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