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The Junction field effect transistor

JFET’s are constructed in two types. They can either be N-channel, or P-channel. In an N-channel JFET, There is a solid layer of N-type semiconductor, with two layers of P-type material attached to the sides. These two P-type materials are calledd the gate, and the two ends of the N-type material are called the source and the drain.

In diagrams, the drain is at the upper end, and the source is at the bottom end. Currrent in the drain circuit flows from the source to the drain.

The JFET is always operated with the gate-source junction reverse-biased. This reverse biasing of the gate-source junction with a negative gate voltage produces a depletion region in the p-n junction, which extends into the N-channel and increases the resistance between the source and the drain terminals.

In an example with two power supplies, one is attached from the drain to the sourceand is called Vdd, and is known as the drain circuit. The negative terminal is connected to ground, as well as to the source of the JFET. The positive end is connected to a series limiting resistor (Rs) and also to the source terminal of the JFET.

The gate supply (Vgg) is connected with the positive end to ground, and the negative end to the gate. This creates a negative gate voltage, which is needed for the reverse biasing of the gate source pn junction.

A greater value of Vgg narrows the channel, which increases the resistance of the JFET, and decreases drain current (Id).

Less Vgg widens the channel, which decreases resistance and increases drain current(Id).

Pinch-off voltage

The Pinch-off voltage (Vp) is the value of voltage from drain to source at which drain current (Id) becomes constant. In this area, known as the constant-current area, drain current will remain constant until it reaches breakdown. Once breakdown occurs, the JFET is being operated out of range and current will increase quite rapidly until it is destroyed.

Cutoff voltage

The value of voltage from the gate to the source that produces a drain current of approximately zero is called the cutoff voltage, or Vgs(off). For N-channel JFET’s, this will be a negative voltage, and this causes the delpetion region to become so large that current flow is stopped.

There is a relation between the pinch-off voltage and the cutoff voltage. Vgs(off) and Vp are always equal, but opposite in sign. That is, if Vgs(off) is -3 volts, then pinch-off voltage is 3 volts.

to be continued…

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I have a Ti 35 (not shown) and a Ti 89 platinum (below). I use TiLP to transfer files to and from my TI 89 and my Ubuntu box.

Links:

Ticalc.org

Backlighting

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The simplest way of describing a diode is a single P-N junction with a lead attached to each end. The end with the N-type material is named the Cathode, and the end with the P-type material is known as the Anode. Diodes (and other semiconductor devices) behave differently than simple resistors due to the fact they are non-linear, which means their current is not directly proportional to their voltage. When you have a simple resistive circuit, current proportional to voltage is plotted on a straight line, and is therefore linear. The graph of a diode has a certain point where it begins to conduct, and also a reverse point where it starts to breakdown.

Starting with the forward region, once the biasing voltage source overcomes the barrier potential, the diode begins to allow electron flow. For a normal doped silicon diode, this is .7 volts. This is also known as the knee voltage, because once .7 volts is achieved, the voltage on the graph turns very sharply up, creating what looks like a knee in the line. Above the knee voltage, diode current increases very rapidly. Once the barrier potential is overcome, all that impedes the flow of current is the resistance of the P and N junctions. This is called the Bulk resistance of the diode and can be calculated from the sum of the resistance of the P and N junctions.

Another thing to consider in the forward region is the maximum DC forward current. This can be found on datasheets. Once this is achieved, the diode will probably be destroyed due to excessive heat. This is usually termed If(max) or Io. Diode datasheets also have a maximum power disspation rating.

When diodes are operated in the reverse region, you get a very small amount of leakage current, and there is a point when the diode will breakdown, due to an effect called avalanche. When so many electrons are being forced onto the diode, the energy that propels them is enough to force other electrons out of their valence band and across the P-N junction. This breakdown voltage is also put on the datasheet. Although some specialty diodes, like zener diodes are meant to be operated in this way, on a normal diode, avalanche is to be avoided.

