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Animating AC Phasors

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(Refresh the browser once more for the equations to load properly) Animating AC Phasors In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation! Sinusoids: Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the ele...
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Transient States in RL Circuits: The Journey to Steady-State

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(Refresh the browser once more for the equations to load properly) Transient States in RL Circuits: The Journey to Steady-State  In this article, we attempt to obtain the complete solution of an RL circuit, including the transient state. We first consider the case of a DC voltage, where the current gradually builds up and after that, we take up the special case of AC Voltage which is often skipped in elementary courses. Fig: Growth of current in cases of DC and AC voltages DC Voltage: A DC Voltage refers to a steady voltage that does not vary with time. The below diagram shows a simple DC circuit involving a resistor `R` and an inductor `L`. A cell acts as the source of a DC emf E. Fig: A simple RL circuit with DC emf From Kirchhoff's Voltage Law, we know that the magnitude of the net Voltage drop across a loop is equal to the magnitude of the net emf in that loop. The voltage drop due to the resistor is `iR` and due to the inductor is `L \frac{di}{dt}`. Thus we m...
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Derivative of Products & Quotients

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Derivative of Product and Quotient The derivatives are widely used throughout Mathematics and the Sciences, and the Identities simplifying computations are invaluable tools. In this post, our intent is to derive the multiplication and division identities of derivatives using the very first principles. Derivative Suppose there be a continuous real function ` y = f(x) ` defined over an open interval `(a, b) \subseteq \mathbb{R}`. Then the derivative of the function `f(x)` at some point `c \in (a, b)` represented by, `f'(c) or \frac{d}{dx}f(x)` at `x = c` gives the rate of change of the function at the point `c` with respect to the independent variable. This idea however is still vague and we need a precise mathematical definition.   Right-hand & Left-hand Rate: In the neighbour of `c`, let us define an interval `I = [c - h, c + h]` for some ` h > 0`. Then we may define the right-hand average rate of change of the function in the interval as,  `\frac{f(c+h) - f(c)...
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