Question

How does the phase constant ϕ intervene in the equation of a sine wave?

Answer #1

According to the equation of Sine wave. x = A sin ( ω t + ϕ )

describes the displacement motion of a *passive* linear
harmonic oscillator without loss. In other words there is no
*input* or driving function. Whatever motion the oscillator
exhibits is solely due to its initial conditions. ϕ in this case
provides a point of reference in space for the oscillations.

But for the driven oscillator, ϕ provides a more significant role in terms of how efficiently energy is transferred from the driver to to the oscillator (system). If the driving force is in perfect phase with the system and pointing in the right direction, maximum energy is transferred at the harmonic resonant frequency. Either side of this point either leads or lags, decreasing the efficiency of energy transfer.

a)Draw a sine wave and point out the Amplitude, Peak Amplitude,
Peak-to-Peak Amplitude, and the period of the wave.
b) What is the formula for a sine wave? – the Period and the
Frequency?
c) Draw two sine waves where one is 90 deg phase shifted from
the other and label the leading and lagging wave. Why are sine
waves important in electronics?

The motion of an underdamped oscillator is described by equation
x=Ae−(b/2m)tcos(ω′t+ϕ).
Let the phase angle ϕ be zero.
Part A
According to this equation, what is the value of x at
t=0?
Express your answer in terms of the variables A,
b, m, ω, and ϕ.
x(0) =
A
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Part B
What is the magnitude of the velocity at t=0?
Express your answer in terms of the variables A,
b, m, ω, and ϕ.
vx(0) =...

For the transformer, sketch the input and output waveforms given
a sine wave input voltage. Note: the vertical axis represents
voltage and the horizontal axis represents time. Compare and
contrast the amplitude, shape, phase, and period. Repeat for a
square wave.

The wave function of a wave is given by the equation
D(x,t)=(0.2m)sin(2.0x−4.0t+π),
where x is in metres and t is in seconds.
a. What is the phase constant of the wave?
b. What is the phase of the wave at t=1.0s and x=0.5m?
c. At a given instant, what is the phase difference between two
points that are 0.5m apart?
d. At what speed does a crest of the wave move?

A sine wave trough 10 meters in amplitude is located at x = 3m at t = 4s. The wave travels to the left at 47 m / s. If the oscillation frequency of the middle points is 2.1 Hz:
a. What is the equation of the wave?
b. When will the point at x = 2m be at a position y = - 5m, the second time after t = 4s?

Consider a 10 Hz sine wave sampled at 35 Hz. If the samples are
plotted and displayed using linear interpolation, you will see that
the reconstructed waveform will not resemble the original
continuous-time sine wave. Show how sinc interpolation can be used
to reconstruct the original continuous-time signal. (Hint: In this
question you are being asked to resample the sine wave to a higher
sampling rate of, say, 105Hz, by sinc interpolation – i.e.,
convolution with a sinc function)
Please...

if sin wave starts at zero does it mean phase angle is
always zero
or When Vo=0 then phase angle is zero?
in a problem Vo was not zero and the phase angle was
zero by looking sin wave graph
I didn't understand

1) State the heat equation and hence use the relationship
between the heat and wave equation to solve
(du/dt)=(d2u/dt2) x>0,t>0
2) Define half range sine or cosine series and hence expand sinx
0<x<pie in Fourier cosine series

A single phase AC induction motor need to be controlled by
varying its input frequency. The supply frequency is a single phase
with 50 Hz and the input frequency to the motor is controlled to be
12.5 Hz. Design a Full-wave rectifier (Cycloconverter), show the
conversion of 5 cycles, and explain its operation. And show how the
output voltage of Cycloconverters can be controlled to get a
waveform shape near sine wave.

Use the Fourier sine transform to derive the solution formula
for the heat equation ut = c2 uxx
on the half-infinite bar (0 ≤ x < ∞) with Dirichlet boundary
condition u(0, t) = a, for some constant a, and initial condition
u(x, 0) = f(x).

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