Understanding the Voltages
In this scenario, we have two AC voltages:
- v1 = V₀ * cos(ωt)
- v2 = 2V₀ * cos(ωt + π/2)
The second voltage, v2, is phase-shifted by π/2 radians (90 degrees) from the first voltage.
Phase Shift and Representation
- The term "cos(ωt + π/2)" can be rewritten using the sine function because cos(θ + π/2) = -sin(θ).
Thus, v2 can be expressed as:
- v2 = 2V₀ * (-sin(ωt)) = -2V₀ * sin(ωt)
Finding the Resultant Voltage
To find the resultant voltage (v_r), we need to add the two voltage functions:
- v_r = v1 + v2
Substituting the expressions:
- v_r = V₀ * cos(ωt) - 2V₀ * sin(ωt)
Using Phasor Representation
- Convert these expressions into phasors for easier addition.
- v1 corresponds to the phasor V₀ at angle 0°.
- v2 corresponds to the phasor 2V₀ at angle 90°.
Calculating the Resultant Phasor
- The resultant phasor can be calculated using vector addition:
- The real part (x-axis) = V₀ (from v1)
- The imaginary part (y-axis) = -2V₀ (from v2)
- Use the Pythagorean theorem to find the magnitude:
Resultant magnitude = √(V₀² + (-2V₀)²) = √(V₀² + 4V₀²) = √(5V₀²) = V₀√5
Final Resultant Voltage Expression
- The resultant voltage can be expressed as:
- v_r(t) = V₀√5 * cos(ωt + φ)
Where φ is the phase angle determined by the arctangent of the imaginary and real components.
This gives the complete understanding of how to combine two AC voltages with a phase difference.