**Shiawassee Amateur Radio Association ~ W8QQQ **

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Frequency, Wavelength & Bands
| Ohm's Law
| Capacitance
| Inductance
| Resonance

# SARA's Electronics Reference Page

Shiawassee Amateur Radio Association presents Reference Information which we hope is useful to our viewers.

We also have a mathematics reference page ~ Mathematics Reference Page.

## Frequency, Wavelength & Bands

The speed of light is the constant rate of propagation of electromagnetic waves. Using the value in atmospheric environment (air) propagation at a frequency can be ratioed to determine the length cooresponding to one complete wave. Further, a 'rounded approximate' value for the speed of light is 300,000,000 is generally used for establishing frequency band limits using radio frequencies. Thus "Lamda" (wavelength) is speed of light [*c*] divided by frequency in hertz [*Hz*]. The unit of hertz is named after Henrich Rudolf Hertz [1857 ~ 1894] and is defined as one cycle per second. Wavelength for a given frequency or frequency for a given wavelength are represented as: { c=300,000,000 }

or

A band is a small section of the spectrum of radio communication frequencies, in which divisions are defined by various regulations.

To prevent interference and allow for efficient use of the radio spectrum, similar services are allocated in the various bands. For example, broadcasting, mobile radio, or navigation devices, will be allocated in non-overlapping ranges of frequencies. As a matter of convention, bands are divided at wavelengths of 10^n metres, or frequencies of 3 x 10^n hertz { 10^n is in integer powers of 10 }. For example, 30 MHz or 10 m divides shortwave (lower and longer) from VHF (shorter and higher). Further use has offically named the various bands as:

Band Name | Abbreviation | ITU Band | Frequency | Wavelength |
---|---|---|---|---|

Extremely Low Frequency | ELF | 1 | 3 - 30 Hz | 100,000 km - 10,000 km |

Super Low Frequency | SLF | 2 | 30 - 300 Hz | 10,000 km - 1,000 km |

Ultra Low Frequency | ULF | 3 | 300 - 3000 Hz | 1,000 km - 100 km |

Very Low Frequency | VLF | 4 | 3 kHz - 30 kHz | 100 km - 10 km |

Low Frequency | LF | 5 | 30 kHz - 300 kHz | 10 km - 1 km |

Medium Frequency | MF | 6 | 300 kHz - 3000 kHz | 1 km - 100 m |

High Frequency | HF | 7 | 3 MHz - 30 MHz | 100 m - 10 m |

Very High Frequency | VHF | 8 | 30 MHz - 300 MHz | 10 m - 1 m |

Ultra High Frequency | UHF | 9 | 300 MHz - 3000 MHz | 1 m - 100 mm |

Super High Frequency | SHF | 10 | 3 GHz - 30 GHz | 100 mm - 10 mm |

Extremely High Frequency | EHF | 11 | 30 GHz - 300 GHz | 10 mm - 1 mm |

## Ohm's Law

An electrical circuit has a power source with voltage and current, some external circuit attached to the power source with electrical conductors. A direct current source has 'steady state' electrical characteristics. The external circuit restricts the current to a value based on the voltage and the amount of restriction. Under these steady values, the restriction to current flow (charge motion) is identified by the term resistance [*R*]. The electromotive force [*E*] or electric potential is called voltage. The charge motion is called electrical current [*I*].
The relationship between these characteristics was developed by Georg Ohm [1789~1854] and is called "Ohm's Law". The relationship is expressed as:

or or

Electrical power [watts] is the product of voltage [* E *} and current [* I *]. It follows that there are relationships for power using just voltage or current and the resistance values. They are:

## Capacitance

For purposes here we are defining capacitance as a storage of electrical charge between two metal surfaces. They are separated by a distance with an insulating material which is between the metal surfaces (the *dielectric*). The capacitance is directly proportional to the surface area and inversely proportional to the spacing distance. The permittivity of the dielectric material between the plates establihes the capacitance value. Using dry, atmospheric pressure air as a 'base material', which is defined as having permittivity of 1.0, different materials can be classified as modifying the actual capacitance (the area/spacing values get multiplied by the capacitance). For many materials the permittivty is independent of the voltage, thus only a direct multiply is required. Restating: "The capacitance is a function only of the geometry of the design (area of the plates and the distance between them) and the permittivity of the dielectric material between the plates of the capacitor." If the charges on the plates are *+q* and *-q*, and *V* they establish the voltage between the plates. Capacitance [C] is then defined and by:

which gives the voltage/current relationship

The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt.

The energy stored in a capacitor is found by integrating the work W:

The most common subunits of capacitance in use today are the microfarad (μF), nanofarad (nF), picofarad (pF). Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known.

where:

C is the capacitance, in farads;

A is the area of overlap of the two plates, in square meters;

*εr* is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum, *εr* = 1);

*ε0* is the electric constant (ε0 ~ 8.854 x 10^-12 F / m) (Farads per meter) and

d is the separation between the plates, in meters;

Capacitance can vary with applied voltage and/or frequency but these topics are beyond this discussion. If desired go to www.wikipedia.com and search "capacitance" for further details in this area. Also, there are formulas for various shaped plate capacitors and other details that you can review.

The *electric constant* or *vacuum permittivity* or *permittivity of free space* is the capability of a vacuum to permit electric field lines and relates the units for electric charge to mechanical quantities such as length and force. Historically, the parameter *ε0* has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum", "permittivity of empty space", or "permittivity of free space" are generally widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity.

