Which wire is best for current adjustment

Basics of electrical engineering

Danger! - Preliminary version without correction, pictures and internal links!

Preface: These are not instructions that would be suitable for self-study without supervision. It is more of a commented collection of formulas that say the most necessary and really only that.


The basics of the basics of electrical engineering include the safe handling of powers of ten and their abbreviations in order to be able to adapt the units to requirements. The "prefixes for units of measurement" are sufficiently brief and concise in the following Wikipedia article, so that reference is made to them here.

Wikipedia: Unit prefixes

Circuit analysis

Physical basics

Electric field

Magnetic field

Alternating current


(Drawing: mixed circuit, completely labeled, draw in energy transfer)

Electricity flows in the technical direction of flow outside of sources from plus to minus, but inside sources from minus to plus, otherwise it would not be electricitycircle!

If the current and voltage arrows are in the same direction, energy is drawn from the system; if they are opposite, energy must be supplied to the system.

Comparison of hydraulics: pressure = voltage, current = volume flow.


R = resistance
[R] = Ω, Ω = Ohm (Georg Simon Ohm, German physicist, 1787-1854, Ohm's law, 1821)

U = voltage
[U] = V, V = Volt (Alessandro Giuseppe Antonio Anastasio Count von Volta, 1745 - 1827, frog legs, voltaic column, 1800)

I = current
[I] = A, A = Ampère (André Marie Ampère, 1775 - 1836, conductor drawing, 1820, current direction)

(Triangular formula)

Gustav Robert Kirchhoff, German physicist, 1824 - 1887.

The sum of all tensions in a mesh is zero. In simple terms, if there is only one source in the circuit, the following applies: the sum of the consumer voltages is equal to the source voltage. Note: Consumers and sources are mere terms, not physical statements about "energy".
The mesh rule is a physical law of conservation: charge remains in the circuit.

Series connection:
In the unbranched circuit, the current is the same at every point. The partial stresses add up to the total stress.


Combination of resistors in series connection

Rges = R1 + R2 + ... + Rn

Series connections are relatively rare in energy-electronic practice. The classic can be seen once a year in the form of the Christmas tree lighting.


The sum of all currents at a node is zero (in +, out -). The currents flowing in and out are equal in sum.
The knot rule is a physical law of conservation: energy remains in the circuit.

(Drawing: How many stitches and knots are there)

Parallel connection:
In the parallel connection, the voltages of all parallel branches are the same. The partial flows add up to the total flow.


Combination of resistors in parallel connection

3 formulas for different applications / scenarios

Add reciprocal values, then calculate the reciprocal value of the sum (note on calculator operation):
Rges = 1 / (1 / R1 + 1 / R2 + 1 / R3 + ... + 1 / Rn)

for exactly two resistances:
Rges = R1 * R2 / (R1 + R2) (good for a clear calculation in your head)

for any number of exactly the same resistances:
Rges = individual resistance / number


Star-delta conversion in resistor networks

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Electrical power

P = U * I (boundary condition: no phase shift between voltage and current!)
[P] = W = V * A - in watts this is always real power,
W = Watt (James Watt, Scottish inventor, 1736 - 1819, improvement of the steam engine)

[S] = VA - apparent power always in VA,
[Q] = var - reactive power in var.
(Extra chapter on alternating current necessary)

P = I ^ 2 * R = (I * R) * I (U was "replaced" by Ohm's law)

P = U ^ 2 / R = U * (I / R) (I was "replaced" by Ohm's law)

Square connection! - That means that with 80% voltage only 80% current flows, so that results in only 0.8 * 0.8 = 64% power! Important for estimating the voltage drop on lines and for estimating the effective values ​​for alternating currents (since these are defined by the power).

So if you accept a 3% voltage drop, then that means that (0.97 * 0.97 = 0.9409) you are willing to accept around 6% performance losses on the consumer. This 6% performance is implemented on the line!


Electrical work ("energy consumption")

The electrical work is what the (ordinary) "electricity meter" counts. So it sums up the performance over time as work done.

