Q2Electromagnetics Waves

(a) Smith chart and its application: the Smith chart is a polar plot, within the unit circle |Γ|≤1, of the normalized-impedance grid obtained from the bilinear mapping z=(1+Γ)/(1−Γ), consisting of a family of constant-resistance circles and constant-reactance arcs all passing through the open-circuit point Γ=+1. It is used to read VSWR directly (as the radius, or equivalently the real-axis crossing value, of the constant-|Γ| circle through a plotted load point), to find the impedance transformation along a length of line (by rotating around the constant-|Γ| circle by an angle 4πd/λ corresponding to physical distance d), to convert between impedance and admittance (a 180° chart rotation), and to design impedance-matching networks such as single- and double-stub tuners and quarter-wave transformers graphically, without needing repeated complex-number arithmetic — full construction and application details are given alongside a worked Smith-chart diagram in Part C, Q.5 of the companion 2024 (Main/Back) paper on this same subject.

(b) Reflection coefficient and VSWR calculation: with Z0 = 50+j0.01 Ω and ZL = 73−j42.5 Ω:

Multiplying numerator and denominator by the conjugate of the denominator to rationalize:

Result: Γ ≈ 0.274 − j0.251 (magnitude ≈ 0.371, angle ≈ −42.5°), and VSWR ≈ 2.18. This is a moderate mismatch — the small +j0.01 Ω reactive component of Z0 (an unusually specified, very slightly lossy or reactive source/line impedance rather than the ideal purely real 50 Ω) has a negligible effect on the final result compared with using an idealized purely resistive Z0 = 50 Ω, since 0.01 Ω is over three orders of magnitude smaller than the 50 Ω real part; the mismatch is dominated entirely by the substantial reactive component (−j42.5 Ω) of the load impedance itself, consistent with a physical load such as an antenna operated somewhat off its exact resonant frequency (where its reactance is not yet fully tuned out).

Verification using the normalized-impedance route (equivalent to a Smith-chart reading): as a cross-check, normalize the load by the real part of Z0 (≈50 Ω, since the imaginary part is negligible): zL = ZL/Z0 ≈ (73−j42.5)/50 = 1.46 − j0.85. Plotting this point on a Smith chart (roughly in the lower-right quadrant, inside the r=1 to r=2 region and below the real axis in the capacitive-reactance half) and drawing the constant-|Γ| circle through it, the circle's radius read directly off the chart gives |Γ| ≈ 0.37, and its crossing of the positive real axis gives VSWR ≈ 2.18 — matching the algebraic computation above to within normal graphical reading precision, and illustrating exactly how the Smith-chart procedure described in part (a) would be used to solve this same numerical problem graphically rather than by direct complex-number algebra.

Physical significance of the result: a VSWR of 2.18 means that Vmax/Vmin ≈ 2.18 along the line — a clearly measurable, moderate standing wave, corresponding to roughly 13.8% of the incident power being reflected back toward the source (reflected power fraction = |Γ|² ≈ 0.1379) and about 86.2% actually delivered forward past the mismatch point. Such a mismatch, while not severe, would typically be considered unacceptable in a well-engineered RF system (where VSWR below about 1.5, corresponding to under 4% reflected power, is a common design target for antenna feeds and amplifier interfaces), and would normally prompt the insertion of a matching network — such as the single-stub technique detailed in Part B, Q.5 of the companion 2024 (Main/Back) paper, or a quarter-wave transformer section — between the 50 Ω line and this 73−j42.5 Ω load to bring VSWR down closer to unity and recover the lost 13.8% of forward power.

Reflected-power and return-loss figures: two related figures of merit are often quoted alongside VSWR for the same mismatch. The return loss, RL = −20log10|Γ| = −20log10(0.3714) ≈ 8.6 dB, expresses the mismatch severity logarithmically (a higher return loss in dB corresponds to a better match; well-matched RF ports are commonly specified with RL > 15-20 dB, so 8.6 dB here again confirms this is a distinctly imperfect match by typical design standards). The mismatch loss, the fraction of available power actually delivered to the load relative to the matched case, is 1−|Γ|² ≈ 0.862, i.e. approximately 0.65 dB of the available source power is lost to reflection at this interface (10log10(1/0.862) ≈ 0.646 dB) — a modest but non-negligible penalty that would accumulate significantly if multiple such mismatched interfaces existed in cascade along a longer RF signal chain.

Locating the nearest voltage minimum (slotted-line measurement context): the phase angle of Γ, θΓ ≈ −42.5° (or −0.742 rad), also determines exactly where along the line the first voltage minimum from the load occurs, via dmin = (θΓ+π)λ/(4π) (using the convention that a voltage minimum occurs where Γ is real and negative, i.e. its rotated phase equals π). This distance dmin, together with the measured VSWR itself, is precisely the pair of quantities a slotted-line probe measures directly in the classical laboratory technique for determining an unknown load impedance — moving a small probe along a slotted section of the line to locate the standing-wave minima and measure the min/max voltage ratio, then working backward through exactly the Γ-and-Smith-chart relationships used in this solution to recover the complex load impedance ZL, historically an essential technique before vector network analyzers made such measurements direct and automatic.

Summary of the full worked comparison: part (a) of this question established the general graphical machinery of the Smith chart (VSWR reading, impedance transformation along a line, and matching-network design), while part (b) has now applied the identical underlying complex-arithmetic relationships — reflection coefficient Γ=(ZL−Z0)/(ZL+Z0) and VSWR=(1+|Γ|)/(1−|Γ|) — to a specific numerical load, first algebraically and then cross-checked via the equivalent graphical Smith-chart construction, demonstrating concretely how the abstract chart-based procedure of part (a) and the direct complex-number calculation of part (b) are two routes to identically the same physical answer.