Ac Principles, Components And Phasors

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tags:

• Tutoring
• Physics

• Electronics

AC generation

AC Current is generated by electromagnetic induction. A rotating magnet converts mechanical energy to electrical current. As the magnet is a dipole, the polarity is inverted every half cycle. This means that the current, and therefore the voltage swing between a positive and negative maximum.

This rotation is inherent to the nature of the mathematics behind AC currents.

Everything is a sin wave. When we drive a resistive load with AC, the current and voltage oscillate with a frequency equal to the driving voltage.

Current through Components

Resistors:

\(V = IR\) \(V_0sin(\omega t) = iR\) \(i = \frac{V_0}{R} sin(wt)\)

So they are in phase

Cap:

\[V_c = V_s\] \[q/C = V_0 sin(\omega t)\] \[i = dq/dt\] \[q = C V_0 sin(\omega t)\]

\(dq/dt = i = d/dt(C V_0 sin(wt)) = \omega C V_0cos(\omega t)\) from trigonometry:

\[cos(x) = sin( \frac{\pi}{2} - x)\]

and \(sin(\frac{\pi}{2} - x) = sin(x + \frac{\pi}{2})\) \(\therefore i = \omega C V_0 sin(\omega t + \frac{\pi}{2})\) and current leads voltage

Inductor:

Inductors generate emf (voltage) based on current flow through the coil

\[V_L = L\frac{di}{dt}\]

In a pure inductive circuit

\[L\frac{di}{dt} = V_0sin(wt)\]

If we want current, we rearrange and integrate:

\[\int di = \int \frac{V_0}{L}sin(\omega t) dt\] \[i = \frac{V_0}{\omega L}(-cos(\omega t)) + C\]

what is the constant then? set \(V_0 = 0\) which makes:

\[i = C\]

but without voltage driving charge, there is no current:

\[\therefore C=0\]

as \(-cos(x) = -sin( \frac{\pi}{2} - x)\) and

\(sin(-x) = -sin(x)\) \(-cos(\omega t) = -sin(\frac{\pi}{2} - \omega t) = sin(\omega t - \frac{\pi}{2})\)

therefore current lags voltage.

Resistance, Reactance and Impedance

Resistance is the classic opposition to the flow of charge.

Reactance is a opposition to charge flow, but it is not resistance, it is caused specifically by a changing changing electric field.

Inductive reactance: \(X_L = \omega t [\ohm]\) Capacitive reactance:

\[X_C = \frac{1}{\omega C} [\ohm]\]

$\omega$ is a angular frequency, and is equal to

\[\omega = 2\pi f\]

Impedance: all these guys together

Phasor representations

A phasor is a convenient way to represent the components in these circuits

The orientations of these vectors are determined by the nature of the current-voltage relationships for the components. The magnitudes are found from the resistance/reactance of the circuits.

Like any vectors, we can find the magnitude, which is the Impedence of the circuit. This is found from the parallelogram rule.

\[Z = \sqrt{R^2 + (X_L-X_C)^2}\]