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Foundations: Maths

Complex division in polar form

Quotient of two complex numbers written as modulus and angle: the moduli divide and the angles subtract.

Ref: FMA-43
Rev: v1
Field: Foundations: Maths
Result: Re(z₁/z₂)

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} e^{j (θ_{1} - θ_{2})} z_{1} z_{2} = r_{1} r_{2} e^{j (θ_{1} + θ_{2})}

In polar form z = r e^jθ, the modulus r is the size and θ the angle. To divide z₁ by z₂, divide the moduli and subtract the angles; to multiply, multiply the moduli and add the angles. This is much easier than working in the form a + jb. The result here is the real part of the quotient, r₁/r₂ cos(θ₁ − θ₂). The divisor must be nonzero.

Quotient of two complex numbers written as modulus and angle: the moduli divide and the angles subtract. The product, for comparison, multiplies the moduli and adds the angles. The result here is the real part of the quotient.

Calculator

Solve for

Re(z₁/z₂)Real part of the quotientComputed

r₁Modulus of z₁

0.150

θ₁Angle of z₁

−180180

r₂Modulus of z₂ (nonzero)

0.150

θ₂Angle of z₂

−180180

Modulus of the quotient, r₁/r₂3
Angle of the quotient, θ₁ − θ₂40 °
Imaginary part of the quotient1.928
Modulus of the product, r₁r₂12
Angle of the product, θ₁ + θ₂80 °

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.

Real part of the quotient () current value 2.298

r₂Modulus of z₂ (nonzero)Key driver

0.091946

r₁Modulus of z₁

0.038319.2

θ₂Angle of z₂

−33

θ₁Angle of z₁

−2.992.99

Dividing 6∠60° by 2∠20° gives 3∠40°, a real part of 2.298 and an imaginary part of 1.928.
In a circuit, if the voltage phasor is the dividend and the impedance the divisor, the result is the current phasor: its angle is the voltage angle minus the impedance angle.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
r₂ Modulus of z₂ (nonzero)	0.1 to 50	0.09193	45.96	45.87	+10%	−9.09 %
r₁ Modulus of z₁	0.1 to 50	0.0383	19.15	19.11	+10%	+10 %
θ₂ Angle of z₂	−180 to 180 °	−3	3	6	+7.2 °	+9.73 %
θ₁ Angle of z₁	−180 to 180 °	−2.993	2.993	5.985	+7.2 °	−11.3 %

Sweep of Angle of z₁

Angle of z₁ (°)	Real part of the quotient
−180	−2.819
−135	−2.719
−90	−1.026
−45	1.268
0	2.819
45	2.719
90	1.026
135	−1.268
180	−2.819

Sweep one variable

Vary

Real part of the quotient ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Re(z₁/z₂)

Real part of the quotient

Result ·

What the equation gives you. Everything else feeds it.

r₁

Modulus of z₁

· typical 0.1 to 50

The modulus (size) of the number being divided, z₁, a positive number. Raises the real part in direct proportion, because the whole quotient scales with r₁. Raising it by 1 (to 7) moves the result by +16.7 %, from 2.298 to 2.681.

θ₁

Angle of z₁

° · typical −180 to 180

The angle of z₁ from the positive real axis, in degrees. Changes the real part through the cosine of the angle difference; the effect is small while the difference is near 0° and large near 90°. Raising it by 5° (to 65°) moves the result by −7.69 %, from 2.298 to 2.121.

r₂

Modulus of z₂ (nonzero)

· typical 0.1 to 50

The modulus of the divisor z₂, a positive number. It must not be zero. Lowers the real part in inverse proportion, because a larger divisor gives a smaller quotient. Raising it by 0.5 (to 2.5) moves the result by −20 %, from 2.298 to 1.839.

θ₂

Angle of z₂

° · typical −180 to 180

The angle of z₂ from the positive real axis, in degrees. Changes the real part through the cosine of the angle difference, in the opposite direction to θ₁. Raising it by 5° (to 25°) moves the result by +6.93 %, from 2.298 to 2.457.

Worked example with the values above

Divide the moduli: r₁/r₂ = 6 / 2 = 3.
Subtract the angles: θ₁ − θ₂ = 60° − 20° = 40°.
Real part = 3 × cos(40°) = 2.298.

Assumptions and limits

Both numbers are written in polar form with angles in degrees (converted to radians inside the calculation).
The divisor is nonzero.

Common mistakes

Dividing the angles instead of subtracting them.
Mixing degrees and radians in the angle difference.

Statistics and uncertainty

Moduli multiply or divide, so their relative errors combine as in the uncertainty entry; the angle errors simply add or subtract. A 1° error in either angle shifts the quotient angle by up to 1°, and the real part by r₁/r₂ sin(θ₁ − θ₂) × 0.01745 per degree.

Where it is used

Finding a current phasor from a voltage phasor and an impedance.
Dividing transfer-function phasors in control and vibration analysis.

Maths you need

Polar form: modulus and angle
Cosine: horizontal component of a unit vector at an angle

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