Nano Electronics (EC-702 (C)) - Unit 2 Short Notes

UNIT 2: MICROWAVE ENGINEERING AND NANO ELECTRONIC DEVICES

I. TRANSMISSION LINES AND WAVEGUIDES

A. Transmission Line Fundamentals

Primary Constants (per unit length):

• R: Series resistance (Ω/m)

• L: Series inductance (H/m)

• C: Shunt capacitance (F/m)

• G: Shunt conductance (mho/m)

Secondary Constants:

• Characteristic Impedance: $$\displaystyle Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} $$

• Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)} $$

  • $\alpha$: Attenuation constant (Np/m)

  • $\beta$: Phase constant (rad/m)

• Phase Velocity: $$\displaystyle v_p = \frac{\omega}{\beta} $$

[!TIP] For low-loss lines ($R \ll \omega L$, $G \ll \omega C$): $$\displaystyle Z_0 \approx \sqrt{L/C} $$, $$\displaystyle \alpha \approx \frac{R}{2}\sqrt{\frac{C}{L}} + \frac{G}{2}\sqrt{\frac{L}{C}} $$, $\beta \approx \omega\sqrt{LC}$.

Numerical Example (Dec 2024):

Given: $$\displaystyle R = 40\ \Omega/\text{km} $$, $$\displaystyle L = 2.5\ \text{mH}/\text{km} $$, $$\displaystyle C = 0.009\ \mu\text{F}/\text{km} $$, $$\displaystyle G = 0.29\ \mu\text{mho}/\text{km} $$, $$\displaystyle f = 1\ \text{kHz} $$.
$$\displaystyle \omega = 2\pi \times 10^3\ \text{rad/s} $$.
$$\displaystyle Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} = \sqrt{\frac{40 + j15.7}{0.29 + j0.0017}} \approx \boxed{530\ \Omega} $$ (approx).
$$\displaystyle \gamma = \sqrt{(40 + j15.7)(0.29 + j0.0017)} \approx 3.4 + j0.026\ \text{rad/km} $$.

Thus, $\alpha \approx \boxed{3.4\ \text{Np/km}}$, $\beta \approx \boxed{0.026\ \text{rad/km}}$, $$\displaystyle v_p = \omega/\beta \approx \boxed{2.4 \times 10^8\ \text{m/s}} $$.

TEM Mode:

• Definition: Transverse ElectroMagnetic; both $\vec{E}$ and $\vec{H}$ entirely transverse to direction of propagation (no $$\displaystyle E_z $$, $$\displaystyle H_z $$).

• Conditions: Two-conductor transmission line, homogeneous medium, no frequency cutoff.

• Examples: Coaxial cable, parallel plate line.

• Quasi-TEM: In planar lines (microstrip), fields partly in air, partly in dielectric; effective parameters used.

B. Planar Transmission Lines

Microstrip Line:

• Structure: Conducting strip on dielectric substrate (height $d$, width $W$, permittivity $$\displaystyle \epsilon_r $$), ground plane below.

• Effective Dielectric Constant (for $$\displaystyle W/d > 1 $$):

$$\epsilon_{\text{eff}} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12d/W}}$$

• Characteristic Impedance (for $$\displaystyle W/d > 1 $$):

$$Z_0 = \frac{60}{\sqrt{\epsilon_{\text{eff}}}} \ln\left(\frac{8d}{W} + \frac{W}{4d}\right)$$

For $$\displaystyle W/d < 1 $$: $$\displaystyle Z_0 \approx \frac{120\pi}{\sqrt{\epsilon_{\text{eff}}} \left( \frac{W}{d} + 1.393 + 0.667 \ln\left(\frac{W}{d} + 1.444\right) \right)} $$.

Slot Line:

• Structure: Slot in ground plane, strip on opposite side (inverted microstrip).

