Lecture on Phasors and Current Calculation in Circuits

Key Concepts

• Phasors: Phasors are representations of sinusoidal functions in terms of their amplitude and phase. They are used to simplify calculations involving sinusoidal functions.
• RMS Value: The RMS (Root Mean Square) value is often used instead of peak values for practical calculations.

Problem Overview

• Objective: Calculate current $I_T$ in two circuits with given phasor voltage representations.

Circuit 1: Series Resistors

• Components: Two 1 Ohm resistors connected in series.
• Voltage Given: $V_T$ in phasor form as $10 \angle 30^\circ$.
• Key Point: Phasors are typically RMS values, though sometimes confused with peak values in literature.

Calculating Current $I_T$

1. Total Resistance: $1 \Omega + 1 \Omega = 2 \Omega$.
2. Current Calculation:
   • $I_T = \frac{V_T}{2 \Omega}$.
   • Use phasor $V_T = 10 \angle 30^\circ$: $I_{phasor} = \frac{10 \angle 30^\circ}{2} = 5 \angle 30^\circ$ Amperes.
3. Convert to Time Domain:
   • $I_T = I_M \cos(\omega t + \theta)$ where $I_M$ is the amplitude.
   • Derived $I_M = 5\sqrt{2}$ Amperes, $\theta = 30^\circ$.
   • Final Answer: $I_T = 5\sqrt{2} \cos(\omega t + 30^\circ)$ Amperes.

Circuit 2: Sinusoidal Voltage

• Voltage Given: $V_T = 6 \sin(\omega t + 10^\circ)$.
• Phasor Representation: $6 \angle 10^\circ$.

Calculating Current $I_T$

1. Current Calculation:
   • $I_{phasor} = \frac{6 \angle 10^\circ}{2} = 3 \angle 10^\circ$ Amperes.
   • Note: Here $3$ is the maximum value of the current.
2. Convert to Time Domain:
   • $I_T = I_M \sin(\omega t + \theta)$.
   • Final Answer: $I_T = 3 \sin(\omega t + 10^\circ)$ Amperes.

Conclusions

• Phasor representation allows simpler calculations of currents in AC circuits.
• Distinguishing between RMS and peak values is crucial to avoid confusion in circuit calculations.
• Depending on whether sine or cosine is the parent signal, the formula for $I_T$ will vary accordingly.