Phasor / Rotating Vector Representation of Sinusoidal Quantities - Interactive Simulation

R, Ω	L, mH	C, μF	ν, Hz

Plot CONTROLS: Click the legends above to hide/unhide individual lines/curves. Double click to show exclusively.
Vary the parameters ( R, L, C, ν ) in the above table and see the effects.

Animation stop/resume: -- uncheck to stop animation, check to resume

Reload the page to reset-restart!!

The animated plot named "Phasors" on the left shows five vectors (phasors, refer description given below) - each rotating at an angular speed of ω ;

1. V representing v ( t )
2. I representing i ( t )
3. V _R representing v _R ( t )
4. V _L representing v _L ( t )
5. V _C representing v _C ( t )

It can be seen that, among the above five, the second and the third phasors are co-linear and in same direction. Also, the fourth and the fifth phasors are co-linear and in opposite directions. The end-points of these five rotating vectors are projected horizontally on to the five respective sinusoidal waves on the animated plot on the right titled 'Sinusoids: v ( t ), i ( t ) etc.'. In the animation, the constant parameter is the input voltage phasor:

V = 100 + j 200 Volts     [0]

R, L, C and ν can be varied individually via the input table.

Phasor representation of sinusoidally varying quantities is commonly used in Science and Engineering. For example, voltages and currents in ac (alternating current) circuits, forces and displacement mechanical vibration systems, and, several kinds of quantities in automatic control systems.

The above interactive simulation / visualization shows the phasor representation of four voltage sinusoids and one current sinusoid involved in the steady state analysis of a series RLC circuit excited by an ac supply v ( t ).

The applied voltage, v ( t ) at a time t , is,

v ( t ) = V sin ( ω t + φ )    [1]

Here, ω (rad/s) is its angular frequency and φ (rad) is its initial ( t = 0 ) phase angle. And, the frequency ν is,

ν = ω / 2 π     [2]

Using Euler's formula , the v ( t ) can also be represented as,

v ( t ) = Im{ V e ^{j ( ω t + φ )} }    [3]

with j being the imaginary number √ -1 . For brevity, the notation Im{.} is dropped and,

v ( t ) = V e ^{j ( ω t + φ )}     [4]

Further, separating the time-invariant and time-dependent parts,

v ( t ) = V e ^jωt     [5]

Where,

V = V e ^jφ     [6]

Applying Kirchhoff's voltage law to the above shown series RLC circuit,

v ( t ) = v _R ( t ) + v _L ( t ) + v _C ( t )    [7]

Assuming i ( t ) to be the current in the (series) circuit, such that,

i ( t ) = I e ^{j ( ω t + φ _i )}     [8]

or,

i ( t ) = I e ^jωt     [9]

Where,

I = I e ^{jφ _i}     [10]

As per the definitions of R-L-C elements,

v _R ( t ) = iR     [11]

v _L ( t ) = j ( iX _L ), with inductive reactance X _L = ωL     [12]

v _C ( t ) = - j ( iX _C ), with capacitive reactance X _C = 1/ ωC     [13]

After some algebraic manipulations, one gets,

I = V / Z    [14]

in which, the total impedance of the series RLC circuit is,

Z = R + ( X _L - X _C )    [15]

Credits:
This article uses Plotly.js - "A free open source interactive JavaScript graphing library".

Transverse Vibration of Strings - Interactive simulation of the Wave Equation

Wave #	v	ν	λ	k	ω	Y	T	φ
1
2

Transverse vibration of strings is modeled by the following relation [1, 2, 3]:
(Please reload the page if equations do not display properly)
\begin{equation} y(t, x) = Y \sin (\omega t - k x + \phi) \end{equation}
Here, \(y\) is the transverse displacement at time \(t\) at a location identified by the coordinate \(x\).
\(\omega\) is the angular frequency of vibration of the string.
\(k\) is the propagation constant,
\begin{equation} k = \frac{2\pi}{\lambda} \end{equation}
\(\lambda\) is the wavelength.
Other important relations are:
Frequency \(\nu\),
\begin{equation} \nu = \frac{\omega}{2\pi} \end{equation}
Time period \(T\),
\begin{equation} T = \frac{1}{\nu} \end{equation}
Wave velocity \(v\),
\begin{equation} v = \nu \lambda \end{equation}
\(Y\) is the amplitude of vibration.
\(\phi\) is the initial (\(t = 0\)) phase angle at (\(x = 0\)).

The simulation shows 2 different waves - pre-initialised with all parameters having same values, except for \(Y\) and \(\phi\).
Change the values and see the effects.
Noticeably, certain values are coupled.

References

[1]   http://www.pstcc.edu/nbs/WebPhysics/Chapter016.htm

[2]   http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/wavsol.html

[3]   http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/AnalyzingWaves.htm