When you have a sum of two sinusoidal voltages or currents, the resulting amplitude depends on the relative phases and magnitudes of the individual sinusoidal components.
Let's consider two sinusoidal signals:
V₁ = A₁*sin(ωt + φ₁)
V₂ = A₂*sin(ωt + φ₂)
where:
- V₁ and V₂ are the individual sinusoidal voltages or currents,
- A₁ and A₂ are the amplitudes of the respective sinusoidal components,
- ω represents the angular frequency,
- t is the independent variable representing time, and
- φ₁ and φ₂ are the phase angles or phase shifts of the sinusoidal components.
To find the amplitude of the sum of these two sinusoidal signals, you can use the principle of superposition. Simply add the two individual sinusoidal signals together:
V = V₁ + V₂
V = A₁sin(ωt + φ₁) + A₂sin(ωt + φ₂)
To simplify the expression, you can use trigonometric identities to rewrite it in a more compact form. One such identity is the sum-to-product identity:
sin(α) + sin(β) = 2*sin((α + β)/2)*cos((α - β)/2)
Applying this identity to the equation:
V = A₁sin(ωt + φ₁) + A₂sin(ωt + φ₂) = 2*sin((ωt + φ₁ + ωt + φ₂)/2)cos((ωt + φ₁ - ωt - φ₂)/2) = 2sin((2ωt + φ₁ + φ₂)/2)*cos((φ₁ - φ₂)/2)
The resulting equation consists of two terms: one representing the amplitude of the sinusoidal component with the sum of the phase angles and another representing the amplitude of the sinusoidal component with the difference of the phase angles.
Therefore, the amplitude of the sum of two sinusoidal voltages or currents is given by the formula:
Amplitude of sum = 2√(A₁A₂)*cos((φ₁ - φ₂)/2)
This formula shows that the resulting amplitude depends on the magnitudes of the individual components (A₁ and A₂), as well as the phase difference between them (φ₁ - φ₂). The cosine term can range from -1 to +1, affecting the magnitude of the resulting amplitude.