Alternating Current
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General Learning Resources
AC and DC Current:
A current that changes its direction periodically is called alternating current (AC). If a current maintains its direction constant, it is called direct current (DC).
Root Mean Square Value:
Root Mean Square Value of a function, from $t_{1}$ to $t_{2}$, is defined as
$f_{\text{rms}} = \sqrt{\frac{\int_{t_{1}}^{t_{2}} f^{2} dt}{t_{2} - t_{1}}}.$
Power Consumed or Supplied in an AC Circuit:
Average power consumed in a cycle:
$\langle P \rangle = \frac{\int_{0}^{\frac{2 \pi}{\omega}} P dt}{\frac{2 \pi}{\omega}} = \frac{1}{2} V_{m} I_{m} \cos\phi$
$\langle P \rangle = \frac{V_{m}}{\sqrt{2}} \cdot \frac{I_{m}}{\sqrt{2}} \cdot \cos \phi = V_{rms} I_{rms} \cos \phi.$
Here, $\cos \phi$ is called the power factor.
Some Definitions:
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The factor $\cos \phi$ is called the power factor.
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$I_{m} \sin \phi$ is called the wattless current.
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Impedance $Z$ is defined as $Z = \frac{V_{m}}{I_{m}} = \frac{V_{rms}}{I_{rms}}$.
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$\omega L$ is called the inductive reactance and is denoted by $X_{L}$.
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$\frac{1}{\omega C}$ is called the capacitive reactance and is denoted by $X_{C}$.
Purely Resistive Circuit:
$I = \frac{v_{s}}{R} = \frac{V_{m} \sin \omega t}{R} = I_{m} \sin \omega t$
$I_{m} = \frac{V_{m}}{R}$
$I_{rms} = \frac{V_{rms}}{R}$
$\langle P \rangle = V_{rms} I_{rms} \cos \phi = \frac{V_{rms}^{2}}{R}$
Purely Capacitive Circuit:
$I = \frac{V_{m}}{1 / \omega C} \cos \omega t = \frac{V_{m}}{X_{C}} \cos \omega t = I_{m} \cos \omega t.$
$X_{C} = \frac{1}{\omega C} \quad \text{and is called capacitive reactance.}$
$I_{C}$ leads $v_{C}$ by $\pi / 2$.
Diagrammatically (phasor diagram), it is represented as shown below.
Since $\phi = 90^{\circ}$, $\langle P \rangle = V_{rms} I_{rms} \cos \phi = 0.$
Resonant Frequency:
$\omega_0 = \frac{1}{\sqrt{L C}}$
Quality Factor:
$ Q = P_{stored}/P_{dissipated} = I^2 X/ I^2 R Q = X/R $
BETA
