EE2025 Engineering Electromagnetics, July - Nov 2024

We consider an AC source connected to a transmission line, such as a coaxial cable. The voltage wave travels the length of the TL at a finite speed. Due to this finite speed, the voltage at different points of the line can be different. In fact, the voltage some length away from a reference point is merely the time delayed version of the reference voltage. This time delay is also called the transit time, equal to length divided by the propagation velocity, \(t_r = L/v\), where \(v = \lambda f\) and L is the length of line. Thus if the voltage a point A is \(v_A(t) = \cos(wt)\), then at point B it is \(v_B(t) = \cos(w(t-t_r))\), and thus there is a phase difference between the two voltages = \(w t_r = 2 \pi f L / (\lambda f) = 2 \pi L/\lambda\). Thus, if lambda is very large (i.e. low frequency) then there is no significant phase difference and \(v_A(t) \approx v_B(t)\). In this scenario, we don’t need to worry about the transit time effect very much as the voltage would not significantly depend on position on the line. On the other hand, if lambda is small (i.e. high frequency) then traversing the length of the line will give significant phase difference and thus different voltages. When we see a voltage difference between two points on the line, we are expecting to see some circuit element between those points, such as R/L/C elements. However, in the high frequency case, we see that without any explicit lumped ckt elements connected between A and B, we still had a voltage difference. This suggests the presence of some kind of ckt elements, just not lumped. Since the voltage difference seems to depend on the length AB, we say that the ckt element is distributed along the length, rather than lumped in one place. Thus, to understand wave propagation along a TL we need to move to a picture of impedance being distributed along length, rather than being lumped.

Having established that the impedances on a TL can be viewed as distributed, we seek to build a ckt picture of a TL. Since any current carrying wire generates a magnetic field (therefore can be modelled as an inductor), and a pair of conductors will have an electric field between them (therefore can be modelled as a capacitor), the simplest ckt model of a section \(\Delta x\) of a TL is of an inductor L/m in series and a capacitor C/m in parallel. To make this picture more realistic in terms of losses, we add a resistor (value = R) in series with the L, and a resistor in parallel with the capacitor (value = 1/G). Then we write out the KCL and KVL equations for this simple ckt and on taking the limit \(\Delta x \to 0\), we get a pair of coupled differential equations linking voltage V and current I, called the Telegrapher’s equations. They are uncoupled by taking another derivative, which leads to \(d^2 v/dx^2 = (R+jwL)(G+jwC)v = \gamma^2 v\), similarly for I. This eqn has two solutions, \(\exp(\gamma x)\) and \(\exp(-\gamma x)\), so a general solution can be written as a linear combination of the two, as \(v(x) = v^+ \exp(-\gamma x) + v^- \exp(\gamma x)\), where \(v^+\) and \(v^-\) are arbitrary constants.

We solved the Telegraphers equations to express the voltage and currents on the TL. The propagation constant \(\gamma\) is expressed as \(\alpha + j \beta\) where \(\alpha,\beta\) are the attenuation and phase constants, respectively. We use the phasor approach to transform the volage and current expressions into the time domain and write the wave expressions in terms of forward (form: \(\cos(wt-\beta x)\)) and backward (form: \(\cos(wt+\beta x)\)) travelling waves. It is found that the ratio of the forward travelling wave’s voltage and current is constant; this constant is called the characteristic impedance of the TL, \(Z_0\). The corresponding ratio for the backward travelling wave is \(-Z_0\).

We start by writing the equations for the voltage and current phasors on the TL as a function of \(x\), noting both forward and backward travelling waves. We introduce a reflection coefficient, which is the ratio of the backward wave phasor to the forward wave phasor as \( \Gamma (x) = \frac{v^- \exp(\gamma x)}{v^+ \exp(-\gamma x)}\). Using this, we can write the wave impedance, \(Z(x)\), at any point on the TL in terms of \(v^+ , \Gamma (x)\), as \( Z(x) = \frac{ v(x) }{i(x)} = Z_0 \frac{ 1 + \Gamma(x)}{1 - \Gamma(x)}\). This expression can be turned around to obtain the reflection coefficient as \(\Gamma(x) = \frac{Z(x) - Z_0}{Z(x) + Z_0}\). At the end of a TL, \(x=x_0\), we typically connect a load, for e.g. a lumped element or a circuit. If this element’s impedance, \(Z(x=x_0)=Z_L\) is equal to \(Z_0\), we can see that \(\Gamma(x_0)=0\), i.e. there will be no reflections as this \(\implies v^- =0\). Such a load is called a matched load. Finally, we shift the coordinate system from the source end to the load end by the transformation \(x = x_0 - l\) such that the load appears at \(l=0\). This is done because it makes TL analysis easier and more natural (as we will see in examples ahead). Introducing \(u^+ = v^+ \exp(-\gamma x_0)\) and \(u^- = v^- \exp(\gamma x_0)\) , we can rewrite the previous relations in terms of \(l\) instead of \(x\). Denoting the reflection coefficient for the line as \( \Gamma_L = \Gamma (l=0)\), we get \( \Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}\) and \(Z(l) = \frac{ v(l)} { i(l) } = Z_0 \frac{ u^+ \exp(\gamma l) + u^- \exp(-\gamma l) }{ u^+ \exp(\gamma l) - u^- \exp(-\gamma l) } \), which can be rewritten in terms of \(\Gamma_L\) as \( Z(l) = Z_0 \frac{\exp(\gamma l) + \Gamma_L \exp(-\gamma l)}{\exp(\gamma l)- \Gamma_L \exp(-\gamma l)}\).

