Instantaneous and Average Power in Resistors

Instantaneous Power in Resistors

• Instantaneous power dissipated in a resistor is calculated using the formula ( p = i^2 R ).
• For alternating current (AC), this can be expressed as ( p = i_m^2 R \sin^2(\omega t) ).
• The average power over a complete cycle is represented as ( \langle p \rangle = \langle i_m^2 R \sin^2(\omega t) \rangle ).

Average Power Calculation

• Since ( i_m^2 ) and ( R ) are constants, the average power simplifies to ( \langle p \rangle = i_m^2 R \langle \sin^2(\omega t) \rangle ).
• The average value of (\sin^2(\omega t)) over a full cycle is ( \langle \sin^2(\omega t) \rangle = \frac{1}{2} ).
• Consequently, average power becomes ( \langle p \rangle = \frac{1}{2} i_m^2 R ).

Root Mean Square (RMS) Current

• The RMS (root mean square) current, denoted as ( I_{\text{rms}} ), is defined for AC systems to equate the power with DC systems.
• The effective current ( I_{\text{rms}} ) yields the same average power loss as direct current (DC).
• AC voltage across a resistor is given by ( v = v_m \sin(\omega t) ), where ( v_m ) is the peak voltage.

Alternating Current and Ohm's Law

• Kirchhoff’s loop rule applied to an AC circuit gives ( v_m = i_m R ).
• The relationship between AC current and voltage mirrors that of DC: ( V = IR ).
• RMS voltage is often indicated in specifications (e.g., household voltage is typically 220V).

Example Problem for a Light Bulb

• For a light bulb rated at 100W on a 220V supply:
  • Resistance can be calculated as ( R = \frac{V^2}{P} ).
  • Peak voltage can be calculated using ( v_m = \sqrt{2} \times V_{\text{rms}} ).
  • RMS current through the bulb can be calculated using ( I_{\text{rms}} = \frac{P}{V_{\text{rms}}} ).

Behavior of Inductors in AC Circuits

• Inductors exhibit self-inductance ( L ) and follow the equation ( v - L \frac{di}{dt} = 0 ).
• Voltage and current in purely inductive AC circuits relate similarly to resistive circuits, modifying through reactance.

Capacitive Reactance

• Capacitive reactance is denoted by ( X_c = \frac{1}{\omega C} ).
• Current amplitude in a capacitive circuit is found as ( i_m = \frac{v_m}{X_c} ).
• Capacitive reactance is proportional to capacitance and inversely proportional to frequency.

Phase Relationship in AC Circuits

• In capacitive circuits, the current leads the voltage by ( \frac{\pi}{2} ).
• Conversely, in inductive circuits, the current lags the voltage by ( \frac{\pi}{2} ).
• Average power through capacitors is zero since no net energy is transferred over a cycle.

Observations with Capacitors and DC Sources

• When a DC source charges a capacitor, current flow ceases once charged, resulting in a non-glowing lamp.
• Changes in capacitance do not alter the current flow in a DC circuit once the capacitor is charged.