# 3: Transmission Lines

- Page ID
- 3915

- 3.5: Telegrapher’s Equations
- In this section, we derive the equations that govern the potential and current along a transmission line that is oriented along the \(z\) axis. For this, we will employ the lumped-element model.

- 3.7: Characteristic Impedance
- Characteristic impedance is the ratio of voltage to current for a wave that is propagating in single direction on a transmission line. This is an important parameter in the analysis and design of circuits and systems using transmission lines. In this section, we formally define this parameter and derive an expression for this parameter in terms of the equivalent circuit model.

- 3.10: Coaxial Line
- Coaxial transmission lines consists of metallic inner and outer conductors separated by a spacer material. The spacer material is typically a low-loss dielectric material having permeability approximately equal to that of free space and permittivity that may range from very near air. The outer conductor is alternatively referred to as the “shield,” since it typically provides a high degree of isolation from nearby objects and electromagnetic fields.

- 3.11: Microstrip Line
- A microstrip transmission line consists of a narrow metallic trace separated from a metallic ground plane by a slab of dielectric material. This is a natural way to implement a transmission line on a printed circuit board, and so accounts for an important and expansive range of applications. The reader should be aware that microstrip is distinct from stripline, which is a very different type of transmission line.

- 3.13: Standing Waves
- A standing wave consists of waves moving in opposite directions. These waves add to make a distinct magnitude variation as a function of distance that does not vary in time.

[m0085_Reflection_Coefficient_for_Various_Terminations]

We now consider values of \(\Gamma\) that arise for commonly-encountered terminations.

**Matched Load (\(Z_L=Z_0\)).** In this case, the termination may be a device with impedance \(Z_0\), or the termination may be another transmission line having the same characteristic impedance. When \(Z_L=Z_0\), \(\Gamma=0\) and there is no reflection.

**Open Circuit.** An “open circuit” is the absence of a termination. This condition implies \(Z_L\rightarrow\infty\), and subsequently \(\Gamma\rightarrow +1\). Since the *current* reflection coefficient is \(-\Gamma\), the reflected current wave is \(180^{\circ}\) out of phase with the incident current wave, making the total current at the open circuit equal to zero, as expected.

**Short Circuit.** “Short circuit” means \(Z_L=0\), and subsequently \(\Gamma=-1\). In this case, the phase of \(\Gamma\) is \(180^{\circ}\), and therefore, the potential of the reflected wave cancels the potential of the incident wave at the open circuit, making the total potential equal to zero, as it must be. Since the *current* reflection coefficient is \(-\Gamma=+1\) in this case, the reflected current wave is in phase with the incident current wave, and the magnitude of the total current at the short circuit non-zero as expected.

**Purely Reactive Load.** A purely reactive load, including that presented by a capacitor or inductor, has \(Z_L = j X\) where \(X\) is reactance. In particular, an inductor is represented by \(X>0\) and a capacitor is represented by \(X<0\). We find \[\Gamma = \frac{-Z_0+j X}{+Z_0+j X}\] The numerator and denominator have the same magnitude, so \(\left|\Gamma\right|=1\). Let \(\phi\) be the phase of the denominator (\(+Z_0+j X\)). Then, the phase of the numerator is \(\pi-\phi\). Subsequently, the phase of \(\Gamma\) is \(\left(\pi-\phi\right)-\phi=\pi-2\phi\). Thus, we see that the phase of \(\Gamma\) is no longer limited to be \(0^{\circ}\) or \(180^{\circ}\), but can be any value in between. The phase of reflected wave is subsequently shifted by this amount.

**Other Terminations.** Any other termination, including series and parallel combinations of any number of devices, can be expressed as a value of \(Z_L\) which is, in general, complex-valued. The associated value of \(\left|\Gamma\right|\) is limited to the range 0 to 1. To see this, note: \[\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} = \frac{Z_L/Z_0-1}{Z_L/Z_0+1}\] Note that the smallest possible value of \(\left|\Gamma\right|\) occurs when the numerator is zero; i.e., when \(Z_L=Z_0\). Therefore, the smallest value of \(\left|\Gamma\right|\) is zero. The largest possible value of \(\left|\Gamma\right|\) occurs when \(Z_L/Z_0\rightarrow\infty\) (i.e., an open circuit) or when \(Z_L/Z_0=0\) (a short circuit); the result in either case is \(\left|\Gamma\right|=1\). Thus,

\[0\le\left|\Gamma\right|\le 1\]