P = W / t<==> W = P * t

W = E = Q; Work = energy = amount of heat, physically the same in each case in a different form.

Units: [W] = Nm = [E] = Ws = [Q] = J

The "hour" is not a metric but a non-decimal unit. The factor 3,600,000 lies between the watt second (Ws) and the kilowatt hour (kWh).

1 kWh = 3,600,000 Ws<==> 1 Ws = 1 / 3,600,000 kWh

With the very common megajoules (MJ) it is "only" a factor of 3.6:

1 kWh = 3.6 MJ<==> 1 MJ = 1 / 3.6 kWh


Mechanical work and torque

Mechanical work:
W = F * s with condition F || s, "F parallel s", s is a segment

M = F * l with condition F ⊥ l, "F perpendicular to l", l is the lever arm

Danger! - Mechanical work and torque have the same unit "Nm", but the numerical value is not the same because of the direction of the vectors!


Torque and power

P = 2 * Pi * M * n
[P] = W = Nm / s

W = 2 * Pi * M
[W] = Nm = [M] = Nm !!!

The term 2 * Pi turns the lever arm (radius of the circle, perpendicular to the force) into a segment (circumference of the circle, parallel to the force)!

especially for electrical machines:
P = M * n / 9549
[P] = kW = 2 * Pi * Nm * 1 / min / (60 * 1000) Important because of the "normal machine" units!
9549 = 60 * 1000 / (2 * Pi)



The dimensionless efficiency η is the ratio of output power to input power in a system. It is always less than 1.

η = Pab / Pzu
[η] = 1

Pzu - Pab = P loss

Efficiencies of energy conversion systems connected in series multiply to form the overall efficiency.

ηtotal = η1 * η2 * η3 * ... * ηn

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The conductance is the reciprocal of the resistance and vice versa. Although the resistor appears "somehow more familiar", it is an unusual quantity because it indicates "how badly" something is working. The conductance indicates how well a leader conducts.

G = 1 / R
[G] = S (Siemens)


Specific resistance "ρ"

The specific resistance is a material constant. The answer, so to speak, to the question of how great the resistance of a one meter long piece of sample material with a cross section of one mm ^ 2 is.

ρ = (R * A) / l
[ρ] = (Ohm * mm ^ 2) / m = µOhm * m

Examples (the higher the value, the worse the ladder):

Cu ==> 0.0175 (Ohm * mm ^ 2) / mAl ==> 0.0265 (Ohm * mm ^ 2) / mAg ==> 0.01587 (Ohm * mm ^ 2) / mAu ==> 0 , 02214 (Ohm * mm ^ 2) / mFe ==> 0.15 (Ohm * mm ^ 2) / m

Electrical conductivity "γ" ("specific conductance")

The electrical conductivity is the reciprocal of the specific resistance, i.e. the reciprocal of the same material constant.

γ = (G * l) / A
[γ] = (S · m) / mm ^ 2 = MS / m

There are also the symbols σ and κ, here γ is used because:
R resistance and ρ (the Greek small r) specific resistance and
G conductance and γ (the Greek small g) electrical conductivity
are easy to remember.

Examples (the greater the value, the better the ladder):

Cu ==> 58 (S m) / mm ^ 2Al ==> 37 (S m) / mm ^ 2Ag ==> 61 (S m) / mm ^ 2Au ==> 45 (S m) / mm ^ 2Fe ==> 10 (S m) / mm ^ 2

Explain unit: [κ] = MS / m = mm ^ 2 / (Ω * m)


Note on conductor materials

As you can see, silver conducts the best of all metals, the copper used in industry is already the second best conductive metal. Gold is only third in the ranking, aluminum is already fourth.

