Electronic Engineering - Line transmission

Thomas Auriel, 21th December 2024

Introduction

When it is question to transmitting a signal through a line, the first considerations are whether the line is capable of carrying that signal. Understanding the line's ability to handle the physical aspects of a signal is key to ensuring reliable communication and preventing signal degradation.

In this article, we will analyse the physical layer of signal transmission, focusing on the properties that influence how a signal propagates through a transmission line. Signals themselves are essentially variations in voltage, and they can range from very-low-frequency or even steady-state signals to high-frequency signals that operate at several gigahertz.

While digital or numerical signals are important topic in communication systems, this article will not look at the content or information carried by the signal. Instead, we will analyze the physical properties of signals, such as their rise and fall times and frequency. These factors play a vital role in determining whether a line can support numerical or analogical signals.

Illustration from Thomas Lin Pedersen ©

Notation

In the next equations, all the equation are time dependent and complex. However, for simplicity the reading, this dependence is omitted in the notation. $V(t) = V e^{j \omega t}$ become simply $V$.

A simple line

An isotropic linear line can be represented as:

with:

• $R$: the resistance of the line
• $L$: the inductance of the line
• $C$: the capacity of the line
• $G$: the conductance of the insulator of the line. It is the reverse of the resistance of this insulator ($G = \frac{1}{R_{Insulator}}$)

This representation applies to each infinitesimally small segment of the line. For analytical purposes, the elements are considered to be of infinitesimal length.

Wave Propagation and Propagation Constant

With the transmission line framework established, the behavior of the line under dynamic signals can be analyzed. We focus exclusively on sinusoidal oscillations of tension and current. By Fourier decomposition, these results can be generalized to arbitrary signals.

The propagation constant characterizes how a wave is affected as it propagates through the transmission line, accounting for both attenuation and phase changes caused by the line's properties.

Telegraph Equations

The tension drop across an infinitesimal segment of the line is given by:

$$\frac{d U}{dx} = -R I - L \frac{d I}{dt}$$

Similarly, the current leakage through the line's capacitance and conductance is:

$$\frac{d I}{dx} = -G U - C \frac{d U}{dt}$$

This two equations are called the telegraph equations. They can be derived directly from Maxwell's equations.

Waves equations

By algebraically manipulating the equations, it is possible to express the tension and the current only with the line's parameters.

$$\frac{d U}{dx} = -( R + \omega L) I$$

$$\frac{d I}{dx} = -( G + \omega C) U$$

$$\frac{d^2 U}{dx^2} = -( R + j \omega L) \frac{d I}{dx}$$

$$\frac{d^2 I}{dx^2} = -( G + j \omega C) \frac{d U}{dx}$$

$$\frac{d^2 U}{dx^2} - ( R + j \omega L) (G + j \omega C) U = 0$$

$$\frac{d^2 I}{dx^2} - ( R + j \omega L) (G + j \omega C) I = 0$$

The expression $(R + j \omega L)(G + j \omega C)$ is often denoted as:

$$\gamma^2 = \alpha + i \beta = ( R + j \omega L) (G + j \omega C)$$

with:

• $\gamma$: the propagation constant
• $\alpha$: the line attenuation constant
• $\beta$: the phase constant

These equations are known as the wave equations. Their general solutions are:

$$\frac{d^2 U}{dx^2} - \gamma^2 U = 0$$

$$U = U_0^+ e^{-\gamma x} + U_0^- e^{\gamma x}$$

$$I = I_0^+ e^{-\gamma x} - I_0^- e^{\gamma x}$$

This solutions describe two waves traveling in opposite directions along the line. $U_0^+$ and $I_0^+$ corresponds to waves entering on one side and $U_0^-$ and $I_0^-$ are entering from the other. They are determined by the boundary conditions.

The subtraction of the current terms derives from a physical convention (generator/load convention). Some authors will consider the opposite convention where current are added. Only the sign of $I_0^-$ will change in numerical applications.

Characteristic impedance

We define the characteristic impedance of the transmission line. This quantity represents the intrinsic relationship between tension and current for waves propagating through the line. It plays a critical role in determining how energy is transmitted and reflected at boundaries.

It is possible to revise the telegraph equation with the identified solutions:

$$\frac{d U}{dx} = -( R + \omega L) I$$

$$\frac{d I}{dx} = -( G + \omega C) U$$

Since we know a solution we can apply find a second expression to $\frac{dU}{dx}$ and $\frac{dI}{dx}$:

$$\frac{dU}{dx} = -\gamma (U_0^+ e^{-\gamma x} - U_0^- e^{\gamma x})$$

$$\frac{dU}{dx} = -\gamma U$$

Then:

$$-( R + \omega L) I = -\gamma U$$

$$I = \frac{\gamma}{R + \omega L } U$$

$$I = \frac{ \sqrt{( R + j \omega L) (G + j \omega C)} }{R + \omega L } U$$

$$I = \sqrt{ \frac{ G + j \omega C } {R + \omega L }} U$$

The characteristic impedance $Z_0$​ is defined as the ratio of tension to current:

