Complex Number
Understanding Complex Number
A complex number extends the ordinary real number line into a two-dimensional plane. Any complex number z can be written in rectangular form as z = a + jb, where a is the real part (written Re{z}) and b is the imaginary part (written Im{z}). The symbol j is the imaginary unit defined by j² = −1. Mathematicians write this unit as i, but electrical and RF engineers use j so that it is never confused with the symbol for instantaneous current. Plotted on the complex plane, a is the horizontal coordinate and b is the vertical coordinate, so every complex number corresponds to a single point or, equivalently, to a vector from the origin.
The reason complex numbers are so central to RF engineering is that a sinusoid at a fixed frequency carries two independent pieces of information: its amplitude and its phase. A single real number can describe only one of them. A complex number describes both at once. When a voltage or current is represented this way it is called a phasor: the magnitude of the complex number equals the signal amplitude and the angle equals the phase relative to a reference. This representation rests on Euler's formula, which links the exponential and trigonometric worlds and lets a rotating sinusoid be treated as a static complex value at a given instant of analysis.
Rectangular and Polar Forms
The same complex number can be expressed in two equivalent forms. Rectangular form, a + jb, is convenient for addition and subtraction because the real parts add and the imaginary parts add independently. Polar form, written M∠θ or M·ejθ, gives the magnitude M and the angle θ directly and is convenient for multiplication and division, where magnitudes multiply and angles add. Converting between the two is a routine first step in almost every RF calculation, from combining series impedances to cascading the gain and phase of amplifier stages.
Complex Numbers in Impedance and S-Parameters
Impedance is inherently complex: Z = R + jX, where R is resistance and X is reactance. The reactance carries the sign and size of the phase shift introduced by inductance or capacitance, information that no real number could hold by itself. The reflection coefficient Γ, the S-parameters of a network, the input admittance of an antenna, and the transfer function of a filter are all complex quantities. Vector network analyzers measure both the magnitude and the phase of these quantities precisely because the underlying physics is complex. Working in complex arithmetic lets an engineer cascade components, de-embed fixtures, and predict matching behavior with simple algebra instead of cumbersome trigonometry.
Why the Algebra Matters
Representing time-harmonic signals as complex phasors converts differentiation into multiplication by jω and integration into division by jω, where ω is the angular frequency. This single substitution turns the differential equations of circuits and transmission lines into linear algebra. As a result, frequency-domain analysis, impedance matching on a Smith chart, stability circles, and noise calculations all reduce to operations on complex numbers. A firm grasp of complex arithmetic is therefore a prerequisite for design work at microwave and millimeter-wave frequencies, where phase relationships dominate system behavior.
Key Properties
- Imaginary unit: j = √−1, with j² = −1, j³ = −j, and j⁴ = 1
- Magnitude (modulus): |z| = √(a² + b²), the distance from the origin
- Phase (argument): θ = atan2(b, a), measured from the positive real axis
- Complex conjugate: z* = a − jb, reflecting z across the real axis
- Addition: done part by part in rectangular form
- Multiplication: magnitudes multiply and angles add in polar form
Core Equations
z = a + jb (j = √−1, j² = −1)
Magnitude and phase:
|z| = √(a² + b²) θ = atan2(b, a)
Polar form and Euler's formula:
z = |z|·ejθ = |z|(cosθ + j·sinθ)
Complex conjugate and product with conjugate:
z* = a − jb z·z* = a² + b² = |z|²
Where a = real part Re{z}, b = imaginary part Im{z}, j = imaginary unit, |z| = magnitude (modulus), θ = phase (argument) in radians or degrees, z* = complex conjugate. Example: z = 3 + j4 gives |z| = 5 and θ = 53.13°.
Worked Reference Values
| Rectangular (a + jb) | Magnitude |z| | Phase θ | Conjugate z* | RF interpretation |
|---|---|---|---|---|
| 1 + j0 | 1.00 | 0° | 1 − j0 | Real, in phase with reference |
| 0 + j1 | 1.00 | +90° | 0 − j1 | Pure inductive reactance |
| 0 − j1 | 1.00 | −90° | 0 + j1 | Pure capacitive reactance |
| 3 + j4 | 5.00 | +53.13° | 3 − j4 | Z = 3 + j4 Ω (R with inductive X) |
| 50 + j0 | 50.0 | 0° | 50 − j0 | Matched 50 Ω system, Γ = 0 |
| 0.707 + j0.707 | 1.00 | +45° | 0.707 − j0.707 | Unit phasor at 45° phase |
Frequently Asked Questions
What is a complex number?
A complex number is a value with two parts, a real part and an imaginary part, written z = a + jb, where j is the imaginary unit equal to the square root of negative one (electrical engineers use j instead of i to avoid confusion with current). The real part a and imaginary part b are both ordinary real numbers. In RF engineering a complex number compactly stores both the amplitude and the phase of a sinusoidal quantity, which is why it appears in nearly every signal, impedance, and network calculation.
Why do RF engineers use complex numbers instead of real numbers?
A sinusoidal signal at a fixed frequency has two independent properties: amplitude and phase. A single real number can only carry one of them, but a complex number carries both at once. Representing a signal as a phasor, a complex number whose magnitude is the amplitude and whose angle is the phase, turns differentiation and integration of sinusoids into simple multiplication and division by jω. This converts differential circuit equations into algebra, so impedance, reflection coefficient, S-parameters, and frequency response all become straightforward complex arithmetic rather than trigonometry.
How do you convert a complex number between rectangular and polar form?
Rectangular form a + jb gives the real and imaginary parts directly. Polar form M∠θ gives magnitude and phase. To go from rectangular to polar, magnitude M equals the square root of (a squared plus b squared) and phase theta equals atan2(b, a). To go from polar to rectangular, a equals M times cos(theta) and b equals M times sin(theta). Use addition and subtraction in rectangular form, and use multiplication and division in polar form because magnitudes multiply while angles add. For example, 3 + j4 has magnitude 5 and phase 53.13 degrees.