Special purpose diodes

Rectifier diodes are constructed to allow current in only one direction. when used with an AC voltage source, this cuts off one side of the sine wave, and creates a pulsating DC wave. Say the circuit is connected so only the positive alternations are passed, once the sine wave reaches 0 volts, it remains there until the wave reaches 0 volts again, and then continues on passing a positive sine.

This arrangement of a diode in a circuit is known as a half-wave rectifier.

where:
Vp is your peak voltage
Vs is your voltage source
and .7v is the voltage drop across the diode (silicon).

If you arrange a diode this way on both ends of a AC voltage circuit and then combine them, you get what is known as a full-wave rectifier. Only the positive waves are passed, and they are 180 degrees out of phase with each other. The end effect is as the positive sine wave of the first signal drops to zero, the other side pulses and completes it’s sine, and so on. the result ends up looking like a regular sine wave, with the negative alternations flipped positive. Full-Wave rectifiers can be used in power supplies, where an AC signal is provided and a DC voltage is desired. Full wave rectifiers must use a center tapped transformer.

Since the negative alternations are simply dropped, normal full-wave rectifiers are wasteful. When designing a rectifier circuit, it is better to use a bridge rectifier. Bridge Rectifiers have two ways for the current to flow, so there is a path on each alternation. Most power supplies use this configuration. Since a center tap is not needed, the rectified voltage is twice what a full wave recifier would create.

In bridge rectifiers, another thing to consider is since you have two diodes dropping voltage on each path, the voltage is calculated by:

where:
Vp is your peak voltage
Vs is your voltage source
and 2(.7v) is two .7 voltage drops across the diodes (silicon).

If you connect a DC Voltmeter across the load, it will indicate the average value of the full wave signal, which is:

which is equivalent to

.636 * peak voltage.

The frequency of a full wave signal is double the input frequency, since a waveform completes it’s cycle as soon as it repeats. For a 60 hertz input:

Time = 1/Frequency

Time = 1/60

Time = 16.7ms

The rectified voltage has a period of

Time2 = 16.7ms/2

Time2 = 8.33ms

Frequency2 = 1/8.33ms

Frequency2 = 120 hertz.

Another way to put this simply is to say:

Fout = 2Fin

where:
Fin is Frequency In
Fout is Frequency Out

Another type of diode is the Zener diode. Most diodes are never operated in the breakdown region beacuse it would damage them. A zener is manufactured to be operated in the reverse region, and to have a specific voltage where it will begin to conduct. Zener’s are available in many different voltages. A zener diode is sometimes referred to as a zener voltage regulator becuase they can be used in parallel to allow a certain voltage to pass to the load, and then begin to conduct once the zener voltage is reached, therefore passing the remaining voltage through the zener and bypassing the load. A series resistor is always used in this configuration to limit current flow.

Maximum power through a zener diode is found by:

Zener Impedance can be found through:

The change in Zener voltage (^Vz) can be found by:

LED‘s or light emitting diodes are another specialty diode. As the electrons cross the P-N junction and fall into holes, they radiate energy. LED’s are constructed to show this as visible light. By using elements like arsenic and phosphorus, LED’s can be manufactured in red, green, yellow, blue, orange and even infrared. The exact voltage drop across LED’s depends on the color. The typical voltage drop is 1.5 to 2.5 volts for currents between 10 and 50 milliamps.

All diodes have an associated capacitance, due to the way they are constructed. The P and N regions can be thought of as the plates, and the depletion region is the dielectric. Varactor diodes are built to take advantage of this, and are used in tuning circuits where a variable resonant frequency is desired. As the voltage is varied, the depletion region expands and contracts, causing the capacitance to change. You can connect a varactor in parallel with an inductor to get a resonant circuit, and then vary the biasing voltage to achieve specific resonant frequencies.

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Since conductors have a single valence electron, and insulators have a full valence ring of eight electrons, it makes sense that semiconductors such as silicon have four valence electrons. This also means that there is four spots for valence electrons in a silicon atom. When atoms of silicon combine they create covalent bonds. Co as in shared, and valent, meaning valence electrons. The result is a silicon crystal, which can be thought of as a lattice of silicon atoms, all connected by their shared electrons.

