = Number of turns in wire coil (straight wire = 1)

μ = Permeability of core material (absolute, not relative)

μr = Relative permeability, dimensionless (μ0=1 for air)

μ0 = 1.26 x 10 -6 T-m/At permeability of free space

A = Area of coil in square meters = πr2

l = Average length of coil in meters

Wheeler’s formulas for inductance of air core coils which follow are useful for radio frequency inductors. The following formula for the inductance of a single layer air core solenoid coil is accurate to approximately 1% for 2r/l < 3. The thick coil formula is 1% accurate when the denominator terms are approximately equal. Wheeler’s spiral formula is 1% accurate for c>0.2r. While this is a “round wire” formula, it may still be applicable to printed circuit spiral inductors at reduced accuracy.

L = N2r2/9r + 10⋅l             L = 0.8N2r2/6r + 9⋅l + 10c                L = N2r2/8r + 11c

Where:

L = Inductance of coil in microhenrys

N = Number of turns of wire

r = Mean radius of coil in inches

l = Length of coil in inches

c = Thickness of coil in inches

The inductance in henries of a square printed circuit inductor is given by two formulas where p=q, and p6=q.

L = 85⋅10-10DN5/3                                                          L = 27⋅10-10(D8/3/p5/3)(1+R-1)5/3

Where:                                                             Where:

D = dimension, cm                                          D = coil dimension in cm

N = number turns                                            N = number of turns

p=q                                                                  R= p/q

The wire table provides “turns per inch” for enamel magnet wire for use with the inductance formulas for coils.

 AWG        turnsgauge       inch AWG        turnsgauge       inch AWG        turnsgauge       inch AWG        turnsgauge       inch 10           9.6 20           29.4 30            90.5 40            282 11           10.7 21           33.1 31            101 41            327 12           12.0 22           37.0 32            113 42            378 13           13.5 23           41.3 33            127 43            421 14           15.0 24           46.3 34            143 44            471 15           16.8 25           51.7 35            158 45            523 16           18.9 26           58.0 36            175 46            581 17           21.2 27           34.9 37            198 18           23.6 28           72.7 38            224 19           26.4 29           81.6 39            248

### Time constant equations

##### Value of time constant in series RC and RL circuits:

Time constant in seconds = RC         or          Time constant in seconds = L/R

##### Calculating voltage or current at specified time:

Universal Time Constant Formula

Change = (Final-Start)(1 – (1/e)

Where:

Final = Value of calculated variable after infinite time (its ultimate value)

Start = Initial value of calculated variable

e = Euler’s number (2.7182818)

t = Time in seconds

τ = Time constant for circuit in seconds

Calculating time at specified voltage or current:

t = -τ (1n (1- (Change/Final-Start)))

## AC circuit equations

##### Inductive reactance:

XL = 2πfL

Where:

XL = Inductive reactance in ohms

f = Frequency in hertz

L = Inductance in henrys

##### Capacitive reactance:

XC = 1/2πfC

Where:

XC = Inductive reactance in ohms

f= Frequency in hertz

##### Impedance in relation to R and X:

ZL = R + jXL

ZC = R – jXC

Ohm’s Law for AC:

E = IZ                               I = E/Z                              Z = E/I

Where:

E = Voltage in volts

I = Current in amperes (amps)

Z = Impedance in ohms

##### Series and Parallel Impedances:

Zseries = Z1 + Z2 + . . . Zn

Zparallel = 1/ (1/Z1 + 1/Z2 + …1/Zn)

NOTE: All impedances must be calculated in complex number form for these equations to work.

##### AC power

P = true power                        P = I2R                        P = E2/R

Measured in units of Watts

Q = reactive power                 Q = I2X                       Q = E2/X

Measured in units of Volt-Amps-Reactive (VAR)

S = apparent power        S = I2Z        S = E2/Z            S = IE

Measured in units of Volt-Amps

P = (IE) (power factor)          S = √P²+Q²

Power factor = cos (Z phase angle)

## Antennas

f = 1/T

Where:

f = frequency

T = period

##### Wavelength

λ = c/f

Where:

c = (speed of light) = 299,792,458 meters/sec (This is usually rounded to 300,000,000 m/s) = 186,000 mph

f = frequency measured in Hertz (cycles per second)

##### Resonance

fresonant = 1/2π

NOTE: This equation applies to a non-resistive LC circuit. In circuits containing resistance as well as inductance and capacitance, this equation applies only to series configurations and to parallel configurations where R is very small.