$$Z_0 = \frac{U}{I}$$

$$Z_0 = \sqrt \frac {R + j \omega L}{ G + j \omega C}$$

The equation the currents in the line is defined as:

$$I = I_0^+ e^{-\gamma x} - I_0^- e^{\gamma x}$$

$$I_0^+ = Z_0^{-1} U_0^+$$

$$I_0^- = - Z_0^{-1} U_0^-$$

The characteristic impedance applies to all the waves in the line and the ratio between tension and current is always:

$$Z_0 = \frac{U_0^+}{I_0^+} = - \frac{U_0^-}{I_0^-}$$

S-Parameters

An S-parameter describes the reflection and transmission characteristics of a two-port (or multi-port) network. They quantify how much of the incident wave is reflected back to the source or transmitted to the load due to impedance mismatching between the input impedance ($Z_0$) and the load impedance ($Z_L$).

These coefficients are essential in analyzing impedance mismatching, wave propagation, reflection, and transmission in communication systems and power transmission lines.

The $S_{ij}$ coefficient is the representation of wave living by the port $i$ while coming from $j$. The most commonly used S-parameters are the $S_{11}$, $S_{21}$, $S_{12}$, and $S_{22}$ coefficients, which represent the reflection and transmission properties for each port when a unit wave is incident on that port.

• The $S_{11}$ coefficient represents the reflection coefficient for port 1 when a unit wave is incident on that same port.
• The $S_{21}$ coefficient represents the forward transmission coefficient or the transfer gain from port 1 to port 2 when a unit wave is incident on port 1. It describes how much of the incident power is transmitted to port 2.
• The $S_{12}$ and $S_{22}$ coefficients represent the reflection and transmission properties for ports 1 and 2 when a unit wave is incident on port 2.

A generic network to transmit signal looks like this:

• a sender applies the signal $U_0$ on one end.
• The line transmit it to the receiver. This line as an characteristic impedance $Z_0$.
• The receiver get the signal $U_0$. This signal is applied on the impedance $Z_L$ which is where the signal is read.

The junction represents the point where the tension $U$ is located. The junction represent the end of a line and the start another. It is then the boundary of two lines. By the principle of continuity of electrical field and current, the tension at a $t$ on the junction is the same from the line and the load. However, the propagation speed can be different in both which lead to different frequencies inside the line and the load.

$$U_{0} = U_{L}$$

$$i_{0} = -i_{L}$$

It is important to note the opposing signs of the current. This occurs because the current flows into the junction from one side and exits from the other, resulting in the negative sign.

Using the previous solution to the telegraph equations, it is possible to describe the tensions and tensions as the superposition of wave going in one direction and the other.

Focusing on the left side, we see that on one hand the tension forward-propagating wave the function is described by $U_0^+$ and $i_0^+$ and on the other hand the reflected wave is described by $U_0^-$ and $i_0^-$. Therefor, and following the previous convention, the tension and current in the junction are:

$$U = U_0^+ + U_0^-$$

$$I = i_0^+ - i_0^-$$

Let us consider the scenario in which the impedance are matching throughout the system:

$$Z_0 = Z_L$$

By algebraic operations, it is possible to express the tensions $U_0^+$ and $U_0^-$ as follow:

$$U_0^- = U - U_0^+$$

$$I = \frac{U_0^+}{Z_0} - \frac{U_0^-}{Z_0}$$

$$Z_0 I = 2U_0^+ - U$$

$$U_0^+ = \frac{U + Z_0 I}{2}$$

By a similar modifications:

$$U_0^- = \frac{U - Z_0 I}{2}$$

Normalized signals and power

For next operations, the signals are normalized using the square root of the characteristic impedance:

$$a_0 = \frac{U_0^+}{\sqrt{Z_0}} = \frac{U + Z_0 I}{2\sqrt{Z_0}}~,~b_0 = \frac{U_0^-}{\sqrt{Z_0}} = \frac{U - Z_0 I}{2\sqrt{Z_0}}$$

• $a_0$ represents the normalized signal propagating from the sender toward the junction.
• $b_0$ represents the normalized signal reflected back to the sender due to impedance mismatch.
• $a_L$ represents the normalized signal propagating from the load toward the function. In the current case, no signal come from the receiver.
• $b_L$ represents the signal going to the load. This is the signal transmitted by the junction.

Their is several advantage to use this normalized values:

• The tension and current are defined as:

$$U_i = \sqrt{Z_0}(a_0 +b_0)$$

$$i_i = \sqrt{Z_0}(a_0 - b_0)$$

• The power carried by the signals can be expressed as:

$$P_{a_0} = \frac{1}{2}|a_0|^2~,~P_{b_0} = \frac{1}{2}|b_0|^2$$

• The total power can be expressed as:

$$S = \frac{1}{2} U I^*$$

Furthermore, the real power is given by:

$$P_i = \frac{1}{2} \Re ( U_i Ii^*)$$

$$P_i = \frac{1}{2} (a_i a_i^* - b_i b_i^*)$$

Impedance mismatch

It is important to consider the scenario in which there is an impedance mismatch between the characteristic impedance of the transmission line, $Z_0$, and the load impedance, $Z_L$.

