TRANSMISSION LINE EQUATIONS Transmission line is often schematically represented as a two-wire line. Figure 1: Voltage and current definitions. The transmission line always have at least two conductors. Figure 1 can be modeled as a lumped-element circuit, as shown in Figure 2.

TRANSMISSION LINE EQUATIONS Supposed a T-Line is used to connect a source to a load, as shown below At position x along the line, there exist a time-varying voltage v(x,t), and current, i (x,t)

TRANSMISSION LINE EQUATIONS For a small section between x and x+Δx equivalent circuit for this section can be represented by the distributed elements Applying Kirchoff’s voltage and current law to this equivalent circuit, 2 equations are produced

TRANSMISSION LINE EQUATIONS Dividing the above eqs by Δx, and taking the limits as Δx -> 0;

TRANSMISSION LINE EQUATIONS Differentiating eq 2.18 wrt x and eq2.19 wrt t;

TRANSMISSION LINE EQUATIONS By substituting eq2.19, eq2.21 into eq2.20, one can eliminate and. If only the steady state sinusoidally time varying solution is desired, phasor notation can be used to simplify these equations Here, v and i can be expressed as: Where: Re [ ]is the real part ω is the angular frequency

A final equation can be written as: ** Note that eq2.24 is a wave eq, and γ is the wave propagation constant given by TRANSMISSION LINE EQUATIONS Where: α = attenuation constant (Np/m) β = phase constant (rad/m)

The general solution to eq2.24 is Eq2.26 gives the solution for voltage along the transmission line. The voltage is a summation of forward wave (V + e- γx ) and reflected waves (V - e +γx ) propagating in the +x and –x directions, respectively TRANSMISSION LINE EQUATIONS

The current I(x) can be found from eq2.18 in the freq domain: Where; TRANSMISSION LINE EQUATIONS

The characteristic impedance of the line is defined by; For a lossless line, R=G=0, and we have; TRANSMISSION LINE EQUATIONS Where: λ g = guided wavelength β = propagation constant

If a coaxial line with inner and outer radii of a and b, respectively, as shown below: TRANSMISSION LINE EQUATIONS Where: λ g = guided wavelength β = propagation constant