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inkyvoyd
Reactance help please! (electrical engineering)
Wikipedia and all about circuits give the following formula for reactance of an inductor: \(\large X_L=2\pi fL\). This is given in ohms. My question is about the concept of reactance.
More specifically, I am wondering about its relation to resistance and why it is imaginary.
Unlike resistance, reactance " impedes" the flow of AC current but does not consume energy. It effectively stores and returns energy during a cycle. Reactance is real but but is used as the imaginary part in the complex number formalism that is useful in AC circuit analysis. Reactance comes naturally out of the differential equation for describing AC circuits with capacitors and inductors.
How does the usage of complex numbers to represent impedance get incorporated into describing the phenomenon of impedance? I mean, when do they use j^2=-1?
Unlike DC circuits with resistive loads AC voltages or currents do not rise and fall at the same time in fact voltages or currents at different parts of the circuit do not rise and fall at the same time. In capacitors current is rising before voltage and in inductors voltage is rising before current. They are said to be out of phase. This phase difference has significant consequenses in the behavior of the circuit. In AC circuits the voltages and currents have a sinusoidal form e.g. \[V=V _{m}\sin(\omega t+\phi) \]where w=2pi*f and f is the frequency and phi is a phase shift relative to a reference. Since we need to multiply and divide currents and voltages in circuit analysis doing so with sines and cosines with different arguments can be exceedingly tedious. There is a representation of cosine and sines in and exponential form given by Eulers Theorem. \[e ^{jx}=\cos(x) + jsin(x)\] so \[\cos(x)= real(e ^{jx}) \] and \[\sin(x)= Imag(e ^{jx)}\] In actual circuits the voltages and currents are complex but after the arithmetic is finished you only use the real part of the result. With this approach the math becomes simple complex arithmetic. When the final results are obtained the complex notation for current and voltage are converted back to sinusoidal notation.The validity of this approach is verified by comparing the results to those using the fundamental differential equation to solve the circuit. This brief discussion may not be very clear until you work through some examples To show how j gets into the discussion. Take a simple circuit of a capacitor driven by a voltage source of frequency f. For a capacitor\[ I = C(dV/dt)\] where \[V=\Re e(V _{0}e ^{j2\pi ft})\] so \[I=\Re e(V _{0}e ^{j2\pi f}/(-j/2\pi fC) )\] Ohms law is valid for reactance. So \[X _{c}=-j/(2\pi fC)\] You can do the same for inductance or a combination of a R, C and L. where you find that Impedance Z can be written as\[Z = R +j(X _{L}-X _{C})\].