• Comparison with Microstrip:

  | Feature | Microstrip | Slot Line | |---------|------------|-----------| | Conductor | Strip on top | Slot in ground | | Mode | Quasi-TEM | Quasi-TEM with $$\displaystyle E_z $$ | | Loss | Lower | Higher (radiation) | | Integration | Easy with series elements | Easy with shunt elements |

Strip Line:

• Structure: Strip sandwiched between two ground planes, dielectric filling.

• Modes: Dominant mode is TEM (if strip centered). Higher-order modes possible if thickness large relative to width.

C. Waveguides

Rectangular Waveguide ($a \times b$, $$\displaystyle a > b $$):

• TE$$\displaystyle _{mn} $$: $$\displaystyle E_z = 0 $$, $$\displaystyle H_z \neq 0 $$. Cutoff wavelength: $$\displaystyle \lambda_c = \frac{2}{\sqrt{(m/a)^2 + (n/b)^2}} $$.

• TM$$\displaystyle _{mn} $$: $$\displaystyle H_z = 0 $$, $$\displaystyle E_z \neq 0 $$. Same $$\displaystyle \lambda_c $$.

• Wave Impedance:

  • TE: $$\displaystyle Z_{\text{TE}} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$

  • TM: $$\displaystyle Z_{\text{TM}} = \eta \sqrt{1 - (f_c/f)^2} $$

  where $$\displaystyle \eta = \sqrt{\mu/\epsilon_0} $$ (free space impedance $\approx 377\ \Omega$).

• TM$$\displaystyle _{11} $$ Example (Dec 2024, Nov 2023): $$\displaystyle a=3\ \text{cm} $$, $$\displaystyle b=2\ \text{cm} $$, $$\displaystyle f=10\ \text{GHz} $$.

  $$\displaystyle \lambda_c = 2/\sqrt{(1/0.03)^2 + (1/0.02)^2} \approx 0.036\ \text{m} = 3.6\ \text{cm} $$.

  $$\displaystyle f_c = c/\lambda_c \approx 8.33\ \text{GHz} < 10\ \text{GHz} $$, so propagation.

  $$\displaystyle Z_{\text{TM}_{11}} = \eta \sqrt{1 - (\lambda/\lambda_c)^2} = 377 \sqrt{1 - (0.03/0.036)^2} \approx \boxed{195\ \Omega} $$.

Circular Waveguide (radius $r$, diameter $D$):

• Dominant Mode: TE$$\displaystyle _{11} $$ (no TM$$\displaystyle _{01} $$ as dominant due to higher cutoff).

  • Cutoff wavelength: $$\displaystyle \lambda_c = \frac{\pi D}{p'_{11}} $$, $$\displaystyle p'_{11} \approx 1.841 $$ → $$\displaystyle \lambda_c \approx 1.706 D $$.

  • Cutoff frequency: $$\displaystyle f_c = \frac{p'_{11} c}{\pi D} $$.

  • Guided wavelength: $$\displaystyle \lambda_g = \frac{\lambda}{\sqrt{1 - (f_c/f)^2}} $$.

• Energy Transmission: Possible for TE/TM modes if $$\displaystyle f > f_c $$.

• Characteristic Impedance for TM$$\displaystyle _{mn} $$: $$\displaystyle Z_{\text{TM}} = \eta \sqrt{1 - (f_c/f)^2} $$ (similar to rectangular).

D. Standing Waves and VSWR

• Standing Wave: Result of interference between incident and reflected waves on a lossless line. Voltage/current vary sinusoidally with position.

• Voltage Standing Wave Ratio (VSWR):

$$S = \frac{V_{\text{max}}}{V_{\text{min}}} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

where $$\displaystyle \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$ is reflection coefficient.

Derivation: $$\displaystyle V(z) = V^+ (e^{-j\beta z} + \Gamma e^{j\beta z}) $$.

$$\displaystyle |V_{\text{max}}| = |V^+| (1+|\Gamma|) $$, $$\displaystyle |V_{\text{min}}| = |V^+| (1-|\Gamma|) $$ → $$\displaystyle S = (1+|\Gamma|)/(1-|\Gamma|) $$.

