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Telegrapher's Equation
IntroductionA transmission line can be represented as an infinite series of cascaded identical two port networks each representing an infinitely small section of the transmission line. The small networks represent:
Fig 1 shows the small networks. R,L,G, and C may be frequency dependent. In practical transmission lines at HF and above the following assumptions are often appropriately used:
DerivationThe line voltage V(x) and the current I(x) where x is displacement can be expressed in the frequency domain as: \[\frac{\partial V(x)}{\partial x}=(R+j \omega L)I(x)\] \[\frac{\partial I(x)}{\partial x}=(G+j \omega C)V(x)\] Differentiating both: \[\frac{\partial^2 V(x)}{\partial x^2}=\gamma^2V(x)\] \[\frac{\partial^2 I(x)}{\partial x^2}=\gamma^2 I(x)\] where: \[\gamma=\sqrt{(R+j \omega L)(G+j \omega C)}\] \[Z_0=\sqrt{\frac{R+j \omega L}{G+j \omega C}}\] γ is the transmission line complex propagation constant, and Z_{0} is a complex value known as the characteristic impedance of the line. A solution for V(x) and I(x) is: \[V(x)=V_f e^{\gamma x}+V_r e^{ \gamma x}\] \[I(x)=I_f e^{\gamma x}I_r e^{ \gamma x}\] where x is the displacement from the load, negative towards the source, and Vf, Vr, If and Ir are forward and reflected voltages and currents respectively at the load end of the line. ApplicationThe above expressions can be rewritten as: \[V(x)=V_f( e^{\gamma x}+ \Gamma e^{ \gamma x})\] \[I(x)=I_f( e^{\gamma x} \Gamma e^{ \gamma x})\] where Γ is the complex reflection coefficient at the load. Transmission line behaviour is described by these equations and the boundary conditions imposed by the load. Given load impedance Z_{l}=V/I: \[\Gamma=\frac{Z_lZ_0}{Z_l+Z_0}\] \[\Gamma(x)=\Gamma \frac{e^{\gamma x}}{e^{ \gamma x}}\] These equations fully describe the behaviour of a transmission line with a given load impedance. From these, the relationships for rho; and VSWR can be developed: \[\rho=\Gamma\] \[VSWR=\frac{1+\rho}{1\rho}\] We can write \(Z_l \text{ in terms of } Z_0 \text{ and } \Gamma \): \[Z_l=Z_0\frac{1+\Gamma}{1\Gamma}\] Input impedance Zin of line of length l can be calculated from the load impedance: \[Z_{in}=Z_0 \frac{Z_l+Z_0 tanh(\gamma l)}{Z_0+Z_l tanh(\gamma l)}\] In the special case of a lossless line, input impedance Zin of line of length l can be calculated from the load impedance: \[Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\] The following relationships exist for the two port network equivalent of a transmission line: \[V_1=V_2 cosh(\gamma l)+I_2 Z_0 sinh(\gamma l)\] \[I_1=\frac{V_2}{Z_0} sinh(\gamma l)+I_2 cosh(\gamma l)\] where V1 and I1 are the voltage and current at the input port, and V2 and I2 are the voltage and current at the output port. LINKSGlossary
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