Doping is the process of adding impurities to silicon (or other semiconductors) to alter the electrical characteristics of the semiconductor. If we think of a pure, or intrinsic piece of silicon, there is ideally no free electrons, and no free “holes” in the valence bands for electrons to go.

We add an atom with 3 valence electrons. Elements with 3 valence electrons are aluminum, boron, gallium, and indium. This creates a tri-valent bond and leaves on open “hole” for an electron to flow in and out of. Doping a semiconductor this way creates a P-type material. To remember, you can think of the “P” as positive. There is a deficiency of one electron, so the 3 valence atom added is known as a acceptor impurity element.

The other method of doping is to add an atom with 5 valence electrons to a piece of silicon. Elements with 5 valence electrons are arsenic, antimony and phosphorus. This creates a crystal with an extra electron that is free to move around and is known as a penta-valent bond. This is known as a N-type material and can likewise be remembered that the “N” is for negative. There is an extra electron, so the 5 valence atom is known as a donor impurity element.

As you can see, the amount of impurities added directly effects the electrical characteristics, and can be used to regulate the amount of electrons moving through the material. Simply having a P or N type material on their own might have some uses, but when the two are used together, a P-N junction is formed, and is the basis of many electronic devices used today.

When the two materials are put together, they repel each other. The free electrons spread out, and some of them diffuse across the junction. This is known as the depletion region. Each time an electron crosses over, it leaves a pentavalent ion, with a relative positive charge. This electron in turn falls into a hole in the P-type material, and causes a negatively charged trivalent ion. It has space for one electron, and when it is filled we can say that it has gained a relative negative charge. This region, with positively charged extra electron ions, and negatively charged electron deficient ions, creates a potential difference between them. This is known as barrier potential. The barrier potential is usually .7v for silicon, and varies for other types of semiconductors. The basic idea is exactly the same though.

The barrier potential must be overcome to allow electron flow in the P-N type material. Biasing a P-N junction is the process of adding a voltage source to either allow or prevent the flow of electrons.

When the N-type is negative with respect to the P-type material, the electrons easily flow from the power supply, to the junction, then from one side to the other. The N-type material constantly feeds the electrons to the P-type, and the electrons flow from the p-type back to the power supply. This is known as forward bias. The P-N junction is arranged in the circuit to allow electron flow.

Reverse biasing is done by changing the polarities of the voltage source, so the negative terminal is connected to the P-type, and the positive terminal is connected to the N_type material. This causes the depletion layer to widen, because the negative terminal attracts the free “holes” and the positive terminal attracts the free electrons. Current is not allowed to flow.

So far we have been looking at this in a perfect world, but in reality, there are a few holes in a N-type material and likewise there is a few extra electrons in a P-type material. These are known as the minority carriers, and are mostly caused by thermal energy, or heating of the P-N junction. Under normal operating temperatures, this amount is negligible. Datasheets are invaluable when seeking the maximum temperatures, voltages, and dissipation of power for any electronic device.

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RLC circuits are named after the components that they contain. Resistors (R), Inductors (L) and Capacitors (C). In these circuits, there are two separate reactances, both opposing each other. In series RLC, the inductive voltage (Vl) is leading the current (I) by up to 90 degrees. The capacitive voltage (Vc) is lagging the current by up to -90 degrees. So what we have is two reactances, working to cancel each other out. Luckily enough, this is exactly how we calculate them.

The lower of the two reactances is subtracted from the higher one. Say Xl is 100 ohms, and Xc is 75 ohms, the resulting reactance (Xnet) is 25 ohms, and since the resulting reactance is Xl, we say the circuit is acting inductively. Likewise, if Xc is greater than Xl, after we subtract one from the other, we say the circuit is acting capacitively.

