Library / Foundations: Maths / FMA-41
Foundations: Maths
Modulus of a complex number
Size of a complex number a + jb: the length of the arrow from the origin to the point (a, b) on the complex plane.
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- FMA-41
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- Foundations: Maths
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- |z|
A complex number z = a + jb is a point (a, b) on the complex plane. Its modulus |z| is the length of the arrow from the origin to that point, √(a² + b²), by Pythagoras. For a phasor representing a sinusoid, |z| is the amplitude, and the angle is the phase.
Size of a complex number a + jb: the length of the arrow from the origin to the point (a, b) on the complex plane. For a phasor it is the amplitude.
Calculator
- Argument (phase) of z53.13 °
- Squared modulus, a² + b² (equals z multiplied by its conjugate)25
- Modulus of the conjugate a − jb (unchanged)5
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The number 3 + j4 has modulus 5: the 3–4–5 right triangle when the parts are 3 and 4. Its conjugate has the same modulus, because only the sign of the imaginary part changes.
- Doubling both parts doubles the modulus (to 10) and leaves the angle unchanged, which is why scaling a phasor changes its amplitude but not its phase.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Imaginary part (coefficient of j) | −20 to 20 | 3 | 20.22 | 17.22 | +0.8 | +13.2 % |
| a Real part | −20 to 20 | 4 | 20.4 | 16.4 | +0.8 | +10.3 % |
Sweep of Real part
| Real part | Modulus |
|---|---|
| −20 | 20.4 |
| −15 | 15.52 |
| −10 | 10.77 |
| −5 | 6.403 |
| 0 | 4 |
| 5 | 6.403 |
| 10 | 10.77 |
| 15 | 15.52 |
| 20 | 20.4 |
Sweep one variable
Variables and how each one contributes
Modulus
What the equation gives you. Everything else feeds it.
Real part
The real part a of the complex number a + jb, a plain number. It is the horizontal coordinate on the complex plane. Raises |z|, because a larger real part lengthens the arrow; the effect is smaller when b is already large. Raising it by 1 (to 4) moves the result by +13.1 %, from 5 to 5.657.
Imaginary part (coefficient of j)
The imaginary part b of the complex number a + jb, the number multiplying j (where j² = −1). It is the vertical coordinate on the complex plane. Raises |z| in the same way as the real part, because the modulus treats the two parts symmetrically. Raising it by 1 (to 5) moves the result by +16.6 %, from 5 to 5.831.
Worked example with the values above
- Square each part: a² = 3² = 9 and b² = 4² = 16.
- Add them: 25.
- Take the square root: |z| = 5.
Assumptions and limits
- The real and imaginary parts are in the same units.
- The number is a single value, not a function of frequency; a frequency response needs the modulus at each frequency.
Common mistakes
- Forgetting to square the imaginary part without its j: j² is −1 in the algebra, but the modulus uses b², not −b².
- Adding moduli to find the modulus of a sum: |z1 + z2| is not |z1| + |z2|.
Statistics and uncertainty
If a and b are measured independently with errors ua and ub, the error in |z| is (a ua + b ub)/|z| at first order, always smaller than the larger of the two. A measurement error in the smaller part matters less.
Where it is used
- Amplitude of a voltage or current phasor.
- Magnitude of a transfer function or impedance at one frequency.
Maths you need
- Pythagoras: the square of the hypotenuse is the sum of squares
- Complex plane: real part across, imaginary part up