Conversely, $$\displaystyle |\Gamma| = \frac{S-1}{S+1} $$.

[!TIP] VSWR is real and $\geq 1$. For matched load, $$\displaystyle S=1 $$, $$\displaystyle \Gamma=0 $$.

E. Slotted Line

• Construction: Section of transmission line (coaxial or waveguide) with a longitudinal slot. A probe moves along the slot to sample the electric field (voltage).

• Working: Probe detects voltage maxima/minima.

• Measurements:

  1. VSWR: $$\displaystyle S = V_{\text{max}}/V_{\text{min}} $$.

  2. Wavelength: Distance between two consecutive minima (or maxima) = $$\displaystyle \lambda_g/2 $$.

  3. Impedance: From VSWR $S$ and distance $d$ from load to first voltage minimum:

     $$\displaystyle Z_L = Z_0 \frac{S + j \tan(\beta d)}{1 + j S \tan(\beta d)} $$ (for lossless line).

II. MICROWAVE ACTIVE DEVICES

A. Vacuum Tubes

Klystron Amplifiers:

• Two-Cavity Klystron:

  • Velocity Modulation: Electrons from cathode accelerated through buncher cavity gap. RF voltage modulates electron velocity → density modulation (bunching). Bunched electrons pass through catcher cavity, induce RF voltage → amplification.

  • Structure: Input (buncher) cavity, drift space, output (catcher) cavity.

• Reflex Klystron:

  • Construction: Single cavity, reflector electrode (negative voltage) behind cathode.

  • Working: Electrons accelerated into cavity, oscillate, reflect back, form bunches. Used as oscillator.

  • Mode Curve: Plot of reflector voltage vs. output current. Oscillation occurs at specific voltages (modes) where bunching is optimal.

[!TIP] Reflex klystron efficiency low (~1%), used in low-power sources.

Traveling Wave Tube (TWT):

• Interaction Mechanism: Electron beam interacts continuously with RF wave on a slow-wave structure (helix or coupled cavities). RF wave travels at nearly same speed as electrons. Velocity modulation causes bunching, which reinforces RF wave → amplification. Wide bandwidth, high gain.

Magnetron:

• Types: Pulsed (most common), continuous wave (CW).

• Oscillation Mechanism:

  1. Permanent magnet creates axial magnetic field $B$.

  2. Cathode emits electrons, accelerated by radial electric field.

  3. $E \times B$ causes electrons to spiral (cyclotron motion).

  4. Electrons interact with resonant cavities (anode block) in π-mode (adjacent cavities 180° out of phase) → bunching and oscillation.

  5. Output taken from one cavity via antenna.

B. Solid-State Devices

Gunn Diode:

• Principle (Gunn Effect): In n-type GaAs/InP, at high electric fields (>3.3 kV/cm), electrons transfer from high-mobility valley to low-mobility valley → negative differential resistance (NDR).

• Domain Formation: High-field domain (space charge) forms near cathode, propagates to anode, causes current pulse.

• Modes:

  • Gunn Mode: Fundamental, domain travels entire length.

  • LSA Mode: Limited Space Charge Accumulation, higher frequency.

  • Transit-Time Mode: Domain forms and collapses within device.

[!TIP] Gunn diodes are transferred-electron devices, not PN junctions.

IMPATT and TRAPATT Diodes:

• IMPATT (Impact Ionization Avalanche Transit Time):

  • Reverse-biased PN junction. High field causes avalanche multiplication → plasma. Plasma drifts, causes current delay → NDR. High power, high noise.

• TRAPATT (Trapped Plasma Avalanche Transit Time):

  • Similar but plasma is "trapped" near junction, leading to more efficient charge removal. Higher efficiency than IMPATT, still noisy.

• Difference: IMPATT relies on avalanche and transit time; TRAPATT uses plasma injection and trapping.

Schottky Barrier Diode:

• Structure: Metal-semiconductor junction (e.g., Pt on n-GaAs).

