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:

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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:

• The factor $\cos \phi$ is called the power factor.

• $I_{m} \sin \phi$ is called the wattless current.

• Impedance $Z$ is defined as $Z = \frac{V_{m}}{I_{m}} = \frac{V_{rms}}{I_{rms}}$.

• $\omega L$ is called the inductive reactance and is denoted by $X_{L}$.

• $\frac{1}{\omega C}$ 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 $