The hydrogen sulfide contained in the air corrodes silver to silver sulfide (the typical black coating), which is electrically non-conductive. Therefore, silver is a very good conductor, but a bad contact material. With gold, the opposite is true: gold does not conduct as well as copper, but it does not oxidize at all, so that contact surfaces made of gold conduct very well. Aluminum is used (because of its much lower price compared to copper) in underground cables with a large cross-section and also in overhead lines. However, it is not easy to process because it is coated with non-conductive aluminum oxide in the air (a so-called "passive layer"), which must be carefully removed before a clamping process (i.e. contacting).

In the GDR, even after the Second World War, many installations were made of aluminum. The electricians there had a small tool that looked like a sharpener and that scraped off / sanded off the oxide layer, after which the wire was protected from renewed corrosion by grease.



Ohm's law
U = R * I
This applies to every resistor, so only a partial voltage drops across a partial resistance, which is called ΔU (Delta is the large Greek D, for "difference"). The current is the same everywhere in an unbranched circuit, all partial voltages add up to "zero" (mesh rule).

This consideration neglects the fact that power distribution systems involve alternating current, which generally drives complex loads (i.e. inductive and capacitive resistances). That would require a correspondingly complex AC calculation and, above all, precise knowledge of the load.

According to DIN 18015 Part 1, the voltage drop between the meter and the sockets or device connection terminals should not be more than 3%.
This 3% is the reason for the frequently mentioned 7 V for ΔU (only valid in 230 V networks).

Because of the better numerical values, the conductance G is used here as a test

G = (γ * A) / l
[A] = mm ^ 2 and
[l] = m (this corresponds to the expectations regarding the units)

But be careful:
l is the entire length of the conductor in the circuit, so the outward and return lines must be taken into account.
l = 2 * le
le is the "simple cable length" or the "effective" cable length of a normal multi-core cable.

Here is an example that can be calculated in your head:

Example 10 m (i.e. 20 m!), 2.5 mm ^ 2 Cu, e.g. NYM 3G 2.5 mm ^ 2

Calculation of the conductance
G = (58 (S m) / mm ^ 2 * 2.5 mm ^ 2) / 20 m
G = 58 * 2.5 / 20 = 145/20 = 7.25 S

Calculation of the voltage drop
ΔU = R * I = I / G
ΔU = 20 A / 7.25 S = 2.76 V

The result is calculated with a pocket calculator, but since 21/7 = 3, it was already clear from an estimate that it is below 7 V and therefore allowed. Calculating with guide values ​​may be unusual, but makes sense in practice, as the numerical values ​​allow estimates to be made relatively quickly. Even typing it into the calculator makes the numbers less prone to typing errors.
The value 20 A was chosen here because of the maximum permissible current carrying capacity.

But the calculation is easy to vary for 16 A:
ΔU = 16 A / 7.25 S = 2.21 V

Another variation on 18 m usable (!) Cable length (because that might explain why many electricians use this value as a limit value of 18 m):

G = 145/36 = 4.03 S (i.e. 4 S)

ΔU = 16 A / 4 S = 4 V

This leaves a 3 V voltage drop on the 1.5 mm ^ 2 part of the system. The voltage drop is proportional to the cable length for a given cross-section and a given current strength.

Perhaps one could also take a practice-oriented approach in another way and still expect resistance, which one "prepares" a bit:

For one meter of cable, i.e. le = 1 m (that's l = 2 m!) 2.5 mm ^ 2 Cu cable:
ρ = (R * A) / l
R = (ρ * l) / A

R = 0.0175 * 2 / 2.5 = 0.014 ohms

So in practice one can say:
2.5 mm ^ 2 copper wire has a resistance of 0.014 Ohm / m

Again for 1.5 mm ^ 2

R = 0.0175 * 2 / 1.5 = 0.023 ohms

So: 1.5 mm ^ 2 copper wire has a resistance of 0.023 Ohm / m

Continue for 18 m (le!) 2.5 mm ^ 2:
18 m * 0.014 ohm / m = 0.252 ohm

ΔU = R * I = 0.252 ohms * 16 A = 4.032 V = 4 V.
(so this result matches the results before)

But you can also ask the question the other way around:
With a given current and cross-section, how long can a cable be until a certain ΔU is exceeded?