• Working: Majority carriers only (no minority storage) → fast response.

• Applications:

  • Mixer: Nonlinear I-V, down-converts RF to IF (RF + LO → IF).

  • Detector: Rectifies RF signal to DC.

Microwave BJT:

• Construction: npn or pnp, small emitter, base, collector.

• Working: Small base current controls large collector current.

• Frequency Limitations:

  • Transit Time: Base transit time $$\displaystyle \tau_b \approx W_b^2/(2D_n) $$.

  • Junction Capacitances: $$\displaystyle C_\pi $$ (base-emitter), $$\displaystyle C_\mu $$ (base-collector).

  • Cutoff frequency $$\displaystyle f_T = \beta/(2\pi C_\pi) $$.

• Applications: Amplifiers up to ~10 GHz.

Microwave FET:

• Basic Relations:

  • Drain current $$\displaystyle I_D = I_{DSS}(1 - V_{GS}/V_p)^2 $$ (for JFET/MESFET).

  • Transconductance $$\displaystyle g_m = \partial I_D/\partial V_{GS} $$.

  • Higher input impedance than BJT, better high-frequency performance.

Other Devices:

• BARITT (Barrier Injection and Transit Time): Similar to IMPATT but uses barrier injection (e.g., p-i-n). Lower noise, lower power.

• Tunnel Diode: Heavy doping → tunneling. Modes: oscillator, amplifier, mixer, switch (based on NDR region).

III. MICROWAVE PASSIVE COMPONENTS AND NETWORKS

A. Impedance Matching Networks

• Purpose: Match load impedance $$\displaystyle Z_L $$ to line $$\displaystyle Z_0 $$ for max power transfer, minimize reflection.

• Techniques:

  1. L-section: One series, one shunt element (L or C). Two configurations (high-to-low or low-to-high).

  2. π-section: Two shunt, one series.

  3. Quarter-wave Transformer: $$\displaystyle Z_{0T} = \sqrt{Z_1 Z_2} $$, narrowband (only at $\lambda/4$).

B. Impedance Transformers

• Single-section: $$\displaystyle Z_{01} = \sqrt{Z_1 Z_2} $$. Bandwidth limited.

• Multi-section: Cascaded quarter-wave sections with intermediate impedances.

  • Binomial design: Maximally flat response.

  • Chebyshev design: Equiripple, wider bandwidth.

  [!TIP] More sections → wider bandwidth, but higher loss.

C. Directional Couplers and Tees

Directional Coupler (4-port):

• Coupling Factor: $$\displaystyle C = 10 \log_{10}(P_1/P_3) $$ (dB).

• Directivity: $$\displaystyle D = 10 \log_{10}(P_3/P_4) $$ (dB).

• S-Matrix (matched, reciprocal, symmetric):

$$\mathbf{S} = \begin{bmatrix} 0 & S_{12} & S_{13} & 0 \\ S_{12} & 0 & 0 & S_{14} \\ S_{13} & 0 & 0 & S_{12} \\ 0 & S_{14} & S_{12} & 0 \end{bmatrix}$$

with $$\displaystyle |S_{12}|^2 + |S_{13}|^2 = 1 $$. For 3 dB coupler: $$\displaystyle |S_{12}| = |S_{13}| = 1/\sqrt{2} $$.

Hybrid Tee (Magic Tee):

• Structure: Combines E-plane tee (series) and H-plane tee (parallel). Ports: 1 (input), 2 & 3 (collinear), 4 (isolated).

• Properties:

  • Signals into ports 2 and 3 add at port 1 (in-phase), cancel at port 4.

