Phasor Diagram Calculator: Calculate V & W

A professional tool to add or subtract two phasors (V and W) with real-time visualization on a phasor diagram.

Resultant Phasor (R)

R = 12.81 ∠ 56.99°

The formula for phasor addition/subtraction involves converting to rectangular coordinates, performing the operation, and converting back to polar form.

Intermediate Values

Resultant (Rectangular)12.81 + j10.74

Phasor V (Rectangular)8.66 + j5.00

Phasor W (Rectangular)0.00 + j8.00

OperationV + W

Phasor Diagram

Phasor V,
Phasor W, and
Resultant R.

What is a Phasor Diagram?

A phasor diagram is a graphical way of representing the magnitude and phase relationship between two or more sinusoidal quantities of the same frequency. Phasors are essentially rotating vectors, where the length of the vector represents the amplitude (e.g., peak voltage or current) and the angle it makes with a reference axis represents its phase angle. This calculator helps you calculate v w using phasor diagram principles, specifically for vector addition and subtraction.

This method is invaluable in electrical engineering and physics for analyzing AC circuits. Instead of dealing with complex sinusoidal equations in the time domain, engineers can convert them into static vectors (phasors) in the complex plane. This simplifies addition, subtraction, and other operations into straightforward vector arithmetic.

The Formula to Calculate V & W Resultant

To add or subtract two phasors given in polar form (Magnitude ∠ Angle), the most reliable method is to first convert them to rectangular form (a + jb), perform the operation, and then convert the result back to polar form.

1. Convert to Rectangular Coordinates:

   For a phasor P with magnitude M and angle θ:

   Real Part (x) = M × cos(θ)

   Imaginary Part (y) = M × sin(θ)

   So, V(M_v, θ_v) → V(x_v, y_v) and W(M_w, θ_w) → W(x_w, y_w).

2. Add/Subtract Rectangular Components:

   The real and imaginary parts of the resultant phasor R(x_r, y_r) are found by adding or subtracting the corresponding parts of V and W.

   Addition: x_r = x_v + x_w;   y_r = y_v + y_w

   Subtraction: x_r = x_v – x_w;   y_r = y_v – y_w

3. Convert Resultant back to Polar Form:

   Magnitude (M_r) = √(x_r² + y_r²)

   Angle (θ_r) = atan2(y_r, x_r)

For more detailed calculations, our Ohm’s Law Calculator can be a useful related tool.

Variables Table

Variables used in phasor calculations.

Variable	Meaning	Unit	Typical Range
Mv, Mw	Magnitude of Phasors V and W	Volts, Amps, etc. (context-dependent)	0 to ∞
θv, θw	Angle of Phasors V and W	Degrees (°) or Radians (rad)	-360° to 360°
x, y	Real and Imaginary Components	Same as magnitude	-∞ to ∞
j	Imaginary Unit (√-1)	Unitless	N/A

Practical Examples

Example 1: Phasor Addition

Suppose you need to find the total voltage by adding two voltage phasors in an AC circuit.

• Phasor V: 10 V ∠ 45°
• Phasor W: 15 V ∠ -30°

Calculation Steps:

1. V → (10 × cos(45°)) + j(10 × sin(45°)) = 7.07 + j7.07
2. W → (15 × cos(-30°)) + j(15 × sin(-30°)) = 13.00 – j7.50
3. R = (7.07 + 13.00) + j(7.07 – 7.50) = 20.07 – j0.43
4. Result: Magnitude = √(20.07² + (-0.43)²) = 20.07 V. Angle = atan2(-0.43, 20.07) = -1.23°. So, R = 20.07 V ∠ -1.23°.

Understanding these interactions is key, much like in our Series RLC Circuit Calculator.

Example 2: Phasor Subtraction

Imagine you want to find the voltage drop across a component, which requires subtracting one phasor from another.