Let's take the remaining allowable 3V on the 1.5mm ^ 2 line:

To do this, one has to solve for R:
R = ΔU / I = 3 V / 16 A = 0.1875 ohms

le (!) = 0.1875 Ohm / 0.023 Ohm / m = 8.15 m

This means that the consumer may only be a total of 26 m (18 m + 8 m) away from the distribution.

In all the calculations it can be clearly seen that the rule "maximum 3% voltage drop" very quickly represents a much stronger restriction with regard to the permissible current carrying capacity than the heat development. Nowadays (with an FI circuit breaker!), However, the voltage drop is no longer as "safety-critical" as it used to be (without an FI circuit breaker). Only the devices don't run that well and you pay for the "consumption" on the lines (which get warm).

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simplified according to Nils Bohr


Positive protons and neutral neutrons in the nucleus

Negative electrons in the shell.
An electron carries the elementary charge e = -1.602 * 10 ^ -19 As.

Nucleus: very small, very heavy, makes up the mass of the atom.
Shell: very large, very light, forms the shape of the atom.
If the nucleus is the size of an orange, the first electron orbits it at a distance of 100 m, with empty space in between.

Periodic table of the elements provides all information:

  • Atomic number = number of protons and type of atom (i.e. name of the element)
  • Atomic mass = number of protons + number of neutrons.

The decimal places of the atomic mass can be ignored "for domestic use", they come from the statistical mean that occurs on the basis of so-called isotopes, that is, atoms with the same atomic number (i.e. the same element) but a number of neutrons that deviates from the mean. Well-known isotopes are e.g. the fissile uranium 235 (normal is uranium 238) as well as the isotopes of hydrogen deuterium (1P + 1N) and tritium (1P + 2N), which form so-called "heavy water" in connection with oxygen. Isotopes only differ physically in terms of their mass; they cannot be distinguished chemically. Isotopes can be separated by centrifugation.

Electrons in the outermost shell are called valence electrons (they determine the chemical "valence" of an element). With metals, they are freely movable and can move from one atom to the other as long as their total number in the atomic association remains "appropriate" to the protons of the nuclei. Because of this free mobility, one speaks of an "electron gas" in the case of metals, which surrounds the atoms and ensures their cohesion, namely the metal bond. Electron gas explains the ductility of metals and their conductivity for electrical current as well as their good thermal conductivity.



Electron conduction

  • all metals (all metals are PTCs, their resistance increases with increasing temperature)
  • Carbon (in the form of graphite; physically a semiconductor, but electrotechnically always used as a conductor! - Carbon is an NTC, its resistance decreases with increasing temperature)

Ionic conduction 1

  • Salts in aqueous solution or in melt
  • Acids in aqueous solution
  • Bases in aqueous solution

These substances are chemically decomposed by the conduction process (electrolysis). Ions are whole atoms that are electrically charged, i.e. have one or more electrons too many (-) or too few (+).

Positive ions are called cations because they are deposited on the cathode (negative electrode).
Negative ions are called anions because they are deposited on the anode (positive electrode).

Electrolysis also occurs when alternating current flows through the electrolytes, so we strongly advise against consumption of food heated by direct current flow (the famous sausage between the fork prongs) and drinks (two raiser blades in a glass of water)!

Ionic conduction 2

  • diluted gases (diluted = low pressure)

Non-conductors (here technically: insulators)

  • air
  • Ceramics, glass
  • Plastics: PVC (problematic due toChlorine -> flue gases contain hydrochloric acid gas and dioxin), PE / PP (relatively expensive), PS, ABS, PU (mostly orange line for heavy loads), silicone (for high temperatures)
  • "Rubber" = synthetic rubber
  • resin-impregnated hard paper ("Pertinax", brown printed circuit boards)
  • glass fiber reinforced epoxy resin (green circuit boards)
  • Paper, soaked in oil
  • Insulating oils (formerly mineral oils containing PCBs, today mostly silicone oils)
  • Insulating gas sulfur hexafluoride (SF6)

English class

silicon (Silicon Valley) = silicon (silicon valley)

silicone = silicone (long-chain silicon compound)



Semiconductors are substances that are non-conductors under normal conditions, but become conductive under certain conditions (temperature, light / radiation, B-field, E-field, current, pollution).