##### Length of antenna in feet

Length of wire in feet = 468/f                        or                     Length = 246/f

Note: This does not account for the velocity factor 468 or 246 should be multiplied by the velocity factor of the wire.

Where:

468 is for 1/2 wave length (0.95 vf usually works for initial lengths giving λ = 445/f as the formula)

246 is for ¼ wave length (0.95 vf usually works for initial lengths giving λ = 234/f as the formula)

f = frequency in Hertz (cycles per second)

##### Helpful Math Trick to Refine Above Formula

Here’s a simple math trick to figure out if 468 isn’t correct for your area. When the dipole is installed at the length of the standard equation determine the SWR at resonance. For example, resonance happens to be14.234 Mhz.  Slightly higher than calculated. Here’s the math to figure out a new constant.

where:

Original Length: 33ft.

New Resonant Freq: 14.234 MHz

33 x14.234 ̃=470

Now the equation for a dipole at your location is: Lfeet = 470/ƒMHz

Do the calculation one more time using the new constant to determine the new length and the new dipole will be resonant at the design frequency.

##### Decibels

AV(dB) = 20 log AV(ratio)                                                 AV(ratio) = 10 Av(dB)/ 20

AI(dB) = 20 log AI(ratio)                                       AI(ratio) = 10 AI(dB)/20

AP(dB) = 10 log AP(ratio)                                                  AP(ratio) = 10 AP(dB)/10

## DC Network Analysis

The major methods/theorems for DC network analysis are given below.

##### Branch Current Method

The branch current method is a network analysis technique in which branch current directions are assigned arbitrarily, and then Ohm’s law and Kirchhoff’s current and voltage laws are applied systematically to solve for the unknown currents and voltages.

##### Millman’s Theorem

In Millman’s Theorem, the circuit is re-drawn as a parallel network of branches, each branch containing a resistor or series battery/resistor combination. Millman’s Theorem is applicable only to those circuits which can be redrawn accordingly.

##### Superposition Theorem

The superposition theorem states that any linear circuit with more than one power source can be analyzed by summing the currents and voltages from each individual power source.

##### Thevenin’s Theorem

Thevenin’s theorem states that any linear circuit, no matter how complex, can be simplified to an equivalent circuit consisting of a single voltage source with a series resistance connected to a load.

##### Norton’s Theorem

Norton’s theorem states that any linear circuit can be simplified to an equivalent circuit consisting of a single current source and parallel resistance that is connected to a load.

##### Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than the maximum.

##### Δ-Y and Y-Δ Conversions

In many circuit applications, components are encountered connected together in one of two ways to form a three-terminal network: the “Delta,” or Δ (also known as the “Pi,” or π) configuration, and the “Y” (also known as the “T”) configuration.

It is possible to calculate the proper values of resistors necessary to form one kind of network (Δ or Y) that behaves identically to the other kind, as analyzed from the terminal connections alone. That is, if we had two separate resistor networks, one Δ and one Y, each with its resistors hidden from view, with nothing but the three terminals (A, B, and C) exposed for testing, the resistors could be sized for the two networks so that there would be no way to electrically determine one network apart from the other. In other words, equivalent Δ and Y networks behave identically.

There are several equations used to convert one network to the other:

To convert a Delta (Δ) to a Wye(Y)        To convert a Wye (Y) to a Delta (Δ)

RA = RABRAC/RAB + RAC + RBC                           RAB = RARB + RARC + RBRC/RC

RB = RABRBC/RAB + RAC + RBC                    RBC = RARB + RARC +RBRC/RA

RC = RACRBC/RAB + RAC + RBC                           RAC = RARB + RARC + RBRC/RB