To find the resultant waveform $V$ from the vector addition of two sinusoidal waveforms, $V_1$ and $V_2$, we will use the principle of phasor addition. Both waveforms are sinusoidal, and we can represent them as vectors in the complex plane (phasors). The general form of a sinusoidal waveform is:

$V(t) = A \sin(\omega t + \phi)$

where:

• $A$ is the amplitude,
• $\omega t$ is the angular frequency times time,
• $\phi$ is the phase shift.

Given:

• $V_1 = 5 \sin(\omega t)$
• $V_2 = 3.2 \sin(\omega t - 40^\circ)$

To perform vector addition, we first express each sinusoidal waveform in terms of its amplitude and phase shift as phasors:

1. Phasor representation of $V_1$: Since $V_1 = 5 \sin(\omega t)$, it has an amplitude of 5 and a phase of 0°. In phasor form, this is: $V_1 = 5 \angle 0^\circ$

2. Phasor representation of $V_2$: For $V_2 = 3.2 \sin(\omega t - 40^\circ)$, the amplitude is 3.2 and the phase shift is -40°. In phasor form: $V_2 = 3.2 \angle -40^\circ$

Step 1: Add the phasors

We now add the two phasors $V_1$ and $V_2$ using vector addition. To do this, we first need to express the phasors in rectangular form (real and imaginary components), then add them, and finally convert the result back to polar form to get the amplitude and phase of the resultant waveform.

Converting to rectangular form:

1. For $V_1 = 5 \angle 0^\circ$: $V_1 = 5 (\cos(0^\circ) + i \sin(0^\circ)) = 5 + 0i$

2. For $V_2 = 3.2 \angle -40^\circ$: $V_2 = 3.2 (\cos(-40^\circ) + i \sin(-40^\circ))$ Using the cosine and sine values for -40°: $V_2 \approx 3.2 (0.766 + i(-0.643)) = 2.4512 - 2.0576i$

Adding the phasors:

Now we add the rectangular components of the two phasors:

$V_{\text{total}} = (5 + 0i) + (2.4512 - 2.0576i) = 7.4512 - 2.0576i$

Step 2: Convert back to polar form

Now, we convert the resulting phasor back to polar form to determine the magnitude (amplitude) and phase.

1. Magnitude $|V|$: $|V| = \sqrt{(7.4512)^2 + (-2.0576)^2} \approx \sqrt{55.47 + 4.23} \approx \sqrt{59.7} \approx 7.73$

2. Phase $\theta$: $\theta = \tan^{-1}\left(\frac{-2.0576}{7.4512}\right) \approx \tan^{-1}(-0.2766) \approx -15.5^\circ$

Step 3: Resultant waveform

The resultant waveform $V(t)$ can now be written as:

$V(t) = 7.73 \sin(\omega t - 15.5^\circ)$

Final Answer:

The resultant waveform is:

$V = 7.73 \sin(\omega t - 15.5^\circ)$

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Would you like more details on any of the steps or have further questions? Here are a few more related concepts you could explore:

1. How to convert between polar and rectangular form for phasors.
2. The relationship between phasor addition and the principle of superposition.
3. The role of phase shifts in waveform addition.
4. How to analyze more complex waveforms with different frequencies.
5. The impact of phase differences on the overall waveform shape.

Tip: When adding sinusoidal waveforms with different phases, always convert them to phasors to simplify the vector addition process!