### Capacitive Reactance

An opposition to the change in voltage across a capacitive device is defined as ** Capacitive Reactance**. Capacitive Reactance is inversely proportional to the frequency and the capacitance. As either the capacitance or frequency get larger, the reactance value gets smaller. Reactance acts like a "resistance value" when applied voltage varies at a frequency [AC]. At low frequencies a capacitor is an open circuit so no current flows in the dielectric.

A DC voltage applied across a capacitor causes positive charge to accumulate on one side and negative charge to accumulate on the other side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

A capacitor driven by an AC supply only accumulates a limited amount of charge before the voltage changes polarity and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current (reactance).

Reactance and resistance have similar effects and similarity of the mathematics results, thus making it easier for you to remember the equations involved. There is a choice on mathematical signs in various equations (more advanced concepts get involved). For our purposes, it easiest to assume capacitive reactance has a negative value - so that is what we will do!

A DC voltage applied across a capacitor causes positive charge to accumulate on one side and negative charge to accumulate on the other side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Capacitive reactance, when involved with inductive reactance, is where all the fun gets started ~ stay tuned! [pun intended]

## Inductance

The following is a direct quote from Wikipedia, the free encyclopedia:

= = = = = = = Start quote = = = = = =

In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current flowing through it induces an electromotive force in both the conductor itself and in any nearby conductors by mutual inductance.

These effects are derived from two fundamental observations of physics: a steady current creates a steady magnetic field described by Oersted's law, and a time-varying magnetic field induces an electromotive force in nearby conductors, which is described by Faraday's law of induction. According to Lenz's law, a changing electric current through a circuit that contains inductance induces a proportional voltage, which opposes the change in current (self-inductance). The varying field in this circuit may also induce an e.m.f. in neighbouring circuits (mutual inductance).

The term inductance was coined by Oliver Heaviside in 1886. It is customary to use the symbol L for inductance, in honour of the physicist Heinrich Lenz. In the SI system, the measurement unit for inductance is the henry with the unit symbol H, named in honor of Joseph Henry, who discovered inductance independently of, but not before, Faraday.

= = = = = = = End quote = = = = = =

An ** Inductor** is a component used to add inductance to an electricaal circuit. A conductor in a closed loop develops the magnetic field concentration and linkages to nearby conductors (including itself) to cause the effect of inductance. This field devlops a "back EMF" across the inductor that impacts the rate of change of the current through the inductor(s). This impact is directly proportional to the frequency. The relationship between the self-inductance L of an electrical circuit, the voltage v(t), and the current i(t) through the circuit is:

When a change in one inductor induces a voltage in another inductor it is called **"mutual inductance"**. This could be a wanted or unwanted effect. In the case of transformers it is considered a wanted effect.

where:

is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.

N1 is the number of turns in coil 1,

N2 is the number of turns in coil 2,

P21 is the permeance of the space occupied by the flux.

The complete study of inductance versus physical shape(s) and various mutual inductance relationships is beyond this dialog. You can research these on you own time and pace. We will go on to ** Inductive Reactance** and then to resonance effects.

### Inductive Reactance

An opposition to the change in current thru an inductor is defined as ** Inductive Reactance**. Inductive Reactance is directly proportional to the frequency and the inductance value. As either the inductance or frequency get larger, the reactance value gets larger. Reactance acts like a "resistance value" when the applied current varies at a frequency [AC]. At high frequencies an inductor starts to have very large value of reactance {oppostion to current flow}. The equation for calculating inductive reactance is:

Of note between the capacitive and inductive reactance is that one gets smaller and one gets larger as frequency is increased. Or, the reactances 'switch' characteristics as the frequenct gets lower. That starts to imply our next topic ~ *Resonance*.

## Resonance

At the 'middle frequency range' for reactances, say the frequency is equal and we rearrange the equations and set inductive and capacitive reactance equal and then solve for frequency. The equation for "resonance" gets derived:

Setting reactances to equal values means that ω*L* = 1/ω*C*, so the equation above results. Recognize that these are inductance and capacitance and not reactance values. ω = 2πf is a very common substitution in many electronic mathematics operations. If you solve each reactance equation for 2πf and set them equal to each other, the above equation can easily be seen.

If one considers the real world where resistances are required to be accounted for, the result is an R-L-C type of circuit for analysis. The exact topology of the circuit can have different forms from the simple series connection of the resistors, inductors and capacitors (or simple three parallel connections) to quite complex series-parallel connections of many multiple devices. The basic result is that most circuits can be described in "2nd order liner equations" nad solved quite readily (Note: It does not say 'easily solved').

The addition of the resistor term in the R-L-C results in a 'loss of electrical energy' in the circuit if not resupplied by the source. The rate of loss establishes a 'Q factor' for the circuit. We will leave that for you study on your own.

The 'tuned circuit' is used widely in electronics. Oscillators, filters [band-pass filter, band-stop filter, low-pass filter or high-pass filter], and any 'tunning circuit' all use the resonance effect to achieve the desired result. You can study on-line or in books on these topics to get at whatever level of understanding you desire.

We also have a mathematics reference page ~ Mathematics Reference Page.

The equations shown on this page are presented by using *"Latex"* and the equation writing capabilities from the Code Cogs Web Site. Images are then stored "locally" to allow proper HTML5 validation.

Page Navigation:

Frequency, Wavelength & Bands
| Ohm's Law
| Capacitance
| Inductance
| Resonance

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Web Site Contact: W8QQQ at arrl.net

Page Last Updated: 03-March-2017

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