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Q&A

What reactance actually is?

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I saw that equation of reactance is written like this $$I=\frac{V}{|R+X_L j|}$$ $$I=\frac{V}{|R-X_c j|}$$ When there's capacitor and inductor in a single circuit then it is written like : $$I=\frac{V}{Z}$$

Where $Z$ represents impedance. But here we used different equation for the same "term (reactance)", they are actually called capacitive reactance and inductive reactance. Do we use the "reactance" term just to represent obstacle of electrical flow just like resistance? If not, than what is reactance?

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Reactance is the lossless part of impedance that stores and releases energy E=1/2 LI^2 or 1/2CV^2 . Much like a spring stores energy when compressed or expanded.

L is a conductor that stores current with charges flowing. Q=LI.

C is an insulator with some dielectric constant , Dk, relative to air measured with stationary charges Q=CV.

Losses are maybe modelled in series as DCR in L, or ESR in C when current flows with a source and/or load. Without this there may be a shunt leakage loss in C. For L as a conductor, such parallel R might be the work done by any moving solenoid or motor as P=V^2\R such that with no load or friction it is inductive mainly and only if you exclude DCR.

All conductors have inductance per unit length and increases when coiled and amplified when near high permeability material. Since conductors have insulation around them , they also have capacitance to the nearest conductor. Reactive or "characteristic" impedance is known here as Zo=sqrt(L/C) . This applies equally to lumped parts and "all "Transmission Lines", coax,CAT5, UTP etc...

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Your third equation defines impedance. Rearranged to solve for Z, it is:

Z = V / I

Note that this is exactly Ohms law when V and I are real numbers. In the general case of impedance, all three values can be complex numbers. Put another way, impedance is more generalized resistance, which can be complex too. Resistance is always real. Reactance is usually the imaginary part of impedance, with resistance being the real part.

Your top two equations give the magnitude (not the whole complex value) of current for a resistance in series with an inductance (first equation), and a resistance in series with a capacitance (second equation). In both cases, the denominator is the complex impedance.

Impedance can be used just like resistance, except that values of voltage and current in the equation can be complex too. This can be a very handy way of performing computations. The complex numbers take into account the phase angles automatically. Alternatively, you can use real numbers, but track the phase angles yourself separately.

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"React" implies a response to an "event".

• For an inductor, the "event" is a change in current, $\frac{di}{dt}$
• For a capacitor, the "event" is a change in voltage, $\frac{dv}{dt}$



So, an inductor or capacitor will produce a proportional reaction (or response) to something changing. If we say that a capacitor responds to a change in voltage, we then have to define that response and, the only thing it can do is alter its current. Therefore we can say that the transfer function of a capacitor is: its response current ($i$) divided by its input ($\frac{dv}{dt}$) or this: -

$$\dfrac{i}{\frac{dv}{dt}} = \dfrac{\text{A current is produced}}{\text{For a change in voltage}}$$

And we call that transfer function, (unsurprisingly) capacitance. Rearranging we get this: -

$$C = \dfrac{i}{\frac{dv}{dt}}\hspace{1cm} \text{ or}\hspace{1cm} i = C\cdot\dfrac{dv}{dt}\hspace{1cm} \text{ the standard capacitance formula}$$

For an inductor, the reactive output is voltage and, if we went through the same procedure as we did for a capacitor, we would find that: -

$$v = L\cdot\dfrac{di}{dt}\hspace{1cm} \text{ the standard inductance formula}$$

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Here is another "functional explanation" of this phenomenon (see also my RG question).

General feature. Unlike resistors that directly decrease current, both capacitors and inductors do it by subtracting their voltage from the source voltage creating this current. You can think of them as varying through time "rechargeable batteries" that oppose their voltage to the initial voltage. For this purpose, these "batteries" are oppositely connected in series to the voltage source. Figuratively speaking, they behave as "ungrateful elements" which "steal" voltage from the source and then use this voltage to resist the source:)

The difference between the two elements is that the capacitor creates an increasing through time voltage opposition (reactance) while the inductor creates a decreasing through time voltage opposition.

The capacitor is "lazy" - so when the input voltage "jumps", the capacitor does not react in the first moment and the current is unaffected. Then it begins gradually increasing its voltage… the current decreases… and, after a long time, the capacitor becomes a voltage source "producing" an equivalent "anti voltage" contrary to the input voltage change... like the op-amp in the circuit of a voltage follower...

The inductor, in contrast, immediately converts the voltage change to an equivalent "anti voltage" and applies it contrary to the input voltage change. Figuratively speaking, the voltage "produced" by the inductor "jumps" with a magnitude equal to the input voltage change. Thus we have two voltage sources ("original" and "cloned") in series that neutralize each other; as a result, the total (effective) voltage and accordingly the current do not change. Over time this opposition decreases and finally, the voltage across the inductor becomes zero again (it behaves as a piece of wire), and the current increases to the new value.

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