$$Z_0 \neq Z_L$$

The tension on the left and on the right of the junction is the same (continuity). And the current entering the junction on one side (positive), leave it on the other side (negative).

$$I = i_0 = -i_L$$

$$U = Z_L i_L$$

$$i_L = \frac{U_L}{Z_L}$$

$$I = \frac{U}{Z_L}$$

Using the previous definition of $a_0$ and $b_0$:

$$a_0 = \frac{U + Z_0 I}{2\sqrt{Z_0}}$$

$$a_0 = \frac{U + \frac{U Z_0}{Z_L}}{2\sqrt{Z_0}}$$

$$a_0 = \frac{U (Z_L + Z_0)}{2\sqrt{Z_0} Z_L}$$

$$b_0 = \frac{U - Z_0 I}{2\sqrt{Z_0}}$$

$$b_0 = \frac{U - \frac{U Z_0}{Z_L}}{2\sqrt{Z_0}}$$

$$b_0 = \frac{U (Z_L - Z_0)}{2\sqrt{Z_0} Z_L}$$

Therefore $S_{11}$ parameter can be defined using this new equations:

$$S_{11} = \frac{b_0}{a_0} = \frac{Z_L-Z_0}{Z_L+Z_0}$$

Reflection coefficient

The impedance $Z_0$ is often the impedance of the generator and the line and is named $Z_0$. It is generally a standard impedance ($50\Omega$ for instance). On the other hand, $Z_L$ denotes the impedance of the load, which ideally should match the generator impedance ($Z_0$) to prevent reflections. However, various factors can cause $Z_L$ to deviate from this ideal value.

The ratio $\frac{Z_L-Z_0}{Z_L+Z_0}$ is frequently referred to as the reflection coefficient, denoted by $\Gamma$:

$$\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0}$$

Relation between $a_0$, $a_L$, $b_0$, $b_L$ and S-Matrix

It is possible to define the relation between an incoming wave and the outgoing waves. The relation between a wave and the other can be defined as the quotient between a leaving wave and an incoming wave while the other incoming wave is null :

$$S_{11} = \left. \frac{b_0}{a_0} \right|_{a_L=0}~,~S_{12} = \left. \frac{b_0}{a_L} \right|_{a_0=0}~,~S_{21} = \left. \frac{b_L}{a_0} \right|_{a_L=0}~,~S_{22} = \left. \frac{b_L}{a_L} \right|_{a_0=0}$$

These expressions describe how energy is reflected from junction when the wave $a_0$ is incendent.

If both input are not null, it is possible to sum effect such as:

$$b_0 = S_{11} a_0 + S_{12} a_L$$

$$b_L = S_{21} a_0 + S_{22} a_L$$

Which can be summarized as the S-Matrix, also known as an scattering matrix:

$$\vec b = S \vec a$$

The S-matrix characterizes the input-output relationship of a two-port (or multi-port) network, such as a transmission line termination or a microwave component. It provides a compact and systematic way to analyze the reflection and transmission characteristics of the network under various incident wave conditions.

The S-matrix can be empirically measured using a network analyzer. A way is applied on one port, and the reflected the reflected ways and transmitted ways are measured at different frequencies.

T-matrix

The S matrix is straightforward to measure but it is not the most convenient representation to calculate propagation in network. An alternative approach is to convert the scatter matrix into a transmission matrix. This T-matrix always easy calculation of propagation.

The S-matrix represent the relation between incoming waves and outgoing waves.

$$\begin{pmatrix} b_0 \\ b_L \end{pmatrix} = S \begin{pmatrix} a_0 \\ a_L \end{pmatrix}$$

The T-matrix represent the transmission of on side of the junction to the other.

$$\begin{pmatrix} a_L \\ b_L \end{pmatrix} = T \begin{pmatrix} a_0 \\ b_0 \end{pmatrix}$$

For more details: Microwave transistor amplifiers (2nd ed.) analysis and design p. 25.

Sources

The main source of this article is : Microwave transistor amplifiers (2nd ed.) analysis and design

You can find additional information in lectures on the internet. Be aware that notation can change but the math are the same.

Conclusion

In this article, we have examined the behaviour of a transmission line from an engineering perspective, focusing on simpler cases such as a single line with a single receiver. However, the analysis can be expanded to more complex and realistic scenarios, involving multiple lines, multiple receivers and defects. In such cases, the power is distributed among the various lines and receiver and defect are considered as junctions with specific impedance. Both transmission and reflection must be carefully considered at each junction.

It is also important to remember that measurement tools are integral to the network and must be appropriately calibrated to avoid interfering with the network behaviour. A common issue I have encountered is the improper impedance matching of oscilloscopes, which can lead to artefacts in measurements, particularly on communication buses.

This analytical approach is not limited to transmission lines; it can be extended to other systems as well. Using Maxwell's equations, any medium can be analysed in a similar manner, offering a broader application of these principles across various fields of study, including radio frequency. Moreover, these considerations can be generalised to other physical domains that involve wave propagation, such as mechanics.