# The Characteristic Impedance of Lossless and Lossy Transmission Lines

### Key Takeaways

• Circuit elements such as resistance, inductance, conductance, and capacitance (R, L, G, C) are referred to as primary line constants and are dependent on the geometry of the transmission line.

• When circuit elements R = G = 0, the transmission line becomes lossless. With series resistance and series inductance, the transmission line is lossy. Lossless and lossy transmission lines have different characteristics that are dependent upon the impedance in the transmission line.

• If the transmission line is lossy, the characteristic impedance is a complex number. If the transmission line is lossless, the characteristic impedance is a real number. A microwave transmission line

It is physically impossible to attain a perfectly lossless transmission line in any circuit. All transmission lines are lossy, and the percentage of loss varies with each case. Lossless and lossy transmission lines are defined by different characteristics and are discussed in detail in transmission line theory.

Lossless transmission lines offer an excellent approximation to low-loss transmission lines and help engineers study the behavior of low-loss transmission lines from lossless transmission line equations.

Let’s examine the characteristics of transmission line impedance by taking a closer look at lossless and lossy lines.

## Lossless and Lossy Transmission Lines

Any transmission line can be represented with its lumped element equivalent circuit. Circuit elements in the equivalent circuit, such as resistance, inductance, conductance, and capacitance (R, L, G, C), are referred to as primary line constants and are dependent on the geometry of the transmission line.

The effective series resistance per unit length (R) is due to the loss nature of the conductor material. As current through a transmission line increases, the loss associated with resistance also increases. The effective series inductance per unit length (L) is produced by current flowing through the conductor, which is not a perfect conductor.

Imperfections in the medium where transmission lines are present or fabricated generate the effective shunt-conductance per unit length (G).

The effective shunt capacitance per unit length (C) is between the transmission lines. With lumped elements R, L, G, and C, the transmission line voltage V(x) and transmission line current I(x) can be expressed with the Telegrapher’s equation as follows:  When the circuit elements are R = G = 0 (3), the transmission line becomes lossless. With series resistance and series inductance, the transmission line is lossy.

### Lossless Transmission Lines

In lossless transmission lines, the power transmitted from the source and the power delivered at the load are equal. No power is lost between the source end and the load end. Since there is no power loss, the transmission line is made of perfect conductors and the dielectric medium is also perfect.

When the transmission line becomes lossless, the line equations are simplified and it is easier to solve compared to the general equation with all the circuit elements. The lossless transmission line equation and its behavior can be approximated to low-loss transmission lines that physically exist. In low-loss transmission lines, the following condition persists: ### Lossy Transmission Lines

A lossy transmission line consists of an appreciable value of series resistance and shunt conductance where different frequencies travel at different speeds. This is opposite to a lossless transmission line, where the speed of wave propagation is the same for all frequencies. A change in speed in the lossy transmission line produces distortion in waves as it travels towards the load end. The losses in the transmission line change the propagation velocity of the wave and the signal gets attenuated as it travels from the source end to the load end.

## Determining the Characteristic Impedance of Lossless and Lossy Transmission Lines

Characteristic impedance is an inherent property of a transmission line. It is independent of the length of the transmission and the load connected to it. When the load connected is equal to the characteristic impedance, the transmission line can be said to be matched. In a matched transmission line, all the incident energy is absorbed by the load and there is no reflected energy.

Consider the lumped element model in the above figure terminated with characteristic impedance, Zo. The impedance seen by the load or input impedance Zin at t=0 can be given by the equation: As the line is assumed to be infinitely long, the input impedance Zin is equal to the characteristic impedance Zo. From equation (5), the characteristic impedance Zo can be derived as: If the transmission line is lossy, the characteristic impedance is a complex number given by equation (10). If the transmission line is lossless, the characteristic impedance is a real number. In a lossless transmission line, only purely reactive elements L and C are present and it provides an input impedance that is purely resistive.

Characteristic impedance is an important parameter to consider in both lossless and lossy transmission lines. When a transmission line is terminated with a load impedance equal to the characteristic impedance, the line is said to be matched and there is no reflected energy. If you are looking to transfer all the incident energy on a transmission line to the load end, terminate the line with its characteristic impedance.