Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

About the further exercises of Section 6.1

Seventeen further exercises in the spirit of the problems, the home exercises and the worked examples, graded from easy to harder within each group. Exercises 6.1.1–6.1.4 are about uniform plane waves, $k$ , $λ$ , $η$ and the triad, Exercises 6.1.5–6.1.6 about power, energy and loss, Exercises 6.1.7–6.1.10 about polarisation, Exercises 6.1.11–6.1.12 about the Doppler effect, and Exercises 6.1.13–6.1.17 mix power, exposure and polarisation. Besides the relations listed at the head of this subsection they use the lossy forms $k = ω μ ε_{c} = β - j α$ , $ε_{c} = ε (1 - j σ / ω ε)$ , $η = μ / ε_{c} = ∣ η ∣ e^{j θ_{η}}$ and $P_{a v} = \frac{∣ E _{0} ∣ ^{2}}{2∣ η ∣} e^{- 2 α a_{k} \cdot r} cos θ_{η} a_{k}$ (Exercise 6.1.6 only), the average energy density $w_{a v} = \frac{1}{4} (ε ∣ E ∣^{2} + μ ∣ H ∣^{2})$ , Faraday's law $E = - j ω Φ$ , and the polarisation match $∣ p_{w} \cdot p_{a}^{*} ∣^{2}$ . A field factor $e^{- α z}$ is $8.686 α z dB$ .

Exercise 6.1.7

Three waves to classify

Determine the polarisation (type and hand, IEEE convention) and the direction of propagation of

(a) $E (z, t) = 5 cos (ω t - β z + 3 0^{\circ}) a_{x} + 5 cos (ω t - β z - 15 0^{\circ}) a_{y} V/m$ ,
(b) $E (y, t) = 6 cos (ω t - β y) a_{x} + 6 sin (ω t - β y) a_{z} V/m$ ,
(c) $E (z) = [2 a_{x} + (1 + j) a_{y}] e^{j β z} V/m$ . For this wave also find the axial ratio, the tilt of the major axis and the circular components.

Try it yourself first.

Strategy

Bring each wave to the form of the recipe: the direction from the phase, the field at the origin at $ω t = 0$ and $π /2$ , the hand against $a_{k}$ . For (c), which travels towards $- z$ , use a transverse basis $(u, v)$ with $u \times v = a_{k}$ .

$cos (θ - 15 0^{\circ}) = - cos (θ + 3 0^{\circ})$ , so both components have the same time function:

E = 5 cos (ω t - β z + 3 0^{\circ}) (a_{x} - a_{y}) .

A linear wave in disguise, along $(a_{x} - a_{y}) / 2$ , with the peak $52 = 7.07 V/m$ , towards $+ z$ .

Towards $+ y$ , $a_{k} = a_{y}$ . At $y = 0$ : $6 a_{x}$ at $ω t = 0$ , $6 a_{z}$ at $ω t = π /2$ , constant length: circular. The rotation axis is $a_{x} \times a_{z} = - a_{y} = - a_{k}$ : LHCP.

c) Direction and hand

$e^{+ j β z}$ : towards $- z$ . At $z = 0$ , $E_{x} = 2 cos ω t$ and $E_{y} = Re {2 e^{j 4 5^{\circ}} e^{j ω t}} = 2 cos (ω t + 4 5^{\circ})$ . The tip passes $(2, 1)$ at $ω t = 0$ , $(1.41, 0)$ at $4 5^{\circ}$ and $(0, - 1)$ at $9 0^{\circ}$ : clockwise seen from $+ z$ , that is a rotation about $- a_{z} = a_{k}$ . Right-handed elliptical. In the right-handed transverse basis $u = a_{y}$ , $v = a_{x}$ ( $u \times v = a_{k}$ ) the phase of $v$ relative to $u$ is $- 4 5^{\circ} < 0$ : RH, in agreement.

Ellipse

$E_{1} = 2$ (along $x$ ), $E_{2} = 2$ , $δ = 4 5^{\circ}$ : $16 + 4 + 2 \cdot 4 \cdot 2 cos 9 0^{\circ} = 20$ , and

a^{2}, b^{2} = \frac{1}{2} (6 \pm 20) \Rightarrow a = 2.288, b = 0.874 V/m, AR = 2.618 (8.36 dB), tan 2 τ = \frac{2 \cdot 2 2 cos 4 5 ^{\circ}}{4 - 2} = 2, τ = 31. 7^{\circ} .

Circular parts

In the basis $(u, v)$ : $E_{0} = (1 + j) u + 2 v = E_{R} (u - j v) + E_{L} (u + j v)$ , so $E_{R} + E_{L} = 1 + j$ and $j (E_{L} - E_{R}) = 2$ :

E_{R} E_{L} = \frac{1}{2} (1 + 3 j), ∣ E_{R} ∣ = 1.581 V/m, = \frac{1}{2} (1 - j), ∣ E_{L} ∣ = 0.707 V/m .

Answer

(a) linear, + z (b) LHCP, + y (c) RH elliptical, - z, AR = 2.62 (8.36 dB), τ = 31. 7^{\circ}

Check

$(∣ E_{R} ∣ + ∣ E_{L} ∣) / (∣ E_{R} ∣ - ∣ E_{L} ∣) = 2.288/0.874 = 2.618$ , and $∣ E_{R} ∣ > ∣ E_{L} ∣$ . $83 %$ of the power of (c) is RHCP.

**Figure 6.1.35.** Exercise 6.1.7. In (b) the panel is drawn with $a_{k}$ out of the page ( $z$ right, $x$ up, $a_{z} \times a_{x} = a_{y}$ ): the field turns from $x$ to $z$ , clockwise, left-handed. In (c) the drawing is seen from $+ z$ while the wave goes to $- z$ : the clockwise trace is right-handed. Always turn the picture so that the wave comes towards you before naming the hand.

cf. Problem 6.1.1

Exercise 6.1.7: Three waves to classify