Lcr Circuit Graphical Solution Alternating Currents

AC Circuits and LCR Circuits

Alternating Current (AC)

• Time-varying sinusoidal voltage and current.
• Represented by $$v(t) = V_m \sin (\omega t +\theta)$$

Capacitance (C)

• Measure of ability to store charge in an electric field.
• Unit: Farad (F)
• Formula: $$C = \frac{Q}{V}$$

Inductance (L)

• Measure of ability to store energy in a magnetic field.
• Unit: Henry (H)
• Formula: $$L = \frac{\Phi_{mag}}{I}$$

Resistance (R)

• Measure of opposition to current flow.
• Unit: Ohm (Ω)
• Formula: $$R = \frac{V}{I}$$

Phasors

• Complex numbers representing AC voltage and current with magnitude and phase angle.
• For AC voltage/current $$V(t) = V_mcos(\omega t +\theta)$$ $$\underline{V}_m = V_m \angle \theta $$
• Where (V_m) is the phasor amplitude and (\theta) is the phase angle.

Resonance in LCR Circuits

Resonance

• Condition when circuit impedance is minimum and current is maximum.
• Occurs when $$(\omega L)(\frac{1}{\omega C}) = 1$$ $$\Rightarrow \omega = \sqrt{\frac{1}{LC}}=\omega_0 $$ where(\omega_0) is the resonance frequency

Resonance Frequency (ω0)

• Frequency at which resonance occurs.
• Calculated as $$f_0 = \frac{1}{2\pi \sqrt{L C}}$$

Quality Factor (Q)

• Measure of sharpness of resonance.
• $$Q = \frac{\omega_0L}{R}$$

Power in AC Circuits

Average Power

• Time-averaged power delivered by AC source.
• Formula: $$P_{avg} = VI \cos \phi$$

Peak Power

• Maximum instantaneous power delivered by AC source.
• Formula: $$P_{peak} = VI$$

Power Factor (PF)

• Ratio of average power to apparent power.
• Formula: $$PF = \cos \phi = \frac{P_{avg}}{S}$$

Apparent Power (S)

• Product of voltage and current in AC circuits (VA).
• Formula: $$S = VI$$

Reactive Power (Q)

• Power stored and returned to source by reactive elements (VARS).
• Formula: $$Q = VI (\sin \phi )=VS_C = VI_C X_C =VI_LX_L$$

AC Circuit Analysis

Impedance (Z)

• Vector sum of resistance, inductive, and capacitive impedances.
• Formula: $$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

Ohm’s Law for AC Circuits

• Relationship between current, voltage, and impedance in AC circuits.
• Formula: $$I = \frac{V}{Z}$$

Phasor Diagrams

• Graphical representation of AC voltages, currents, and impedances.
• Provides phasor relationships and allows easy calculation of circuit parameters.

RMS (Root Mean Square) Values:

• Effective values of AC voltage and current. Calculated values that are analogous to DC counterparts $$\Rightarrow $$ $$V_{rms}=\sqrt{\frac{1}{2}}V_m$$ and $$I_{rms}=\sqrt{\frac{1}{2}}I_m$$