Sinusoidal Waves: Lagging & Leading - Find Angle

  • Thread starter mugzieee
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In summary, the angle by which i_1 lags v_1 can be found by comparing the sin(theta) and cos(theta) graphs and thinking about the time axis.
  • #1
mugzieee
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Im having trouble comparing sinusoidal waves and their phases.
As a sample problem I was given
v_1=120cos(120*pi*t - 40deg)
and i_1=2.5cos(120*pi*t +20deg)

and I was asked to find the angle by which i_1 lags v_1.
I have no clue on how to go through with this problem, I don't even know where to start..
 
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  • #2
Given two signals S1 and S2 in phasor form:

S1 = [itex]S_1 \angle \theta_1[/itex] and S2 = [itex]S_2 \angle \theta_2[/itex]

where [itex]S_1, S_2[/itex] are the magnitudes of the signals and [itex]\theta_1, \theta_2 \in (-\pi, \pi][/itex] are the phases.

Signal S1 is said to be leading signal S2 if [itex]\theta_1 > \theta_2[/itex]; it is said to be lagging signal S2 if [itex]\theta_1 < \theta_2[/itex]. Otherwise the two signals are said to be in phase.
 
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  • #3
mugzieee said:
I was asked to find the angle by which i_1 lags v_1.
I have no clue on how to go through with this problem, I don't even know where to start..
Keeping lagging and leading straight can be a little confusing, but there is a trick that has helped me a lot. Draw the traditional sine amplitude versus theta graph with amplitude on the vertical axis and the angle theta on the horizontal. The sin(theta) graph of course goes through zero, rises to the right and oscillates along for a couple cycles, going through zero at Pi, 2Pi, etc. Now also draw cos(theta) on the same graph, and it starts at cos(0)=1 of course, and comes down and oscillates along, crossing the horizontal axis at Pi/2, 3Pi/2, etc.

Now look at the two plots, and think of the horizontal axis as a time-related axis (like when theta is a function of time). Time is increasing to the right, so the waveform that is shifted to the right is shifted to later time, which is lagging. When you take the cos(theta) plot and shift it to the right by Pi/2, you get the sin(theta) plot, right? So the sin(theta) function *lags* the cos(theta) function by Pi/2. And since the sin and cos functions have a period of 2Pi, you can also say that the sin(theta) function *leads* the cos(theta) function by 3Pi/2. Makes sense?

And finally, let's write sin(theta) as cos(theta-Pi/2). Look at the argument (theta-Pi/2) -- it is zero when theta is Pi/2. And cos(0)=1, so cos(theta-Pi/2) is a *right* shift of the cos(theta) function. Makes sense?
 
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1. What are sinusoidal waves?

Sinusoidal waves are a type of wave that follows a mathematical pattern known as a sine function. They are characterized by a smooth, repetitive oscillation between a maximum and minimum value.

2. What is the difference between lagging and leading sinusoidal waves?

Lagging and leading sinusoidal waves refer to the phase relationship between two waves. In lagging waves, one wave lags behind the other in time, while in leading waves, one wave leads the other. This can be visualized as one wave being shifted to the left or right of the other on a graph.

3. How do you find the angle of lag or lead between two sinusoidal waves?

The angle of lag or lead can be found by calculating the phase shift between the two waves. This can be done by finding the difference in the horizontal displacement (in radians) between the two waves at a specific point in time.

4. What causes lag or lead in sinusoidal waves?

Lag or lead in sinusoidal waves can be caused by a variety of factors, including differences in frequency, amplitude, or initial phase of the waves. These differences can result in a phase shift between the waves, leading to lag or lead.

5. Why is it important to understand lagging and leading sinusoidal waves?

Understanding lagging and leading sinusoidal waves is important in various fields, including physics, engineering, and electronics. It allows us to analyze and predict the behavior of waves and their interactions with each other, which is crucial in many applications such as signal processing and communication systems.

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