  • Signals into port 1 split equally to 2 and 3 (90° phase difference? Actually: E-plane: in-phase; H-plane: out-of-phase? Standard: Port 1 to 2,3: equal magnitude, 0° phase? Wait, check: In magic tee, E-plane: fields add at port 1, cancel at port 4; H-plane: fields add at port 4, cancel at port 1. So S-matrix:

    $$\displaystyle S_{12} = S_{13} = 1/\sqrt{2} $$ (from port 1 to 2,3),

    $$\displaystyle S_{14} = 0 $$,

    $$\displaystyle S_{21} = S_{31} = 1/\sqrt{2} $$,

    $$\displaystyle S_{24} = 1/\sqrt{2} $$, $$\displaystyle S_{34} = -1/\sqrt{2} $$,

    others zero.

  Derivation: From symmetry and orthogonality.

• Applications: Power divider, mixer, phase shifter.

D. Circulators and Isolators

Circulator (3-port or 4-port):

• Symbol: Triangle with arrows.

• Working (using two magic tees + phase shifter):

  Combine two magic tees with a 90° phase shifter in one arm. Signal entering port 1 goes to port 2, port 2 to port 3, port 3 to port 1.

• Simplified S-Matrix (3-port):

$$\mathbf{S} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$

(ideal, lossless).
Isolator: Circulator with port 2 terminated in matched load. Allows forward transmission only.

E. Resonators and Phase Shifters

YIG Resonator:

• Structure: Yttrium Iron Garnet (YIG) sphere, bias magnetic field $$\displaystyle B_0 $$ from electromagnet. Coupling loops.

• Working: Precession of magnetic moments at resonance frequency $$\displaystyle f_0 = \gamma B_0/(2\pi) $$ (γ gyromagnetic ratio).

• Tuning: Vary $$\displaystyle B_0 $$ → continuous tuning over wide range (GHz).

• Applications: Tunable filters, oscillators, frequency meters.

Diode Phase Shifter:

• Principle: Varactor diode in transmission line. Reverse bias changes capacitance → changes electrical length → phase shift.

• Types: Analog (continuous), digital (discrete states).

F. Mixers and Frequency Converters

Microwave Mixer:

• Principle: Nonlinear device (diode or FET) mixes RF ($$\displaystyle f_{RF} $$) and LO ($$\displaystyle f_{LO} $$) to produce sum/difference frequencies. IF = $$\displaystyle |f_{RF} - f_{LO}| $$.

• Roles:

  • LO: Provides local oscillation.

  • RF: Input signal.

  • IF: Intermediate frequency (easier to amplify).

• Conversion Loss: $$\displaystyle L_c = P_{RF}/P_{IF} $$ (linear) or $$\displaystyle 10 \log_{10}(P_{RF}/P_{IF}) $$ (dB). Typically 6–9 dB.

Frequency Converters:

• Block Diagram (Schottky diodes): RF and LO combined via hybrid coupler → diode pair → IF filter → IF output. Balanced configuration suppresses even harmonics.

Frequency Multipliers:

• Derivation: Nonlinear I-V: $$\displaystyle i = a_0 + a_1 v + a_2 v^2 + \cdots $$. If $$\displaystyle v = V \cos \omega t $$, then $i$ contains $2\omega, 3\omega, \ldots$ terms.

• Operating Principle: Use varactor or step-recovery diode for high efficiency. nth harmonic: $$\displaystyle f_{out} = n f_{in} $$.

IV. SCATTERING PARAMETERS (S-PARAMETERS)

A. Need for S-Parameters at Microwave Frequencies

• Difficulties with Z/Y/h/ABCD:

  • Voltages/currents not measurable due to standing waves.

  • Open/short circuits hard to realize perfectly at high freq.

  • Distributed effects invalidate lumped assumptions.

• Advantages of S-parameters:

  • Use incident/reflected waves, measurable with slotted line or VNA.

  • Well-defined for multi-port networks.

  • Easier to cascade.

B. Definition and Properties

• Definition: For n-port network, $$\displaystyle S_{ij} = b_i/a_j $$ with all other ports matched ($$\displaystyle a_k=0 $$ for $k\neq j$).

  $$\displaystyle a_i $$: incident wave at port i, $$\displaystyle b_i $$: reflected wave.

