# The Transverse Magnetic Mode of Wave Propagation in Rectangular and Circular Waveguides

Key Takeaways

• The modes of wave propagation in a waveguide change with the distribution of the electric and magnetic fields inside it.

• In the TM mode of electromagnetic wave propagation, the magnetic field is purely transverse to the direction of propagation.

•  The dominant TM mode in a rectangular waveguide is TM11. The electromagnetic field distribution depends on the geometry of the waveguide

Electromagnetic wave propagation through waveguides takes different modes. The mode of wave propagation changes with the distribution of the electric and magnetic fields inside the waveguide. The electromagnetic field distribution depends on the geometry of the waveguide as well as the type and frequency of the wave. It is important to excite the right mode of wave propagation in a waveguide, otherwise, the propagation is incurred with the attenuation of signals and losses.

In commonly-used waveguides such as rectangular waveguides and circular waveguides, the transverse electric (TE) and transverse magnetic (TM) modes are excited as per the application requirements. In this article, we will explore these modes of wave propagation further, with a particular focus on the transverse magnetic mode of propagation and how it functions in rectangular and circular waveguides.

## Modes of Wave Propagation

In electromagnetic waves, electric and magnetic fields are perpendicular to each other and travel in the same direction. Assuming the direction of propagation of the wave is in the z-direction, there can be components of electric and magnetic fields in this direction—called transverse components. Depending on the presence of transverse components of the magnetic and electric fields, the modes of propagation are defined.

There are three modes of wave propagation:

1. Transverse magnetic (TM) mode
2. Transverse electric (TE) mode
3. Transverse electromagnetic (TEM) mode

The electric and magnetic fields align in different directions inside a waveguide when a time-varying electromagnetic signal is transmitted through it. This alignment establishes the aforementioned modes in a waveguide.

Next, let’s look at the field distribution for the transverse magnetic mode of wave propagation.

## The Transverse Magnetic Mode of Wave Propagation

In the TM mode of electromagnetic wave propagation, the magnetic field is purely transverse to the direction of propagation (z-axis), but the electric field has components in different directions. The electric field components exist in the transverse and longitudinal directions with respect to the direction of propagation. Only electric fields exist in the direction of propagation.

### In TM, Hz= 0, Ez ≠ 0

For each waveguide, the wave equation can be written with the prevailing conditions of TM mode, and the solution corresponds to electric fields. The solutions are distinguished from each other using mode indexes and are represented as TMmn.

### TM Modes in Rectangular Waveguides

Rectangular waveguides are employed for signal transmission in various devices such as radars, couplers, attenuators, etc. When a wave passes through the rectangular waveguide, total internal reflection occurs and the electromagnetic field aligns inside the waveguide. When the magnetic fields are normal to the direction of propagation, they form the TM modes in a rectangular waveguide.

In the TM mode of electromagnetic wave propagation, the magnetic field is normal to the direction of propagation; however, the electric field is not transverse. The generalized solution of the wave equation giving the electric field present in the TM mode is given by equation (1), where m and n represent mode indexes. In the equation, the length and breadth of the rectangular waveguide are denoted by the letters ‘a’ and 'b’, respectively.

### 1) Ezmn (x,y,z) = emnsin(mπx/ a) sin(nπy / b) e-jkzz

By putting the values for m and n in the above solution, the electric field corresponds to TMmn mode. In a rectangular waveguide, neither m nor n can be equal to zero in TM waveguide mode. Therefore, the dominant mode in the rectangular waveguide is TM11.

### TM Modes in Circular Waveguides

In a circular waveguide, there are several TMmn modes, with m and n representing the radial and circumferential field variations. The active TM modes in the circular waveguides are TM01 , TM02 , TM11, TM12 in which  TM01 is the dominant TM mode.

The transverse magnetic modes in rectangular and circular waveguides are often utilized in RF, microwave, and communication engineering systems. Cadence offers a suite of design tools that help when designing  RF and microwave circuits with waveguides.