It is easier to remember this by looking at vector diagrams. Current is once again our reference vector, with resistance in phase on the horizontal, inductance is plotted upward, and capacitance plotted downward. Whether we are talking voltages (Vl/Vr/Vc/Vt) or resistance (Xl/R/Xc/Z), in series circuits, the vectors look the same.

When applying the pythagorean theorem to series RLC, we simply subtract the reactances, and use the remainder (Xnet) in the formula. First subtract lesser reactance from the greater.

or

and then

same thing for reactances to calculate impedance.

Xl – Xc or Xc – Xl = Xnet

Z = √(R^2 + Xnet^2)

An example to calculate Impedance in Series RLC:

R = 100 ohms
Xl = 75 ohms
Xc = 60 ohms

75 – 60 = 15 ohms = Xnet

√(100^2 + 15^2) = 101.119 ohms = Z

Tan-1( 15 / 100 ) = 8.53 degrees = Phase angle

Say we had an applied AC voltage of 20 volts @ 100Hz . First calculate total current using Ohm’s law. I = V / Z.

20 / 101.119 = 197.787 mA

From there we can calculate the voltage drops across the components

Vr = .197787 * 100 = 19.7787 volts
Vl = .197787 * 75 = 14.834 volts
Vc = .197787 * 60 = 11.8672 volts

As you can see, there is a lot of voltage here, but Vl and Vc are canceling each other out. Working through it again, we can double check our work.

Vl – Vc = Vnet
14.834 – 11.8672 = 2.9668 volts

√(19.7787^2 + 2.9668^2) = 20 volts = Vt

Tan-1( 2.9668 / 19.7787 ) = 8.53 degrees. It checks out.

Parallel RLC circuits

Once again, the reactances cancel each other, the same way as in series, but in parallel circuits, voltage is our reference vector (since the same voltage flows though each branch) and is plotted horizontally, along with resistive voltage (in phase). Capacitive current leads the voltage by up to 90 degrees (ICE) and inductive current lags the voltage by up to -90 degrees.

An example of a parallel RLC circuit. Let’s keep our 20 volts @ 100Hz and use:

R = 55 ohms
Xl = 225 ohms
Xc = 125 ohms

Say for simplicity, that each component is on it’s own branch. In a parallel circuit, we want to calculate branch currents first.

Ir = Vt /R
Ir = 20 / 55
Ir = 363.636 mA

Ixl = Vt / Xl
Ixl = 20 / 225
Ixl = 88.8889 mA

Ixc = Vt / Xc
Ixc = 20 / 125
Ixc = 160 mA

We subtract Ixl from Ixc to get our net reactance current (Inet):

160 – 88.8889 = 71.1111 mA = Inet

find total current through the pythagorean theorem:

It = √(Ir^2 + Inet^2)
It = √(.363636^2 + .07111111^2)
It = 370.524 mA

use arctan to solve for phase angle:

Tan-1( .0711111 / .363636 ) = 11.0649 degrees = phase angle

and lets finish by calculating impedance from Ohm’s law.

Z = Vt / It
Z = 20 / .370524
Z = 53.9776 ohms

References:
Foundations of Electronics, by Russell L Meade

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Complex numbers are an easy way to perform mathematical computations with AC quantities. While you can use Trigonometry and the Pythagorean Theorem to solve for magnitudes and values, two notations exist to make life easier.

In rectangular notation, the complex number has two parts, one real and one imaginary. The real number represents the in phase and resistive element, and is plotted on the X-axis. The imaginary number, is represented on the Y-axis. To input an imaginary number on your calculator, you use the i button. The real number is input, followed by + or – depending on if the imaginary number is a positive or negative angle, then the imaginary number, followed by the i symbol. On my Ti-89, to show an i you press (2nd > Catalog). It ends up looking like this.

3 + 4i

This tells you that 3 is the Adjacent (horizontal axis) and 4 is the Opposite (vertical axis). If you input numbers in this fashion, and put your calculator in Polar form notation, it will give you the hypotenuse, and the phase angle, all at once.