  • Silicon (the second most common element in the earth's crust, but expensive to obtain in its purest form) is the most frequently used semiconductor. Silicon diodes have a characteristic forward voltage of 0.7 V.
  • Germanium is a little-used semiconductor today. Germanium diodes have a characteristic forward voltage of 0.3 V.
  • Carbon conducts under normal conditions, but its conduction mechanism is that of a semiconductor. It can also be recognized by the fact that carbon, like all semiconductors, is an NTC (resistance decreases with increasing temperature).

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PTC - Positive Temperature Coefficient - resistance increases with increasing temperature

NTC - Negative Temperature Coefficient - resistance decreases with increasing temperature

Δϑ = Tnew - Talt
Temperature increase (+), temperature decrease (-)

Change in resistance:
ΔR = R * α * Δϑ
[ΔR] = Ω * 1 / K * K


Thermal expansion does not belong to electrical engineering, but the fact that it follows the same "formulaic laws", the connection is established and shown here.

α electrical and α mechanical are different material properties, but they still have the same unit, namely 1 / K!

The same applies here:
Δϑ = Tnew - Talt
Temperature increase (+), temperature decrease (-)

Linear expansion:
Δl = l * α * Δϑ
[Δl] = m * 1 / K * K

Volume expansion:
ΔV = V * γ * Δϑ
with γ = 3 * α
[ΔV] = m ^ 3 * 1 / K * K

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(Drawing capacitor and field)

Field lines begin and end on a ladder, so they have a start and end point. They only enforce non-conductors (dielectrics). The direction is that in which a positive test charge would move in the field.

Electric field strength

E = U / d
[E] = V / m

F = E * Q

E = F / Q
[E] = V / m = N / As

[Q] = As = C

Charge Q
1 As = 1 C, C = Coulomb (Charles Augustin de Coulomb, 1736 - 1806, forces of electrical charges, static friction)


C = Q / U [C] = As / V = ​​F, F = Farad (Michael Faraday, English physicist, 1791-1867, electrolysis, Faraday cage)

C = εr * ε0 * A / d
[C] = 1 * As / Vm * m ^ 2 / m,
[ε0] = As / Vm = F / m

εr = permittivity number = property of the dielectric.
ε0 = electric field constant.

A charge shift (influence) can occur in the dielectric. The stronger the influence of the dielectric, the higher the dielectric constant, the greater the capacitance of the capacitor.

Simple field courses: pair of conductors over ceiling, coaxial line; more complicated: four-rope high-voltage line

Charge and discharge of a capacitor

Circuit diagram

Charge / discharge curves of voltage and current over time. Voltage remains the same, current reverses in the direction of flow!

τ = R * C
Consider [τ] = s = Ω * As / V: [R] = Ω = V / A

1 * τ = 63% charge / discharge
5 * τ = 99% charge / discharge

Uc =

Capacitor keeps voltage constant, the current on the capacitor can jump. So: first current, then voltage, or: voltage after current.

The charge of an open capacitor is the constant quantity (does not change because it cannot flow away). If the plates are removed from one another, the voltage increases (charges are separated, pulling the plates apart is work that is found in the capacitor).

Surface charge density ("electrical displacement")

This variable is rarely important for the calculations on the capacitor; it is mentioned here because of the comparison with the magnetic field.

D = Q / A
[D] = As / m ^ 2

D = εr * ε0 * E
[D] = As / m ^ 2 = As / Vm * V / m

Influence of the dielectric (capacity calculation)

(Drawing influence)

C = εr * ε0 * A / d
[C] = As / V = ​​1 * As / Vm * m ^ 2 / m

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(Drawing coil and field)

Magnetic field lines are closed, so they have no start and end point - and therefore also no magnetic monopole. Their direction is that in which a sample north pole would move in the field.

