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 $$I_{rms}=\sqrt{\frac{1}{2}}I_m$$