Transmission Line Theory
Understanding Transmission Line Theory
Transmission line theory is the bridge between low-frequency circuit theory (where wires are equi-potential) and full electromagnetic wave theory (Maxwell's equations). It applies whenever conductor lengths approach a significant fraction of the wavelength.
Telegrapher's Equations
- Voltage: -dV/dz = (R + jwL) I
- Current: -dI/dz = (G + jwC) V
- Solutions: V(z) = V+ e^(-gamma z) + V- e^(+gamma z)
Key Transmission Line Parameters
- Z0 = sqrt(L/C): Characteristic impedance (lossless case).
- v_p = 1/sqrt(LC): Phase velocity.
- lambda = v_p / f: Guide wavelength.
- gamma = alpha + j beta: Propagation constant.
Z_in = Z0 (ZL + jZ0 tan(beta l)) / (Z0 + jZL tan(beta l))
Special cases:
l = lambda/4: Z_in = Z0^2/ZL (impedance inversion)
l = lambda/2: Z_in = ZL (impedance repeats)
ZL = Z0: Z_in = Z0 for any length (matched)
Frequently Asked Questions
What is transmission line theory?
TL theory describes wave propagation using distributed R, L, C, G parameters per unit length. It replaces lumped-circuit analysis when conductor lengths exceed ~lambda/10. Key results are characteristic impedance Z0 and propagation constant gamma.
When do I need transmission line theory?
When conductor lengths exceed about 1/10 of a wavelength. At 1 GHz (lambda=30 cm), traces longer than 3 cm require TL analysis. At 10 GHz, traces longer than 3 mm. At 100 MHz, traces longer than 30 cm.
What is the most important TL equation?
The impedance transformation equation: Z_in = Z0(ZL + jZ0*tan(bl))/(Z0 + jZL*tan(bl)). This single equation describes all impedance behavior along a transmission line: matching, transformation, stubs, and resonators.