# How do inductance and reactance differ?

## AC inductor circuits

### AC inductor circuits

#### Chapter 3 - Reactance and Impedance - Inductive

### Resistors to inductors

Inductors do not behave in the same way as resistors. While resistors simply oppose the flow of electrons through them (by dropping a voltage that is directly proportional to the current), inductors resist changes in the current through them by dropping a voltage in direct proportion to the rate of change in the current. In accordance with Lenz's law (which you can read more about here), this induced voltage always has such a polarity that an attempt is made to keep the current at its present value. That is, when the current increases in size, the induced voltage will "push" against the flow of electrons; When the current decreases, the polarity is reversed and the electron current "pushed" to counteract the decrease. This opposition to the current change is known as reactance, not resistance.

In mathematical terms, the relationship between the voltage dropped across the inductor and the rate of current change through the inductor is as such:

### Alternating current in a simple inductive circuit

The term di / dt is one from the calculus, which means the rate of change of the instantaneous current (i) over time in amperes per second. The inductance (L) is in henries and the instantaneous voltage (e) is of course in volts. Sometimes you can find the instantaneous voltage rate expressed as "v" instead of "e" (v = L di / dt), but it means exactly the same thing. To show what happens to alternating current, let's analyze a simple inductance circuit: (Figure below)

Purely inductive circuit: The inductor current is 90 ° behind the inductor voltage.If we were to draw the current and voltage for this very simple circuit it would look something like this: (image below)

Pure inductive circuit, waveforms.Remember that the voltage that drops across an inductor is a response to the change in the current flowing through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current has a peak (zero change or level rise on the current sine wave), and the instantaneous voltage is at a point where the instantaneous current is at its maximum change (the points of steepest slope on the current wave where it crosses the zero line). This leads to a tension wave that rises around 90 ^{°} is out of phase with the current wave. Looking at the graph, the stress wave appears to have a "lead" on the current wave; the voltage "carries" the current, and the current "hangs" behind the voltage. (Picture below)

Things get even more interesting when we calculate the power for this circuit: (Figure below)

In a purely inductive circuit, the instantaneous power can be positive or negativeSince the instantaneous power is the product of the instantaneous voltage and the instantaneous current (p = ie), the power is equal to zero when the instantaneous current or voltage is zero. If the instantaneous current and voltage are both positive (across the line) the power is positive. As with the resistor example, the performance is positive even if the instantaneous current and voltage are both negative (under the line). However, because the current and voltage waves are 90 degrees out of phase, there are times when one is positive while the other is negative, resulting in equally frequent negative instantaneous powers.

### What is the negative power "Inductive Reactance">

The resistance of an inductor to a change in current leads to a contrast to alternating current in general, which by definition always changes in the instantaneous magnitude and direction. This resistance to alternating current is similar to resistance, but differs in that it always leads to a phase shift between current and voltage and consumes zero power. Because of the differences, it has a different name: reactance. Reactance to AC is expressed in ohms, just like resistance, except that its mathematical symbol is X instead of R. To be precise, the reactance associated with an inductor is usually symbolized by the capital letter X with a letter L as an index that: X _{L.}

Because inductors drop voltage in proportion to the rate of change in current, they will drop more voltage for faster changing currents and less voltage for slower changing currents. This means that the reactance in ohms for each inductor is directly proportional to the frequency of the alternating current. The exact formula for determining reactance is as follows:

When we subject a 10 mH inductor to the frequencies of 60, 120 and 2500 Hz, the reactances are shown in the table below.

Reactance of a 10 mH inductor:Frequency (Hertz) | Reactance (ohms) |
---|---|

60 | 3, 7699 |

120 | 7.5398 |

2500 | 157.0796 |

In the reactance equation, the term "2πf" (everything on the right except for the L) has a special meaning. It is the number of radians per second that the alternating current "rotates" if you imagine that one cycle of AC represents the rotation of an entire circle. A radian is a unit of angular measurement: there are 2π radians in a full circle, just like 360 ^{O} in a full circle. If the alternator generating the alternating current is a two-pole unit, it will generate one cycle for every full revolution of shaft rotation, which is every 2π. Radians or 360 ^{° is} . When this constant of 2π is multiplied by the frequency in Hertz (cycles per second), the result will be a number in radians per second known as the angular velocity of the AC system.

### Angular velocity in AC systems

Angular velocity can be represented by the expression 2πf, or it can be represented by its own symbol, the lower case Greek letter omega, which appears similar to our roman lower case "w": ω. Thus, the reactance formula could be X _{L.} = 2? fL also as X _{L.} =? _{L.} to be written.

It must be understood that this "angular velocity" is an expression of how fast the AC waveforms are cyclic, with one full cycle equal to 2π. Is radian. It is not necessarily representative of the actual shaft speed of the alternator. If the alternator has more than two poles, the angular velocity is a multiple of the shaft velocity. For this reason, & ohgr; sometimes expressed in units of radians per second instead of (normal) radians per second to distinguish it from mechanical motion.

However we express the angular velocity of the system, it is obvious that it is directly proportional to the reactance in an inductor. As the frequency (or alternator shaft speed) is increased in an AC system, an inductor will offer more resistance to the passage of current, and vice versa. Alternating current in a simple inductive circuit is equal to the voltage (in volts) divided by the inductive reactance (in ohms), just as either alternating current or direct current in a simple resistor circuit is equal to the voltage (in volts) divided by the resistance (in ohms) . An example circuit is shown here: (Figure below)

Inductive reactance### Phase angle

However, we must note that the voltage and current are not in phase here. As shown before, the voltage has a phase shift of +90 ° compared to the current. (Figure below) If we mathematically represent these phase angles of voltage and current in the form of complex numbers, we find that the resistance of an inductor to current also has a phase angle:

The current in an inductor is around 90^{°}lagging behind.

Mathematically, we say that the phase angle of an inductor with respect to current is 90 °, which means that the resistance of an inductor to current is a positive imaginary quantity. This phase angle of the reactive opposition to the current is of crucial importance for circuit analysis, especially for complex alternating current circuits in which reactance and resistance interact. It will prove beneficial to represent a component's resistance to current in terms of complex numbers rather than scalar quantities of resistance and reactance.

**REVIEW:**- • Inductive reactance is the opposite of an inductance to the alternating current due to its phase-shifted storage and release of energy in its magnetic field. The reactance is symbolized by the capital letter "X" and, like the resistance (R), is measured in ohms.
- • Inductive reactance can be calculated using the following formula: X
_{L.}= 2πfL - • The angular velocity of an AC circuit is another way of expressing its frequency in units of radians per second instead of cycles per second. It is symbolized by the lower case Greek letter "Omega" or ω.
- • Inductive reactance increases with increasing frequency. In other words, the higher the frequency, the more opposed to the alternating current flow of electrons.

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