ELEC 401 MICROWAVE ELECTRONICS Lecture 2 Instructor: M. İrşadi Aksun Acknowledgements: Artworks describing the fields in dielectric and magnetic materials were taken from the Charged Particle Acceleration download site: http://fieldp.com/cpa/cpa.html

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

Review of EM Wave Theory Frequency Domain Representations of Maxwell’s equations Integral form Differential form Differential forms are easier to work with

Review of EM Wave Theory ElectroMagnetic waves (EM waves) are characterized by the vector fields in Maxwell’s equations. What forms of EM waves do we commonly encounter in practice? Electrical signals in high-frequency circuits (voltage and current may not be enough to analyze such circuits), signals transmitted by antennas to carry information in wireless communication systems, data carried by any transmission medium for internet operation or for wired communication systems, optical data carried by fiber-optic cables or in free-space, and lots of other forms

Review of EM Wave Theory How many unknowns do we need to find to characterize a system with EM waves? E, B, H, D are all vector unknowns to be determined; There are 12 unknowns in the form of scalar functions; J and r are related and known source functions. However, Differential form The divergence equations (the last two) are dependent on the curl equations (the first two); No information on the medium has been introduced yet.

Review of EM Wave Theory Therefore, we have two linearly independent vector equations and four vector unknowns !!! Independent Maxwell’s equations no material properties are introduced !!! Material properties are introduced as relations between D and E, and B and H; The simplest relations are obtained for isotropic and homogeneous media, which are the most common type of media used in practice; D = eE, B=mH, where e and m are constants.

Review of EM Wave Theory Constitutive Relations Relations between D and E, and B and H in a material are known as constitutive relations. Fields are modified by the existence of material bodies. Behavior of Dielectric Medium

Review of EM Wave Theory Constitutive Relations Behavior of Magnetic Medium

Review of EM Wave Theory Constitutive Relations Conducting Materials: - the current density J is proportional to the force per unit charge Ohm’s law EM, chemical or gravitational Conductivity of the material Density r of mobile charges in a piece of copper wire of 1mm in diameter (assuming each atom contributes one free-electron): Average electron velocity:

Review of EM Wave Theory Constitutive Relations Dielectric Materials: For anisotropic and inhomogeneous For anisotropic and homogeneous For isotropic and homogeneous constants functions of space coordinates constants

Review of EM Wave Theory Constitutive Relations Dielectric Materials: For isotropic and homogeneous The permittivity of a medium is usually a complex number, whose imaginary part accounts for the loss in the material mr=1 for non-magnetic materials

Review of EM Wave Theory Let us go back to the solutions of Maxwell’s equations: Remember that there are two linearly independent vector equations and four vector unknowns medium parameters are introduced via constitutive parameters Now there are two vector equations and two vector unknowns

Review of EM Wave Theory Wave equation in a source free, homogeneous and isotropic region Wave equation Helmholtz equation

Review of EM Wave Theory Why do we call these equations Wave Equations? Because the solutions represent waves, and let us see on a simple example!!! Let us find the solution of a one-dimensional1 time-domain wave equation. - Assume and the medium is free-space with no source: If the time dependence of the solution is assumed to be f(t), then the functional form of the solutions will be Can you see this? 1 One-dimensional refers to the fields varying in one space coordinate only.

Review of EM Wave Theory Let us consider one of the solutions, represents waves propagating in –z direction

Review of EM Wave Theory Let us do the same calculation in the frequency domain - Assume is the time-harmonic representation of the solution, and the medium is free-space with no source. Helmholts Equation: General solution in frequency domain: General solution in time domain: