Voltage phasor tool
Enter Rectangular Voltage Components
Use matching voltage units and a single basis for both components.
Reference values
Example Data Table
| Rectangular voltage | Magnitude | Phase | Polar voltage |
|---|---|---|---|
| 3 + j4 V | 5.0000 V | 53.1301° | 5.0000 ∠ 53.1301° V |
| 12 − j5 V | 13.0000 V | −22.6199° | 13.0000 ∠ −22.6199° V |
| −8 + j6 V | 10.0000 V | 143.1301° | 10.0000 ∠ 143.1301° V |
Calculation method
Formula Used
Let VR be the real component and VI be the imaginary component. The calculator finds the phasor length first. It then finds the direction while preserving the correct quadrant.
θ = atan2(VI, VR)
V = |V| ∠ θ
Vpeak = VRMS × √2
VRMS = Vpeak ÷ √2
The atan2 operation uses both signs. This protects the phase angle when the real component is negative. A zero magnitude does not have a unique angle.
Simple workflow
How to Use This Calculator
- Enter the real voltage component in volts.
- Enter the imaginary voltage component in volts.
- Select whether your components are RMS or peak values.
- Select degrees or radians for the displayed phase.
- Choose the decimal precision needed for your work.
- Press Calculate Voltage to view the polar result above the form.
- Use Download CSV or Save as PDF when you need a record.
Phasor guide
Understanding Voltage Coordinate Forms
Why Voltage Phasors Matter
Voltage in alternating-current work has size and direction. A rectangular phasor shows both parts directly. The real component sits on the horizontal axis. The imaginary component sits on the vertical axis. Together, they describe one voltage vector. This view helps when adding, subtracting, or balancing circuit quantities. Resistors affect the real portion. Reactance often creates the imaginary portion. Engineers use this format during network analysis because like components combine easily. However, rectangular values can hide the overall amplitude. Polar coordinates solve that problem. A polar voltage lists one magnitude and one angle. The magnitude explains the voltage level. The angle explains timing relative to a chosen reference. Both formats describe the same phasor.
Reading the Rectangular Components
A rectangular voltage is normally written as VR plus jVI. VR is the in-phase component. VI is the quadrature component. Positive VI points upward on a phasor diagram. Negative VI points downward. Signs determine the correct quadrant. A basic inverse tangent calculation can give a misleading angle when VR is negative. The atan2 function avoids that problem. It uses both components to identify the correct quadrant automatically. Enter values with a consistent voltage basis. Do not mix RMS with peak quantities. A 12 volt RMS real component and a 5 volt peak imaginary component cannot form one valid phasor. The calculator preserves that selected basis for the reported magnitude.
Choosing Polar Coordinates
The magnitude is found with the Pythagorean relationship. It is always zero or positive. The angle is measured from the positive real axis. Degrees are familiar for electrical work. A negative angle simply points below the reference axis. An angle above 180 degrees can also be expressed as an equivalent negative value. Polar form is especially useful for multiplication and division. Magnitudes multiply or divide directly. Angles add or subtract directly. This simplifies impedance, current, and power calculations. For addition and subtraction, convert polar values back to rectangular components first. This prevents incorrect magnitude-only arithmetic. A zero voltage has no unique direction. Its phase is therefore undefined. The calculator labels that case clearly instead of presenting a misleading angle.
Practical Checks Before Using Results
Check the entered signs before trusting any result. A missing minus sign can move the voltage into another quadrant. Review the displayed rectangular form. Then compare the magnitude with the largest input component. The magnitude cannot be smaller than either absolute component. Verify the angle direction against the imaginary sign. Positive imaginary voltage should produce an upward angle. Negative imaginary voltage should produce a downward angle. Use enough decimal places for the task. Three or four places suit many classroom exercises. More precision can help during tolerance studies. Fewer places improve quick reports. Keep the selected basis visible in drawings and notes. RMS is common for equipment ratings. Peak is common for waveform limits. Save the result table when documenting a design. Clear, careful phasor notation reduces wiring, testing, and communication mistakes.
Common questions
Rectangular to Polar Voltage FAQs
1. What is rectangular voltage form?
It writes a voltage phasor as a real component plus an imaginary component. A common format is VR + jVI. This makes addition and subtraction of phasors straightforward.
2. What is polar voltage form?
It writes the same phasor as one magnitude and one phase angle. A common format is |V| ∠ θ. This is convenient for multiplication, division, and comparison of phase shifts.
3. Why does the calculator use atan2?
atan2 checks both the real and imaginary signs. It finds the correct angle quadrant. A simple inverse tangent can return the wrong direction when the real component is negative.
4. Are degree and radian angles equivalent?
Yes. They describe the same direction using different units. One full rotation equals 360 degrees or 2π radians. Select the unit that suits your circuit work or software.
5. Can I enter negative voltage components?
Yes. Negative values are important. They position the phasor in the correct quadrant and produce the correct phase angle. Keep the sign that belongs to each measured component.
6. What happens when both components are zero?
The magnitude is zero. The phase angle is undefined because a zero-length vector has no direction. The calculator reports that condition instead of inventing an angle.
7. Should I use RMS or peak voltage?
Use the basis that matches your source data. Equipment ratings commonly use RMS. Waveform limits often use peak. Never combine one RMS component with one peak component in the same conversion.
8. Can phasors be added in polar form?
Not by adding magnitudes and angles directly. Convert polar values to rectangular components, add the components, then convert the sum back to polar form.
9. Do the voltage units affect the phase?
No. Changing volts to millivolts changes the magnitude scale only. The phase stays the same when both rectangular components use the same conversion factor.
10. How many decimal places should I use?
Use enough precision for the measurement and calculation. Three or four places suit many tasks. Use more places for tolerance studies, but avoid implying accuracy beyond your source data.
11. Is this calculator useful for direct current?
It can represent DC as a real-only value with a zero-degree phase. However, phasor notation is most useful for sinusoidal alternating-current analysis and frequency-domain circuit work.