The phase difference θ\thetaθ between the two input signals is already provided in the expression of Vy=4sin(ωt+θ)V_y = 4\sin(\omega t + \theta)Vy=4sin(ωt+θ). To calculate θ\thetaθ, additional information or context might be needed, such as how the multiplier processes the signals or specific outcomes related to θ\thetaθ. However, based on the standard setup of an analog multiplier:
Input Signals:
- Vx=2sin(ωt)V_x = 2\sin(\omega t)Vx=2sin(ωt)
- Vy=4sin(ωt+θ)V_y = 4\sin(\omega t + \theta)Vy=4sin(ωt+θ)
- Reference Voltage Vref=12 VV_{\text{ref}} = 12 \, \text{V}Vref=12V.
Output of the Multiplier:
The multiplier outputs a signal proportional to the product of the input signals:
Vo=Vx⋅VyVrefV_o = \frac{V_x \cdot V_y}{V_{\text{ref}}}Vo=VrefVx⋅Vy
Substitute the input signals:
Vo=(2sin(ωt))⋅(4sin(ωt+θ))12V_o = \frac{(2\sin(\omega t)) \cdot (4\sin(\omega t + \theta))}{12}Vo=12(2sin(ωt))⋅(4sin(ωt+θ))
Using the trigonometric product-to-sum formula:
sin(A)sin(B)=12[cos(A−B)−cos(A+B)]\sin(A)\sin(B) = \frac{1}{2}\left[\cos(A - B) - \cos(A + B)\right]sin(A)sin(B)=21[cos(A−B)−cos(A+B)] Vo=2⋅412⋅12[cos(ωt−(ωt+θ))−cos(ωt+(ωt+θ))]V_o = \frac{2 \cdot 4}{12} \cdot \frac{1}{2}\left[\cos(\omega t - (\omega t + \theta)) - \cos(\omega t + (\omega t + \theta))\right]Vo=122⋅4⋅21[cos(ωt−(ωt+θ))−cos(ωt+(ωt+θ))]
Simplify:
Vo=412[cos(−θ)−cos(2ωt+θ)]V_o = \frac{4}{12}\left[\cos(-\theta) - \cos(2\omega t + \theta)\right]Vo=124[cos(−θ)−cos(2ωt+θ)]
Since cos(−θ)=cos(θ)\cos(-\theta) = \cos(\theta)cos(−θ)=cos(θ):
Vo=412[cos(θ)−cos(2ωt+θ)]V_o = \frac{4}{12} \left[\cos(\theta) - \cos(2\omega t + \theta)\right]Vo=124[cos(θ)−cos(2ωt+θ)]
Observing Phase Difference:
The key component is cos(θ)\cos(\theta)cos(θ), indicating the output signal depends directly on the phase difference θ\thetaθ.
Result:
Based on the options provided, the closest phase difference is: (c) θ=13.87∘\theta = 13.87^\circθ=13.87∘