Background

In RF and microwave engineering, the distributed-element model is used to model and characterize electrical circuits and networks of interconnected electrical components. The distributed model is employed when the wavelength becomes comparable to the physical dimensions of the circuit, rendering the lumped model inaccurate. This phenomenon occurs at high frequencies, where the wavelength is extremely short, or on low frequency, yet very long, transmission lines like overhead power lines.

Transmission lines

Propagation of electromagnetic waves in a device whose physical dimensions are comparable to or smaller than the signal wavelength can be illustrated with a transmission line using the distributed-element model, as shown in the following figure.

Here, R is the resistance per unit distance, expressed in ohms/m. Similarly, L, C and G are distributed inductance, capacitance and conductance per meter of the transmission line. Voltage and current are a function of time, t and position, z. Partial differential equations relating the two are as follows.

Solving these equations leads to the voltage phasor

where

V_+ (z,ω) and V_- (z,ω) represent the complex-valued phasors of the waveforms traveling in the +z (forward direction) and -z directions respectively, and propagation constant γ is

The real part of the propagation constant, α, is called the attenuation constant and the imaginary part, β is called the phase constant.

Similarly, the equations for current are as follows:

with

4B I+z,ω=1Z0V+z0,ωe-γz-z0
4C I-z,ω=-1Z0V-z0,ωe+γ(z-z0)

where

Z0 denotes the characteristic impedance of the transmission line and is equal to the ratio of the voltage and current travelling in one direction along the line.

All symbols and their definitions are summarized below.

Symbol Definition
z By convention, the direction of wave propagation is along the z-axis.
z_0 The point at which the values of forward and reverse waves are known.
ω The frequency of the signal in radians per second.
V(z,ω) The total complex voltage phasor at point z.
V_+ (z,ω) The complex voltage phasor of the forward wave at point z.
V_- (z,ω) The complex voltage phasor of the reverse wave at point z.
I(z,ω) The total complex current phasor at point z.
I_+ (z,ω) The complex current phasor of the forward wave at point z.
I_- (z,ω) The complex current phasor of the reverse wave at point z.
γ The propagation constant.
α The attenuation constant.
β The phase constant.
R Resistance per unit distance (Ωm^(-1) ).
L Inductance per unit distance (Hm^(-1) ).
C Capacitance per unit distance (Fm^(-1) ).
G Conductance per unit distance (℧m^(-1)).
Z_0 Characteristic impedance of the transmission line.

The voltage equation in the time domain can be written as follows.

where V_+ (0,ω)=Re{V_+ (0,ω)} is the amplitude of the forward wave at point z=0 and V_- (0,ω)=Re{V_- (0,ω)} is the amplitude of the reverse wave at the same point.