Polar form notation shows you the length (hypotenuse) and the angle of the resulting vector. The angle symbol on the Ti-89 is generated by the keystrokes (2nd > EE). In this example, you would see

5 ∟ 53.1301°

This can be confirmed by using the sin and cos functions.

5 * sin(53.1301°) = 4
5 * cos(53.1301°) = 3

With your calculator in Rectangular form, you can input polar equations and have them return a real number and an imaginary number. On my calculator I need to enclose the polar notation in brackets.

(5 ∟53.1301°) = 3 + 4 * i

You can also use built in functions on a Ti 89 to do these conversions on the fly. Press the catalog button and you will find â–ºRect and â–ºPolar. You can use these functions without changing the mode of your calc.

(5 ∟ 53.1301°) ►Rect
returns
3 + 4 i

and also
(3 + 4 i) â–ºPolar
returns
5 ∟ 53.1301°

Algebraic Operations in Rectangular notation

Addition (add the in phase terms, and add the out of phase terms)

10 + 10i
15 + 15i
———
25 + 25i

Subtraction (change the sign of the subtrahend and add each)

10 + 10i
-5 – 15i
———
5 – 5i

Algebraic Operations in Polar notation

Multiplication (Multiply the magnitudes and add the angles)

5 ∟ 30°
20 ∟ -15°
———-
100 ∟ 15°

Division (Divide one magnitude by the other and subtract the angles)

35 ∟ 60°
140 ∟ 20°

35 / 140 = .25
60° – 20° = 40°
so the answer is
.25 ∟ 40°

References:
Foundations of Electronics, by Russell L Meade
Basic Electronics, by Bernard Grob

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After looking at Inductors and Capacitors and their reactances, I got to wondering why you would use both inductors and capacitors in a circuit when it seems they counteract each other. The answer is to create resonant circuits. The goal here is to create a circuit with equal inductive reactance (Xl) and Capacitive reactance (Xc).

Lets start with series resonance. In inductive reactance, the voltage leads the current (ELI) by 90 degrees. In capacitive reactance, the current leads the voltage (ICE) by 90 degrees, or to be more precise, the voltage lags the current and creates a negative phase angle. Current is our reference vector and is plotted on the horizontal axis.

Characteristics of series resonance

• Xl = Xc
• Impedance (Z) is at a minimum because the inductive and capacitive elements are counteracting each other. This causes Z to equal Resistance (R)
• Current (I) is at a maximum and equals V/R
• Phase angle is 0 degrees since the circuit is acting completely resistive.

The inductive voltage (Opp at +90 degrees) is leading the current and resistive voltage (our reference vector, the Adj, etc.) and the Capacitive voltage is lagging the current and resistive voltage by -90 degrees (also the Opp).

Remembering the formula used in calculations of frequency when dealing with inductive reactance,

f = Xl/(6.28*L)

Where:
f = frequency in hertz
Xl = inductive reactance in ohms
L = inductance in henrys

and the formula for capacitive reactance.

f = 1/(6.28*C*Xc)

where:
f = frequency in hertz
Xc = capacitive reactance in ohms
C = capacitor value in farads

We need to combine them so we can get a formula to calculate resonant circuits.

fr = 1/(6.28 * √(L*C))

where:
f = frequency in hertz
C = capacitor value in farads
L = inductance in henrys

This, and the variations below, are formulas used to create series and parallel resonant circuits. If you have the Frequency (Fr) of the circuit and the capacitive value (C) you can calculate the inductor(s) needed by transposing the formula to the following.

L = .02533 / Fr^2 * C

and if you have the Frequency and the inductive value, you can use the formula

C = .02533 / Fr^2 * L

Q is known as a magnification factor or figure of merit. In general, the higher the ratio of reactance at resonance to the series resistance, the higher the Q. We can calculate Q from the following formula.

Q = Xl / Rs

where:
Q = Magnification Factor
Xl = Inductive reactance at the resonant frequency
Rs = resistance in series with Xl

Q can also be calculated by

Q = Vout /Vin

where:
Vout = ac voltage measured across the reactive element
Vin = applied voltage

Q will always be a positive number.