Magnetic field strength H

H = I * n / lm
[H] = A / m
n = number of turns, lm = mean field line length

The magnetic field strength is directly proportional to the current. The electricity involved in their generation is called the great theta flow

Flooding Θ

Θ = I * n
[Θ] = A

The current through a wire is counted as often to generate the field as it is actually used. The flow is also called "magnetic tension" (so to speak, the "pressure with which the magnetic field is built up").

Force effect of the magnetic field

Force on iron at the magnetic pole

Force on conductors in a magnetic field

Magnetic flux Φ

The magnetic flux is a measure of the totality of all field lines of a magnetic field. It corresponds to the charge on a capacitor that creates the electric field.

[Φ] = Vs = Wb, Wb = Weber (Eduard Wilhelm Weber, German pysicist, 1804 - 1891)

Magnetic flux density B

If the magnetic flux is divided by the cross-sectional area of ​​the field, the result is the magnetic flux density (the corresponding size for the capacitor is the surface charge density). The magnetic flux density is "the" important variable for the strength of a magnetic field (the corresponding size of the capacitor "surface charge density", on the other hand, is rather unimportant).

B = Φ / A
[B] = Vs / m ^ 2 = T, T = Tesla (Nikola Tesla, Serbian inventor and engineer, 1856 - 1943, alternating current dynamo, motor)

B = μ0 * μr * H

μr = permeability number.
μ0 = magnetic field constant, μ0 = 1.257 * 10 ^ -6 Vs / Am = 1.257 * 10 ^ -6 H / m


L = Φ / I
[L] = Vs / A = H, H = Henry (Joseph Henry, 1797 - 1878, American physicist, self-induction, 1st chairman of the Smithsonian Institution)

L = μr * μ0 * A / lm
[L] = Vs / A = 1 * Vs / Am * m ^ 2 / m
[μ0] = Vs / Am = H / m, "Magnetic field constant"

Permeability number = property of the material penetrated by the magnetic field.

A charge shift (influence) can occur in the dielectric. The stronger the influence of the dielectric, the higher the dielectric constant, the greater the capacitance of the capacitor.

Simple field courses: pair of conductors over ceiling, coaxial line; more complicated: four-rope high-voltage line

Charge and discharge of a coil

Circuit diagram

Charge / discharge curves of current and voltage over time. Current stays the same, voltage reverses in polarity!

τ = L / R
[τ] = s = Vs / A / Ω, consider: Ω = V / A

1 * τ = 63% charge / discharge
5 * τ = 99% charge / discharge

Il =

The coil keeps the current constant, the voltage on the coil can jump. So: first voltage, then current, or: current after voltage ("inductive currents are delayed").

Electrolyte commentary
The motto:
"Inductive currents are delayed"
is certainly very useful, while the opposite sentence, so to speak: "The current rushes at the capacitor" is not exactly loved by me. It is correct, but 1. it has to be emphasized in a very idiosyncratic way and 2. I think it is mnemonic to say something about the current on the capacitor. Because "with capacitors you talk about voltage" and "with coils you talk about current" everything that applies to one thing applies to the other for the other. - So we are looking for a lyrical disguise for the sentence "Voltage lags behind on the capacitor". - Maybe:
"With capacity, voltage comes late".
The alliteration at the end isn't bad, at least.

The constant magnitude of the magnetic circuit is.

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Internal resistance of a real voltage source (e.g. mono cell)

(Equivalent circuit diagram, voltage arrow at Ri in the opposite direction from U0!)

