Adding Alternating Waveforms Calculator
MathCrave tool simplifies the process of adding alternating waveforms, a common task in AC circuit analysis. Input the amplitude and phase angle of two or more sine waves to find the single equivalent waveform, complete with detailed steps.
Waveform Addition Solver
Input your waveforms' parameters to find the resultant amplitude and phase angle. The solver handles both rectangular and polar forms.
Waveform Addition Methods Explained
Learn the theory behind combining waveforms, including phasor diagrams, the use of trigonometric identities, and converting between rectangular and polar coordinates.
Adding Alternating Waveforms (Same Frequency)
Add two sinusoidal waveforms of the same angular frequency:
$y_1(t) = A_1 \sin(\omega t + \alpha_1)$
$y_2(t) = A_2 \sin(\omega t + \alpha_2)$
$y_{sum}(t) = y_1(t) + y_2(t) = A_{sum} \sin(\omega t + \alpha_{sum})$
Wave 1 Parameters:
Phase shift is $\alpha_1 = (\text{slider value}) \times \pi$ radians.
Wave 2 Parameters:
Phase shift is $\alpha_2 = (\text{slider value}) \times \pi$ radians.
Common Angular Frequency:
Resulting waveform properties will appear here...
Trigonometric Problem Solver
Find Angles (0° to 360°)
Analyze Waveform Equation
Frequency & Period Conversion
Basic Trigonometric Waveforms
Visualizing the fundamental shapes of sine, cosine, and tangent functions plotted against an angle (in radians).
- $y = \sin(x)$: Starts at 0, peaks at 1, crosses 0, troughs at -1, returns to 0 over a period of $2\pi$.
- $y = \cos(x)$: Starts at 1, crosses 0, troughs at -1, crosses 0, returns to 1 over a period of $2\pi$. It leads the sine wave by $\pi/2$.
- $y = \tan(x)$: Represents $\sin(x)/\cos(x)$. It has a period of $\pi$ and vertical asymptotes where $\cos(x) = 0$.
Note: Tangent asymptotes occur at $x = \frac{\pi}{2} + n\pi$. The plot shows values approaching $\pm\infty$.
Wave Properties
- Amplitude (A): Peak deviation from the center line (for sine/cosine). $A = \frac{Max - Min}{2}$.
- Period (T): The length (e.g., time or angle) of one complete cycle.
- Frequency (f): Number of cycles per unit time/angle. $f = 1/T$. Unit: Hertz (Hz) if time is in seconds.
- Angular Frequency (ω): Rate of change of phase in radians per unit time/angle. $\omega = 2\pi f = 2\pi/T$. Unit: rad/s or rad/unit.
- Phase Shift (α or C): Horizontal shift relative to a reference wave. Measured in radians or degrees.
General Sinusoidal Form: $y(t) = A \sin(\omega t + \alpha)$
This form allows us to describe modified sine waves:
- $A$: Amplitude (controls height).
- $\omega$: Angular Frequency (controls period/frequency - how compressed the wave is).
- $\alpha$: Phase Angle (controls horizontal shift).
- $+\alpha$: Left shift (Lead) by $\alpha/\omega$.
- $-\alpha$: Right shift (Lag) by $|\alpha|/\omega$.
Relationship: Period $T = \frac{2\pi}{\omega}$, Frequency $f = \frac{\omega}{2\pi}$.
Adjust Parameters:
Phase shift is $\alpha = (\text{slider value}) \times \pi$ radians.
Calculated Properties:
Period (T) = $2\pi / \omega$ =
Frequency (f) = $\omega / 2\pi$ = Hz (if x-axis is time in s)
Horizontal Shift = $-\alpha / \omega$ = units
Harmonic Synthesis
Complex periodic waves can be constructed by adding a fundamental sine wave ($f_1$) and its harmonics (integer multiples of $f_1$, like $2f_1, 3f_1, ...$), each with its own amplitude and phase.
$y_{complex}(t) = C_1 \sin(\omega_1 t + \phi_1) + C_2 \sin(2\omega_1 t + \phi_2) + C_3 \sin(3\omega_1 t + \phi_3) + ...$
This example adds a fundamental and its 3rd harmonic.
Adjust Parameters for $y(t) = A_1 \sin(\omega t) + A_3 \sin(3\omega t + \alpha_3)$:
Phase shift is $\alpha_3 = (\text{slider value}) \times \pi$ radians.
Note:
The resulting complex wave is still periodic with the fundamental period $T = 2\pi / \omega$.
Fundamental Period (T) =