Characteristics of Parallel resonance

In parallel resonance,

• Impedance (Z) is at a maximum
• Tank current is current measured between the inductors and the capacitors and can be thought of as circulating from one to the other.
• Line current is at a minimum
• Xl = Xc
• Phase angle is still 0 degrees

The same formulas for calculating series resonance apply here, just the results differ depending on whether the circuit is series or parallel. To calculate Q in a parallel resonant circuit, use

Q = Zt / Xl

where:
Q = Magnification Factor
Zt = total impedance
Xl = inductive reactance
Selectivity refers to the response curve of a resonant circuit. When resonance is achieved, the sharper the rise (and fall) of the curve, the more selective it is. For series resonance, the two points which current rises and falls to 70.7% of it’s maximum level is known as it’s bandwidth. For parallel resonant circuits, impedance (Z) is used.

Total bandwidth is calculated by the resonant frequency divided by the Q Factor.

BW = Fr / Q

where:
BW = bandwidth in Hertz
Fr = resonant frequency in Hertz
Q = Magnification factor

* Please note in these formulas, I am writing 6.28 as a rounded off version of two times pi. To get a truer number, it would be better to use (2 * pi).

References:
Foundations of Electronics, by Russell L Meade
Basic Electronics, by Bernard Grob

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RL Circuits have a combination of Resistors and Inductors as their name would suggest. What we need to do is combine the resistive elements of our circuit with the reactive ones. Due to the way inductors behave, you cannot just add the resistors and inductive reactance, but you can use the pythagorean theorem, and you can also use vector mapping and basic trigonometry functions.

Series RL

In a series circuit containing resistors and inductors, the current (I) is our reference vector.

Resistors (R) are in phase, at 0 degrees. (Adj)
Inductive reactance (Xl) is at 90 degrees. (Opp)
Impedance (Z) is somewhere in between. (Hyp)

For calculating total voltage and, we use the same vector.

Resistor voltage (Vr) is in phase, at 0 degrees. (Adj)
Inductive voltage (Vl) is at 90 degrees. (Opp)
and total voltage (Vt) is somewhere in between. (Hyp)
The phase angle (Theta) is the angle between R and Z and can be calculated by

Tan-1(Opp/Adj) or in this case Tan-1(Xl/R)

This is most common in electronic circuits. However, sometimes you might only have the impedance and the resistance, and need to find the inductive reactance.

Cos-1(Adj/Hyp) so Cos-1(R/Z)

We can use a 3/4/5 triangle to quickly show how this works. Let’s say we need to find impedance, and Xl = 4, R=3.

Tan-1(4/3) = 53.130 degrees (This is your phase angle)

√(4^2+3^2) = 5 (This is your impedance)

5 * Sin(53.130) = 4 (Checking the angle by requesting the Inductive reactance (Opp).

Parallel RL

In parallel, Total voltage (V) is our reference vector on the horizontal (0 degrees) since the voltage is the same through each branch, and,

Resistive current (Ir) is in phase (0 degrees)

Inductive current (Il) is plotted downward (-90 degrees)

Total current (It) is plotted somewhere in between.

In parallel RL, you just use pythagorean to solve for impedance.

Z = √(Xl^2+R^2)

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I’ve kept an eye on Ebay for the last few weeks, paying special attention to the Business and Industrial: Test equipment (Canada only, 100 dollars and less) and have managed to score some decent multimeters. I got a pair of Mastercraft handheld multimeters (AC/DC voltage, DC current, resistance and diode) that are compact and very nice for the price (~10) and also a Fluke 8010a Bench Multimeter for 11 bucks that works perfectly.

It’s been treated fairly decently, and was calibrated every two years by the Canadian Defense Department. Sweetness.

