Lesson 11

AC Circuits Basics

• AC Circuits: Involves elements that provide alternating voltage.
• Key Parameters:
  • Maximum output voltage: $\Delta v_{max}$
  • Angular frequency: $\omega$
  • Frequency: $f$
  • Period: $T$

Phasor Diagrams

• Phasor: A vector representing the maximum value of a variable (e.g., voltage $\Delta V<em>{max}$, current $I</em>{max}$).

  • Rotates counterclockwise at angular frequency $\omega$.
  • Projection on the vertical axis gives instantaneous values.

• Phase Relationships:

  • Current ($I$) and voltage ($V$) in pure resistive circuits are in phase (same direction).
  • Phasor diagram helps visualize phase relationships.

AC Circuit Behaviour

• Resistors:
  • Average current over one cycle: 0.
  • Directionality of current does not affect the resistance behaviour.
  • Collisions with resistor atoms increase temperature, regardless of electron direction.

RMS Current

• RMS (Root Mean Square) Current:
  • Calculation: $i<em>{rms} = \sqrt{ \frac{1}{T} \int</em>0^T i^2(t) dt}$
  • For AC current, varies as $i^2 \sim \sin^2(\omega t)$.
  • Average current formula: $P<em>{avg} = I</em>{rms}^2 R$.

RMS Voltage

• Use similar RMS approach as for current.
• Example Calculation for AC source voltage:
  • Given $\Delta v = 200 \sin(\omega t)$ and connected to a resistor.

Reactance in AC Circuits

• Inductors:
  • Current $i<em>L$ and voltage $\Delta v</em>L$ are out of phase by $\frac{\pi}{2}$ radians (voltage leads current).
  • Current lags voltage by a quarter cycle.
• Inductive Reactance: $X_L = \omega L$
• Capacitors:
  • Current $i<em>C$ leads voltage $\Delta v</em>C$ by $\frac{\pi}{2}$ (current leads voltage).
• Capacitive Reactance: $X_C = \frac{1}{\omega C}$

RLC Circuits

• Components: Combination of resistors, inductors, and capacitors in AC.
• Impedance (Z):
  • Similar to resistance in DC circuits.
  • Depends on both $X<em>L$ and $X</em>C$.
  • Total impedance: $Z = \sqrt{R^2 + (X<em>L - X</em>C)^2}$.
• Phase Angle:
  • Value varies based on circuit components and their configuration.

Example Problems

• Example 1: Lightbulb brightness at varying frequencies.
• Example 2: Calculate inductive reactance and rms current in a purely inductive circuit.
• Example 3: Effect of increasing the frequency on rms current.
• Example 4: Capacitive reactance and rms current in circuit with capacitor under AC source.