21.5: Transmission Lines as Two-Port Devices
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Figure \(\PageIndex{1}\) shows common ways to represent transmission lines in circuit diagrams. In each case, the source is represented using a Thévenin equivalent circuit consisting of a voltage source \(V_S\) in series with an impedance \(Z_S\). 1 In transmission line analysis, the source may also be referred to as the generator . The termination on the receiving end of the transmission line is represented, without loss of generality, as an impedance \(Z_L\). This termination is often referred to as the load , although in practice it can be any circuit that exhibits an input impedance of \(Z_L\).
Figure \(\PageIndex{1}\): Symbols representing transmission lines: Top: As a generic two-conductor direct connection. Middle: As a generic two-port “black box.” Bottom: As a coaxial cable. © CC BY SA 3.0 Unported (modified)The two-port representation of a transmission line is completely described by its length \(l\) along with some combination of the following parameters:
- Phase propagation constant \(\beta\), having units of rad/m. This parameter also represents the wavelength in the line through the relationship \(\lambda=2\pi/\beta\). (See Sections 1.3 and 3.8 for details.)
- Attenuation constant \(\alpha\), having units of 1/m. This parameter quantifies the effect of loss in the line. (See Section 3.8 for details.)
- Characteristic impedance \(Z_0\), having units of \(\Omega\). This is the ratio of potential (“voltage”) to current when the line is perfectly impedance-matched at both ends. (See Section 3.7 for details.)
These parameters depend on the materials and geometry of the line.
Note that a transmission line is typically not transparent to the source and load. In particular, the load impedance may be \(Z_L\), but the impedance presented to the source may or may not be equal to \(Z_L\). (See Section 3.15 for more on this concept.) Similarly, the source impedance may be \(Z_S\), but the impedance presented to the load may or may not be equal to \(Z_S\). The effect of the transmission line on the source and load impedances will depend on the parameters identified above.
- For a refresher on this concept, see “Additional Reading” at the end of this section.↩