For calculators, there was nothing really spectacular in the used department online, so we checked out all the local stores and I picked up a Texas Instruments TI-89 Titanium. It Hooks up to the computer and I can transfer applications, equations, text files and pictures back and forth to the calculator via USB. Next I need to find the keyboard that plugs into it, so I can take notes on the road without lugging a laptop to school. Here’s a few screeners I took of the Desktop Apps, Word Processor, and a few electronics calculations.

But while I am just rambling on about stuff, I also got my old iBook 700 running OpenBSD 3.7 with X, and I have to admit, I like it alot better than OS X. I’m not trying to compare the two, but for what I need out of a laptop, FVWM and a couple of xterms just does it alot nicer than waiting for the beast that ate windows to start loading (and it runs a lot cooler too!). They didn’t do too much in the way of cooling for these little guys so the HD temperature gauge is my left wrist :P You can just feel that little HD burning away. Add to that it’s a lot more responsive, the CD drive works perfectly where It was quite stubborn before, and upgrades are free (well 50 bucks, cause I like to support the project, and my local store carries OpenBSD disks) and it beats the hell out of the upgrade path Mac had me on (buy!.. it’s obsolete… buy new!.. it’s obsolete, oh ya and your hardware won’t run the newy new. buy hardware and software! hmm.

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Inductance is a measure of magnetic flux for a given current, and it’s measurement is called the Henry (H). When we talk about inductance in electronics, we are talking about the ability to oppose a change in current flow. The lines of magnetic flux are exactly what produces this opposition. Energy is stored by an inductor in the form of a magnetic field surrounding it. Inductors are made of a conductor, coiled upon itself to produce a set amount of magnetic lines of flux. The opposition to the change in current produces a induced voltage. This can be thought of as a series-opposing voltage source. The amount of induced voltage (emf) depends on the rate of cutting the lines of flux. This is known as Faraday’s Law, and it’s formula is represented by:

Vind = d0/dT

Where:
Vind is the induced voltage (emf)
d0 is the rate of cutting the flux (in webers) and
dT is the change in time (seconds)
d is the symbol Delta, which represnts “the change in”

The four important physical elements that affect the amount of inductance are the coil length (number of turns per unit length), the cross sectional area of the coil, the amount of current through the coil, and the type of core material. Inductors can be made with an iron-core, like the ones commonly used for power supplies and audio circuitry, powdered iron-core, ferrite-core, or even just air-core. Each metal (or lack of, in the case of air-core)has a relative permeability and has some effect on the final inductance. iron-core inductors have typical ranges measured in Henry’s, powdered iron-core are usually found measured in milliHenrys (mH) and air-core inductors are usually found in the microHenry (uH) range.

The amount of induced voltage, also known as back-emf or counter-emf is related to the amount of inductance, and the rate of current change. This is more commonly known as self-inductance. The symbol for inductance is L. When we are talking about this oppostion to change, the formula to use is:

Vl = L * di / dt

Vl is our induced voltage in Volts
L is our inductance in Henry’s
di is the change in current, in amperes and
dt is the change in time, in seconds.

When using inductors in a circuit, you can apply many of the concepts you use with resistances in series and parallel. Total inductance in a series circuit is the sum of all inductances in it:

Lt = L1 + L2 + L3 … etc.

When finding the total inductance for a parallel circuit, it is also treted exactly the same as a resistance in parallel. you can either use the product over the sum, or the reciprocal method.

Lt = L1 * L2 / L1 + L2

or as with resistances, with more than two inductances in parallel:

1 / Lt = 1 / L1 + 1 / L2 + 1 / L3 … etc.

These formulas assume of course, that the inductors are non-coupled. This brings up the topic of mutual inductance. To quickly define non-coupled inductors, it refers to the magnetic fields (flux lines) of each inductor acting seperately, and not affecting each other.

Mutual Inductance is when you have two coils located near each other, and it causes the magnetic field of one to interact with the field of the other coil. This is how transformers work. This is usually the only place you want mutual inductance. In the case of seies or parallel inductors, any mutual inductance is undesirable.

References:
Foundations of Electronics, by Russell L Meade

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