Absolute basics of Transmission Line Theory

Telegrapher equation, characteristic impedance and propagation constant

Transmission lines are everywhere whether you realize it or not.

They are used to transport electrical signals over long distances like the high voltage lines that transport electricity to your home. Or they can be designed to carry high frequency signals over microscopic distances like in an integrated circuit.

Either way, having some transmission line fundamentals under your belt will prove useful if you are an electrical engineer.

In this article, you will learn the following:

1. What is a transmission line, and what are some popular kinds?

2. What are telegrapher equations, characteristic impedance and propagation constant?

3. Why is 50Ω the industry standard?

Let's dive in!

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Fundamentals

A transmission line is defined as:

Two or more conducting wires that carry an alternating electrical signal and are physically much larger than the wavelength of the signal.

The relative physical size compared to wavelength (also known as electrical size) is what prevents us from describing transmission lines as a simple circuit network. Due to electrical size, we need to represent transmission lines with a distributed network on a "per-unit-length" basis.

We can associate resistance, inductance, capacitance, and conductance per-unit-length (R, L, G, C) for this transmission line. Now take a long transmission line and zoom in on a tiny segment of length - 𝛥z - and its equivalent circuit representation is shown in Figure 1. We will need this circuit in the next section to define some key parameters.

Fig. 1: Equivalent circuit of a small section of transmission line

Before we get into it, here are several relevant kinds of transmission lines you should know about:

• Coaxial - It has a center core conductor and a metallic shield separated by insulator. It is widely used in CATV distribution, internet access and radio communication equipment.

• Microstrip - A planar structure that has a signal line over a single ground plane separated by a substrate dielectric.

• Stripline - A planar structure whose signal line is surrounded by ground planes on both top and bottom, yet separated by substrate dielectrics.

• Coplanar waveguide - A planar structure whose signal line is surrounded by ground planes on either side of it, and at the same level. Optionally, there can be a third ground plane below the signal line separated by a substrate dielectric.

Fig. 2: Common types of transmission line.

Key parameters defined

When you are dealing with transmission lines, you will need the terminology explained in this section to speak the jargon. My goal is to give you a short explanation of each key parameter of a transmission line with minimal math.

All of these parameters can be derived by solving Kirchhoff's current and voltage equations for the circuit shown in figure 1. The resulting equations are called Telegrapher Equations. Since it is so fundamental to transmission lines, it is worth a mention below.

\(\frac{dV(z)}{dz} = -(R+j\omega L)I(z)\)

\(\frac{dI(z)}{dz} = -(G+j\omega C)V(z) \)

If the RLGC parameters are known for the transmission line based on their physical structure, then several key properties can be calculated. The calculations for a co-axial line are shown in the figure.

Fig. 3: RLGC calculations for a coaxial line

Characteristic Impedance

Let's say you are holding one end of a transmission line which is infinitely long. You use a special meter (because a Radioshack (RIP) ohmmeter just won't do) to measure the impedance of the end of the line you are holding. The reading you see is called the Characteristic Impedance, represented by Z0 and the value depends on the physical construction of the transmission line. This is what makes it a fundamental property for any given type of transmission line.

To keep it simple, assume that the line has no losses so that R=G=0. The formula to calculate characteristic impedance is:

\(Z_0 = \sqrt\frac{L}{C}\)

This simple formula gives us a way to understand how to design the characteristic impedance of the transmission line.

• Higher inductance per unit length, higher the Z0

  • On a planar transmission line, make the signal line thinner, you will increase L and Z0

• Higher capacitance per unit length, lower the Z0

  • On a planar transmission line, move the ground closer to the signal, you will increase C and lower Z0

If there are losses in the line, it affects the characteristic impedance. Additionally, the signal will attenuate as it travels along the line due to dissipative loss.

Propagation constant

It’s interesting that the telegrapher equations mentioned above can be wrangled into a wave propagation equation which we won't get into here (calculus is scary). Traveling electromagnetic waves in any medium have a quantity called propagation constant that describes how the wave travels in any given direction. It describes how fast the wave is and how its amplitude varies as it travels.

Fortunately for us, the propagation constant represented by 𝛾 is calculated from RLGC parameters as follows,

\(\gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)}\)

A few things of note:

1. Propagation constant is a function of frequency since 𝜔 = 2𝛑f, where f is the frequency.

2. Propagation constant is a complex quantity where 𝛼 is called the attenuation constant and 𝛽 is called the phase constant.

Attenuation constant 𝛼 is expressed in Nepers per meter or decibels per meter (a Neper is about 8.7 decibels) and is a measure of how much of the signal energy is dissipated due to losses as it travels through the transmission line.

Phase constant 𝛽 is expressed in radians/meter or degrees/meter and is a measure of how much the phase of the signal changes per unit length as it travels through the transmission line.

For a lossless line,

\(\alpha=0, \beta = \omega \sqrt{LC}\)

If the phase changes by 360 degrees or 2𝜋 radians, then the cycle has completed a full wave and the distance traveled by the wave is a wavelength. Then we calculate wavelength (𝜆) from the phase constant as,

\(\lambda = \frac{2\pi}{\beta}\)

Why is Z0 = 50Ω so common?

In microwave engineering, most systems are designed around a 50Ω impedance (and maybe an occasional 75Ω.) This choice is rather arbitrary and was largely determined by transmission line dimensions that were easy to manufacture. Low (below 20Ω) and high (above 100Ω) characteristic impedance lines required spacings that were too narrow or too wide. Choosing something in between these two extremes was a reasonable choice, and it was the emergence of the test industry and companies like Hewlett Packard that sealed the deal on 50Ω as the standard.

Interesting things happen when you connect a load to a finite length transmission line, but those details are for another post!

Edited to add: A few readers have pointed out that there is an engineering trade-off that the choice of 50Ω satisfies. An air-dielectric coaxial line gives the lowest insertion loss when Z0 is 77Ω, while the peak power handling capability occurs when Z0 is 30Ω. The choice of 50Ω is a compromise that satisfies both power handling and low-loss. There is a lot more detail on this topic and links to many references here.

⭐️Key Takeaways

• A transmission line can be represented with RLGC parameters from which several key properties can be calculated.

• Characteristic impedance is a impedance measured looking into an infinitely long line, and is determined by the physical construction of the transmission line.

• Propagation constant of the transmission line determines the amplitude and phase of the signal changes as it travels down the line.

• 50Ω is a rather arbitrary but popular standard for characteristic impedance.

📚Resources

• Microwave Engineering by David M. Pozar: Chapter 2.1 describes the lumped element model of the transmission line. It describes telegrapher equations in more detail. Chapter 2.2 derives telegrapher equations from field analysis.

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