ELEC 401 MICROWAVE ELECTRONICS Lecture 6 Instructor: M. İrşadi Aksun Acknowledgements: Materials, illustrations, and some examples have been taken from the following book: Modern Microwave Circuits, by N. Kinayman and M. I. Aksun Artech House , Boston, 2005

Outline Chapter 1: Motivation & Introduction Chapter 2: Review of EM Wave Theory Chapter 3: Plane Electromagnetic Waves Chapter 4: Transmission Lines (TL) Chapter 5: Microwave Network Characterization Chapter 6: Smith Chart & Impedance Matching Chapter 7: Passive Microwave Components

Transmission Lines - General In general, circuits and networks are made up of some components that perform operations on the signals, and some others that only carry signals from one place to another with minimum distortion on the signals. These components that are responsible for the transmission of signals are called Transmission Lines (TL), and they come in various shapes and forms:

Transmission Lines - General Coaxial line Two-wire line Strip line Microstrip line

Transmission Lines – General For microwave circuits and antennas, TLs play important role, as apposed to their low-frequency applications. TLs are extremely short for low-frequency circuits in terms of wavelength, and can be modeled by considering only the bulk resistance, which is close to zero in most cases. For microwave and antenna applications, TLs are longer, and hence their influence on the signals passing through them needs to be accounted for.

Transmission Lines – General For the characterization of transmission lines in high-frequency applications, a special method that utilizes circuit theory as well as the propagation aspect of the waves is developed and called the transmission line theory. Although the theory can be developed strictly from the circuit analysis, the major assumptions of the theory, as well as its limitations, will be missing unless its derivation via Maxwell’s equations is introduced. Therefore, a general field-analysis approach is first provided starting from Maxwell’s equations, and then the transmission line theory will be developed.

Field Analysis of General Cylindrical Waveguides General Cylindrical Waveguides (WG) guide signals in a preferred direction, called the longitudinal direction (z-direction); confine the fields in transverse direction (transverse to the direction of propagation, the x-y plane). Assumptions on the structure: it is assumed to be uniform in the direction of propagation; and it has an arbitrary cross-section in the transverse direction.

Field Analysis of General Cylindrical Waveguides WGs vs. TLs: TLs are made up of two conducting materials, while WGs may consists of only one conductor or many conductors; TLs support a special type of waves called transverse electromagnetic (TEM) waves, while WGs can support any type of wave configuration.

Field Analysis of General Cylindrical Waveguides Observations from the geometry: The fields are supposed to be guided along the z-direction inside the waveguide between the two conductors; the waveguide is uniform and infinite in the z-direction; the walls are assumed to be a perfect conductor, and the medium between the walls is filled with a dielectric material or air. Goal is to find governing equations and field expressions, as it may be possible to find a simpler version of the wave equation for such a specific class of geometries.

Field Analysis of General Cylindrical Waveguides To achieve the goal, some common features of the geometry may be utilized in Maxwell’s equations or in the field expressions. For example, because the geometry is uniform and infinite in the longitudinal direction (z-direction), and it is used for the propagation of waves in the z- direction, the electric and magnetic fields should be in the following form:

Field Analysis of General Cylindrical Waveguides Why ? Because the geometry supports propagation in z- direction, variations in z-direction are predicted as exponentials with an unknown factor (propagation constant), there is no geometrical variation along the z-direction, the magnitudes of the fields are expected to be independent of z.

Field Analysis of General Cylindrical Waveguides With this form of the fields, vector quantities in Maxwell’s curl equations can be written as follows: Substituting these into Maxwell’s equations results in transverse longitudinal

Field Analysis of General Cylindrical Waveguides Then, following the manipulations shown below: making use of We obtain

Field Analysis of General Cylindrical Waveguides Let us review what we have achieved: Once the longitudinal components of the fields, (Ez ,Hz), are obtained, the rest of the field components can be calculated from Observations: Since Maxwell’s equations are linear in linear media, it is observed that the fields can be split into three independent modes: Transverse Electric (TE) modes: Transverse Magnetic (TM) modes: Transverse ElectroMagnetic (TEM) mode: Any field configuration in a waveguide can be written in terms of TE, TM, and TEM mode contributions, and they are calculated independently.

Field Analysis of General Cylindrical Waveguides TE Waves - Governing equation for TE waves

Field Analysis of General Cylindrical Waveguides TM Waves - Governing equation for TM waves

Field Analysis of General Cylindrical Waveguides TEM Waves - is there a trivial solution only? Not if Governing equation for TEM waves B.C. on PEC

Field Analysis of General Cylindrical Waveguides TEM Waves - B.C. on PEC transverse the coupling of the electric and magnetic fields is established

Field Analysis of General Cylindrical Waveguides Salient features of TEM Waves: fields of TEM waves are like static fields in the transverse domain, evidenced by Laplace’s equation in the transverse domain; there exists a unique scalar potential, in cases of more than one uniform conductor in the longitudinal direction; B.C. on PEC the scalar potential is equivalent to voltage in electrostatic fields;

Field Analysis of General Cylindrical Waveguides Salient features of TEM Waves (Cont’d): although there seems to be no coupling between the electric and magnetic fields in TEM waves, which is not possible for any wave representation, the dynamic nature of the fields is taken into consideration by the propagation of the fields in the z-direction; TEM waves don’t suffer from DISPERSION Independent of frequency

Dispersion Dispersion causes the shape of a wave pulse to change as it travels Acoustics and Vibration Animations Daniel A. Russell, Ph.D., Physics Department, Kettering University

Dispersion A wave packet built up from a sum of 100 cosine functions with different frequencies. In a non-dispersive medium all of the different frequency components travel at the same speed so the wave function doesn't change at all as it travels. When the medium is dispersive, the wave changes shape. For this example, the dispersion is such that lower frequencies travel faster than higher frequencies. As a result, the wave packet spreads out with the longer wavelengths moving faster and the shorter wavelengths lagging behind. Acoustics and Vibration Animations Daniel A. Russell, Ph.D., Physics Department, Kettering University

Transmission Line Analysis via Circuit Theory Can we model TEM supporting TLs per unit length by a circuit of lumped components?

Transmission Line Analysis via Circuit Theory To devise a circuit model for a transmission line, we need to understand the physical mechanisms involved. Here are the mechanisms and their circuit models: if the conductor of TL has finite conductivity, then there is heat loss, equivalent of a series resistor; current flowing on both conductors in opposite directions induces magnetic flux density between the conductors, which can be modeled as an inductance along the line; two conductors with finite surface areas and finite separation distance can naturally be modeled as a capacitor between the conductors; and finally, finite conductivity of the dielectric medium between the two conductors (i.e., the loss) would cause the current to leak through the media, and modeled as a conductance between the conductors.

Transmission Line Analysis via Circuit Theory What is the governing equations for the voltage and current on the transmission line based on the circuit model of the line? Apply Kirchoffs’ voltage and current laws for the circuit:

Transmission Line Analysis via Circuit Theory Rearranging the last two equations and taking the limit of result in the following expressions: Governing Equations where Z and Y are the series impedance and parallel admittance per unit length on the circuit equivalent of the transmission line, respectively:

Transmission Line Analysis If R=G=0 (lossless line) where

Transmission Line Analysis Define Define as the characteristic impedance of the line

Transmission Line Analysis Let us consider a special case of lossless line (R=G=0): If we introduce a load ZL

Transmission Line Analysis Lossless line (Cont’d): If we introduce a load ZL