• Phasor V: 120 V ∠ 0° (Reference Phasor)
• Phasor W: 80 V ∠ 60°

Calculation Steps:

1. V → (120 × cos(0°)) + j(120 × sin(0°)) = 120 + j0
2. W → (80 × cos(60°)) + j(80 × sin(60°)) = 40 + j69.28
3. R = (120 – 40) + j(0 – 69.28) = 80 – j69.28
4. Result: Magnitude = √(80² + (-69.28)²) = 105.8 V. Angle = atan2(-69.28, 80) = -40.9°. So, R = 105.8 V ∠ -40.9°.

How to Use This ‘Calculate V W using Phasor Diagram’ Calculator

1. Enter Phasor V: Input the magnitude and angle (in degrees) for your first phasor, V.
2. Enter Phasor W: Input the magnitude and angle for your second phasor, W.
3. Select Operation: Choose either ‘Addition (V + W)’ or ‘Subtraction (V – W)’ from the dropdown menu.
4. Interpret the Results: The calculator instantly provides the resultant phasor in polar form (Magnitude ∠ Angle), along with intermediate values in rectangular form.
5. Analyze the Diagram: The SVG phasor diagram visually represents V, W, and the resultant R. For addition, it shows the parallelogram law. For subtraction, it shows the resulting vector from the tip of W to the tip of V.

Key Factors That Affect Phasor Calculations

• Magnitude: The length of the vectors directly impacts the magnitude of the resultant. Larger magnitudes lead to a larger resultant vector, all else being equal.
• Phase Angle: The angle between phasors is critical. If two phasors are in-phase (angle difference is 0°), their magnitudes add algebraically. If they are out-of-phase (180°), their magnitudes subtract.
• Operation (Add/Subtract): Subtraction is equivalent to adding the negative of the second vector. Geometrically, subtracting W from V (V-W) means finding the vector that goes from the tip of W to the tip of V.
• Reference Axis: All angles are measured relative to a zero-degree reference, typically the positive horizontal axis. Consistency is key. Our Vector Addition Calculator provides another perspective on this.
• Frequency: While this calculator doesn’t require frequency as an input, in real AC circuits, all phasors being analyzed must have the same frequency.
• Coordinate System: Calculations are performed in the rectangular (Cartesian) coordinate system for simplicity, while the final representation is often more intuitive in the polar form.

Frequently Asked Questions (FAQ)

What does it mean to ‘calculate v w using phasor diagram’?

This phrase refers to the process of finding the vector sum or difference of two sinusoidal quantities, represented as phasors V and W, by using the geometric or algebraic rules illustrated by a phasor diagram.

Why can’t I just add the magnitudes and angles directly?

Vectors do not add algebraically unless they are perfectly aligned (in-phase). Their direction (angle) must be accounted for. The correct method requires breaking them down into horizontal (real) and vertical (imaginary) components, as this calculator does automatically. For more complex scenarios involving resistance and reactance, see our Impedance Calculator.

What is the ‘j’ in the rectangular form?

‘j’ represents the imaginary unit (√-1). It is used to denote the vertical component of the phasor on the complex plane, which is 90 degrees offset from the real axis.

What is the difference between the diagram for addition and subtraction?

In addition (V+W), the resultant vector R goes from the origin to the opposite corner of the parallelogram formed by V and W. In subtraction (V-W), the resultant vector R is equivalent to V + (-W), where -W is a vector with the same magnitude as W but pointing in the opposite direction (180° phase shift).

Can I use this calculator for current phasors?

Yes. The principles of vector math are the same. ‘V’ and ‘W’ can represent voltage, current, impedance, or any other quantity that can be modeled as a phasor.

What do negative angles mean?

A negative angle represents a clockwise rotation from the positive horizontal axis, whereas a positive angle is counter-clockwise. For example, -90° is the same as +270°.

Does the order matter in subtraction?

Yes. V – W is not the same as W – V. The resultant vector will have the same magnitude but will point in the opposite direction (a 180° phase difference).

What is a practical application of this?

In a series AC circuit with a resistor, inductor, and capacitor, the total voltage from the source is the phasor sum of the individual voltages across each component. To find it, you would use this exact calculation method. Tools like an AC Power Calculator build upon these fundamental concepts.