Ukl = U0 - Ri * I

Determine the number of cells:
I * (Ri * n + Ra) = n * U0

Parallel connection of two voltage sources (battery and charger) with different internal resistance

I = (UL - UB) / (RiL + RiB)

Characteristic curves of voltage sources

U over I (attention with resistors: I over U)

The amount (unsigned) of the slope (which is negative) is Ri! Because:

Ri = ΔU / ΔI

The point of intersection with the I-axis is U0

Voltage, power and current adjustment

A distinction is made between three so-called "adjustments" of voltage or current source and consumer:

Voltage adjustment

Voltage matching is the most common type of relationship between source (Ri) and load resistance (Rl). The source comes close to an ideal voltage source (Ri = 0) because Ri << Rl (minimum condition: Rl = 10 * Ri).

The voltage is therefore almost constant and generally independent of the load. The power is implemented practically exclusively on the load and only to a very small extent on the internal resistance.

Voltage adjustment is available in energy technology and in the transmission of signals for audio signals (low frequency!)

Voltage matching close to the power matching is called over-matching.

Performance adjustment

The power adjustment makes it possible to take the maximum power from a source. Internal resistance and load are the same: Ri = Rl.

Although power adjustment ensures the maximum power transfer from the source to the load, the same power is also implemented at the Ri in the source. That means 50% of the source power is transferred to the load.

Therefore, the power adjustment is mainly used in high-frequency technology. Here the impedances of the source and load are matched in order to avoid reflections. That is the reason why antenna cables (for normal household broadcasting impedance 75 Ω) must not be connected in parallel with a simple terminal.

Current adjustment

The current adjustment is relatively rare. The source comes close to an ideal current source (Ri = infinite) because Ri >> Rl (minimum condition: Ri = 10 * Rl).

The current is therefore almost constant and generally independent of the load. Most of the power is converted into the internal resistance of the source.

Current adjustment is used when charging accumulators (NiCd, NiMH, but not with lead batteries!) And in measurement technology.

Current adjustment close to the power adjustment is called under adjustment.

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Sinus curve diagram alternating current / three-phase current

U (α) = U-roof * sin (α)

α = 360 ° / T * t

U (t) = U-roof * sin (360 ° / T * t) = U-roof * sin (360 ° * f * t)

Three-phase connection

Power in a three-phase circuit (formula)

How does the motor (in the star) have to start?

normal -> directly in front of the motor -> IOff = IMotor / root (3) heavy -> directly after the fuse, in front of the line contactor -> IOut = Imotor excessively long -> in the delta circuit before the delta contactor, motor runs without (!) protection in the star on -> Iout = IMotor / root (3)

Task in between
Two lamps in series, a resistor is connected in parallel to one lamp. What happens to lamp 1 and lamp 2?

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Practical example:
Fluorescent lamp with choke (KVG = conventional ballast)

Basic considerations:
The gas discharge in the fluorescent tube (this is a low-pressure mercury vapor lamp) is a purely ohmic resistance. However, a gas discharge has a negative differential resistance (no "negative resistance", that doesn't exist or is called a "voltage source"). This means that the resistance of the gas discharge depends on the current flowing through it and becomes smaller as the current flow increases. If the resistance R of a gas discharge is plotted against the current I, a "falling" curve results, the slope (the ratio of ΔU / ΔI) is negative!

A series resistor is therefore required to set a stable operating point for the gas discharge. - Without the series resistor, the arc that forms will destroy the tube.

Of course, this series resistor can also be a purely ohmic resistor. However, a not inconsiderable real power is implemented on this.

Current in series connection is the same, voltages add (geometrically!) To the total voltage. Pythagoras.

Calculate the total voltage and its phase position.

S, P, Q, calculate

Calculate cos φ, sin φ, tan φ

Compensation via Q, calculate current, calculate resistance

Determine XC, determine C from it

(Central) compensation in the three-phase network always in a triangle, because the necessary capacities are significantly smaller (but the necessary dielectric strength of the capacitors is greater).

QStrang = Q / 3

QStrang = UStrang ^ 2 / XC

Determine XC and C.

Compensate for certain cos (φ): QDiff / P = tan (φ)

QDiff = P * tan (cos ^ -1 (φ))

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