UY1: Resistors, Inductors & Capacitors In A.C. Circuits

Resistor In An A.C. Circuit Consider a resistor with resistance R through which there is a sinusoidal current: $$i = I \cos{\omega t}$$ From Ohm’s Law, $$\begin{aligned} v_{R} &= iR \\ &= \left( IR \right) \cos{\omega t} \\ &= V_{R} \cos{\omega t} \end{aligned}$$ The current $i$ is in phase with the voltage $v_{R}$ The instantaneous power $p$ delivered to a …

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UY1: Phasors & Alternating Currents

Any device that supplies a sinusoidally varying voltage (potential difference) $$v = V \cos{\omega t}$$ , where V is the voltage amplitude, or current $$i = I \cos{\omega t}$$ , where I is the current amplitude, is an alternating current (ac) source with angular frequency: $$\begin{aligned} \omega &= 2 \pi f \\ &= \frac{2 \pi}{T} \end{aligned}$$ You can represent the …

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UY1: R-L Circuit

A circuit that includes both a resistor and an inductor, and possibly a source of e.m.f., is called an R-L circuit. Suppose both switches are open to begin with, and then at some initial time $t = 0$, we close switch $S_{1}$. Let $i$ be the current at some time $t$ after $S_{1}$ is closed and let $\frac{di}{dt}$ be its …

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UY1: L-R-C Series Circuit

The diagram above shows a L-R-C series circuit. Let’s analyze the circuit: From Kirchhoff’s loop rule: $$\begin{aligned} L\frac{di}{dt} + \frac{q}{C} &=-Ri \\ \frac{d^{2}q}{dt^{2}} + \frac{R}{L}\frac{dq}{dt} + \frac{1}{LC}q &= 0 \end{aligned}$$ The equation above resembles the equation for damped simple harmonic motion, whereby: Overdamped: $$\frac{R^{2}}{L^{2}} > \frac{4}{LC}$$ Critical damped: $$\frac{R^{2}}{L^{2}} = \frac{4}{LC}$$ Underdamped: $$\frac{R^{2}}{L^{2}} < \frac{4}{LC}$$ Solving further: $$\begin{aligned} q &= …

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UY1: Magnetic-Field Energy In Inductor

Establishing a current I in an inductor with inductance L requires an input of energy U. An increasing current i in the inductor causes an emf $\epsilon$ between its terminals, and a corresponding potential difference $V_{ab}$ between the terminals of the source, with point a at higher potential than point b. The source must be adding energy to the inductor, …

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UY1: L-C Circuit

We shall look at an important circuit – one containing an inductor and a capacitor. From Kirchhoff’s loop rule, $$\begin{aligned} -L\frac{di}{dt} &=\frac{q}{C} \\ L\frac{di}{dt} + \frac{q}{C} &= 0 \\ \frac{d^{2}q}{dt^{2}} + \frac{1}{LC}q &= 0 \end{aligned}$$ Solving the differential equation above, will give: $$\begin{aligned} q &= Q \cos{\left( \omega t + \phi \right)} \\ i &=-\omega Q \sin{\left( \omega t + …

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UY1: Self-Inductance & Inductors

Consider a coil of wire with current $i$ flowing through it that is changing at $\frac{di}{dt}$. The current $i$ in the circuit causes a magnetic field $\vec{B}$ in the coil with N turns of wire, and hence an average magnetic flux $\Phi_{B}$ through each turn of the coil. From Faraday’s Law, the induced e.m.f. is given by: $$\epsilon =-N \frac{d\Phi_{B}}{dt}$$ …

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