We write the complex impedance formula conveniently in terms of hyperbolic functions as \(Z(l) = Z_0 \frac{Z_L\cosh(\gamma l)+Z_0\sinh(\gamma l)}{Z_0\cosh(\gamma l)+Z_L\sinh(\gamma l)}\). We can see this as an impedance transformation tool that allows us to express the impedance at a point \(l +l_0\) in terms of the impedance at a point \(l_0\). We then consider the case of low-loss TLs, characterized by \(\alpha \ll \beta\), or equivalently, \(R \ll wL\) and \( G \ll wC\), which gives us approximate expressions for \(\alpha,\beta\). In a lossless TL we have \(\alpha=0\) and \(\beta = w \sqrt{LC}\).

We consider a lossless TL and write out expressions for the voltage and current phasor magnitudes as \(|v(l)|=|v^+ \exp(j\beta l) (1 + \Gamma_L\exp(-2j\beta l))|\) and \(|i(l)|=|(v^+ \exp(j\beta l)/Z_0(1 - \Gamma_L \exp(-2j\beta l))|\). Clearly, the maximum possible value of \(|v(l)|\) occurs when \(arg(\Gamma_L \exp(-2j\beta l)=2n\pi)\), and the minimum when \(arg(\Gamma_L \exp(-2j\beta l)=2(n+1)\pi)\). These values of \(l\) correspond to the places where the \(|i(l)|\) goes to minimum and maximum, respectively. When plotting these phasor magnitudes, we see that the maximas repeat themselves on the \(l\)-axis at intervals of \(\pi/\beta = \lambda/2\). We introduce a new term, voltage standing wave ratio \(\rho = \frac{|v(l)|_{max}}{|v(l)|_{min}} = \frac{1+|\Gamma_L|}{1-|\Gamma_L|}\) and note that \(1 \leq \rho < \infty\).

Considering a lossless TL, we observe that the impedance repeats every \(\lambda/2\) on the TL; further, the normalized imepdance inverts every \(\lambda/4\) on the line. Next, we use the language of phasors to write an expression for complex power in any circuit, observing that the power dissipated on the load is its real part, while the imaginary part captures the reactive power. For a lossless TL, it is found that the power dissipated is position independent. We show how a TL can be used to measure an unknown impedance by measuring the VSWR and carrying out a few simple calculations.

Considering a lossless TL, we discuss various ways of impedance matching. In the first, we examine a section of \(\lambda/4\) length transmission line connected prior to a resistive load as a way of impedance matching to a given TL. This is called a quarter wave transformer (QWT). In case the load is complex, an additional length before the load helps to transform the load to a purely resistive value and then we use the idea of a QWT. Finally, we discuss single stub matching, where a length of open or short circuited TL of length \(l_s\) is connected in parallel with a load \(Z_L\) at a length \(l_0\) away. The length \(l_0\) is chosen such that the real part of the transformed impedance matches the original \(Z_0\) of the line, and the length \(l_s\) is chosen such that the reactance cancels that coming from the load’s transformation by \(l_0\) (in the parallel combination).

We reviewed physical meanings of divergence and curl and a few important results from vector calculus. After that, we recapped Maxwell’s equations.

Starting with the most general equations of Maxwell, we simplified them for the case of vacuum and showed how \(E,H\) are orthogonal to each other and a function of a single variable.

We furthere developed the wave equations and derived some simple properties of the TEM wave such as the ratio of \(E,H\) and the intrinsic impedance of the medium.

We defined polarization of a wave as the locus traced by the tip of the e-field vector. We looked at three different types of polarization; linear, circular, and elliptal and visualized them in space via an animation.

We explore wave propagation in a conducting medium and found that the wave equaiton looks exactly like that of free space, except that by using Ohm’s law, the permittivity got transformed from \(\epsilon\) to \(\epsilon_c = \epsilon - j \frac{\sigma}{\omega}\). This leads to an attenuation of the wave as it travels in the medium and the \(E,H\) phasors being out of phase.

We developed some approximation expressions of \(\alpha,\beta\) for the cases of low loss dielectrics as well as good conductors, discussed the working of a microwave oven, and worked out a problem of probing through ice.

We developed some approximation expressions of \(\alpha,\beta\) for the cases of good conductors, and worked out a problem of communicating underwater. Additionally, we introduced the concept of the Poynting vector for understanding power in an EM wave.

We worked out propagation of EM waves in a good conductor and derived the concept of AC v/s DC resistance.

Starting with writing the free body equation for an electron bound to an atom, we worked out the equation of the damped harmonic oscillator, and saw how it lead to the refractive index and absorption coefficient of a medium being dependent on the frequency.