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3: Transmission Lines

Last updated

Sep 12, 2022
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- 2.8: Electromagnetic Properties of Materials
- 3.1: Introduction to Transmission Lines

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3.1: Introduction to Transmission Lines
Transmission lines are designed to support guided waves with controlled impedance, low loss, and a degree of immunity from EMI.
3.2: Types of Transmission Lines
Two common types of transmission line are coaxial line and microstrip line. Both are examples of transverse electromagnetic (TEM) transmission lines. A TEM line employs a single electromagnetic wave “mode” having electric and magnetic field vectors in directions perpendicular to the axis of the line. TEM transmission lines appear primarily in radio frequency applications.
3.3: Transmission Lines as Two-Port Devices
3.4: Lumped-Element Model
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.
3.6: Wave Equation for a TEM Transmission Line
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.
3.8: Wave Propagation on a TEM Transmission Line
3.9: Lossless and Low-Loss Transmission Lines
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.
3.12: Voltage Reflection Coefficient
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.
3.14: Standing Wave Ratio
3.15: Input Impedance of a Terminated Lossless Transmission Line
3.16: Input Impedance for Open- and Short-Circuit Terminations
3.17: Applications of Open- and Short-Circuited Transmission Line Stubs
3.18: Measurement of Transmission Line Characteristics
3.19: Quarter-Wavelength Transmission Line
3.20: Power Flow on Transmission Lines
3.21: Impedance Matching - General Considerations
3.22: Single-Reactance Matching
3.23: Single-Stub Matching

[m0085_Reflection_Coefficient_for_Various_Terminations]

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

). 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.

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” 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.

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 $\Gamma = \frac{-Z_0+j X}{+Z_0+j X} \nonumber$ 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.

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: $\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} = \frac{Z_L/Z_0-1}{Z_L/Z_0+1} \nonumber$ 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,

$0\le\left|\Gamma\right|\le 1 \nonumber$

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