Delving Deeper Into Resonance and Impedance!

While the physics of antenna resonance can be visualized like a tuning fork, the underlying “engine” of radio theory is Complex Algebra. But you don’t need to be an expert in maths to use these formulas, yet you do need to understand three core mathematical concepts: The Imaginary Unit (j), The Impedance Equation, and the Reflection Coefficient. Actually the imaginary unit is ‘i’. But in electrical discussions as it is likely to be confused with current, ‘j’ is used instead. It is the square root of -1. If any real number is multiplied by it, the product is an imaginary number. Let us go back to our higher secondary school days and recollect the equations! Practical measurement using the NanoVNA is also illustrated, which of course is more important for the amateur radio operator.

1. The Core Equation: Complex Impedance (Z)

In a DC circuit, resistance is simple (R). In an AC circuit (like radio), we use Impedance (Z), which is a complex number (also called a 2D number!).

Z = R + jX

• R (Resistance): The real part. Energy that is actually radiated or turned into heat.
• j: The “imaginary unit” (square root of -1). In radio, it represents a 90-degree phase shift between voltage and current. It means that voltage and current do not rise and fall simultaneously when there is inductance or capacitance, unlike in a purely resistive circuit. There is difference in timing by one fourth of a cycle period. One full cycle is 360 degrees.
• X (Reactance): The “imaginary” part. Energy that is stored in magnetic or electric fields but not radiated.
  • X_L = 2πfL (Inductive reactance: increases with frequency).
  • X_C = 1/(2πfC) (Capacitive reactance: decreases with frequency).

2. The Math of Resonance

Resonance is the specific mathematical state where the “imaginary” parts cancel each other out to zero.

At Resonance: X_L – X_C = 0

When this happens, the equation Z = R + jX simplifies to Z = R + j0. The antenna looks like a pure resistor to the radio.

• Mathematical Result: Voltage and current are perfectly in phase.
• The “But”: Even if jX is 0, R could still be 10 Ω or 500 Ω. Resonance doesn’t mean it’s 50 Ω; it just means it’s “pure.”

3. The Math of Impedance Matching (SWR)

Impedance Matching uses the Reflection Coefficient (Γ) to see how much energy “bounces back” because the load (Z_L) doesn’t match the source (Z_O, usually 50 Ω).

Γ = (Z_L– Z_O)/(Z_L+ Z_O)

The SWR (Standing Wave Ratio) is then derived from the magnitude of that reflection (Γ):

SWR = (1 + Γ)/(1 – Γ)

• If Z_L = 50 Ω: Then Γ = (50- 50)/(50+ 50) = 0. SWR is 1:1, as per the above equation. Perfect match when both load or antenna impedance and source or radio impedance are the same!
• If Z_L is resonant but 10 Ω: Then Γ = (10- 50)/(10+ 50) = -0.66. The SWR would be 5:1 by the same equation. Even though it’s resonant, the math says the match is terrible. Something similar to that occurs in a multi-element Yagi antenna. As the number of elements increase, the impedance of the resonant dipole radiator decreases from the typical 70 Ω of a dipole to a low value. That is why you use a gamma match for the Yagi antenna!

Summary Table for the Mathematically Minded

Concept	Mathematical Condition	Result
Resonance	Xtotal = 0	Pure resistance; no reactive power.
Impedance Match	Zload = Zsource	Zero reflection; maximum power transfer.
Perfect Combo	Zload = 50 + j0	Maximum efficiency AND no reflections.

Let me hope that anyone with a minimal knowledge of mathematics up to the higher secondary school level would have been able to follow the discussion.