• Properties:

  1. Reciprocity: $$\displaystyle S_{ij} = S_{ji} $$ for passive networks.

  2. Losslessness: $$\displaystyle \mathbf{S}^\dagger \mathbf{S} = \mathbf{I} $$ → $$\displaystyle \sum_{k=1}^n |S_{ki}|^2 = 1 $$ for each column $i$.

  3. Symmetry: For reciprocal networks, $\mathbf{S}$ symmetric.

• General Scattering Matrix for Reciprocal and Lossless Two-Port:

  From reciprocity: $$\displaystyle S_{12} = S_{21} $$.

  From losslessness:

  $$\displaystyle |S_{11}|^2 + |S_{21}|^2 = 1 $$ (port 1),

  $$\displaystyle |S_{22}|^2 + |S_{12}|^2 = 1 $$ (port 2).

  Thus $$\displaystyle S_{11} = S_{22} $$ (if symmetric), and $$\displaystyle |S_{11}|^2 + |S_{12}|^2 = 1 $$.

  \boxed{\mathbf{S} = \begin{bmatrix} S_{11} & S_{12} \ S_{12} & S_{11} \end{bmatrix}, \quad |S_{11}|^2 + |S_{12}|^2 = 1}.

C. Applications of S-Parameters

• Network Analysis/Design: Determine gain, stability, matching.

• Oscillator Design:

  • Oscillation Condition: When port 1 input reflection coefficient $$\displaystyle \Gamma_{in} $$ and feedback reflection coefficient $$\displaystyle \Gamma_f $$ satisfy $$\displaystyle |\Gamma_{in} \Gamma_f| \geq 1 $$ and $$\displaystyle \angle(\Gamma_{in} \Gamma_f) = 0 $$ (or $2\pi n$).

  • For two-port with output shorted ($$\displaystyle \Gamma_L = -1 $$), $$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}}{1 - S_{22}} $$.

• Measurement: VNA directly measures S-parameters.

D. S-Parameter Calculations and Examples

Example (Dec 2024):

Given: $$\displaystyle S_{11} = 0.2\angle 0^\circ $$, $$\displaystyle S_{22} = 0.1\angle 0^\circ $$, $$\displaystyle S_{12} = 0.6\angle 90^\circ $$, $$\displaystyle S_{21} = 0.6\angle 90^\circ $$.

(i) Reciprocity: $$\displaystyle S_{12} = S_{21} $$? Yes, both $$\displaystyle 0.6\angle 90^\circ $$. So reciprocal.
Losslessness? Check $$\displaystyle |S_{11}|^2 + |S_{21}|^2 = 0.04 + 0.36 = 0.4 \neq 1 $$. Not lossless.

(ii) Return Loss at Port 1 when Port 2 Short-Circuited:
$$\displaystyle \Gamma_L = -1 $$.
$$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}}{1 - S_{22}\Gamma_L} = 0.2 + \frac{(0.6\angle90)(0.6\angle90)}{1 - 0.1(-1)} = 0.2 + \frac{0.36\angle180}{1.1} = 0.2 - 0.327 = -0.127 $$.
$$\displaystyle |\Gamma_{in}| = 0.127 $$.

Return Loss (RL) = $$\displaystyle -20 \log_{10} |\Gamma_{in}| = \boxed{17.9\ \text{dB}} $$.

V. MEASUREMENT TECHNIQUES AND APPLICATIONS

A. Measurement Setups

• Slotted Line: As described in I.E.

• Power Measurement Bridges:

  • Bolometer: Measures RF power via temperature rise.

  • Thermocouple: RF heating generates thermoelectric voltage.

  • Diode Detector: Rectifies RF to DC, calibrated.

B. Detectors

• Tuned Detector: Resonant circuit (LC) selects specific frequency. Narrowband.

• Broadband Detector: Resistive (e.g., thermistor, diode), wide frequency range.

C. Microwave Applications

1. Radar:

   • Transmits pulse, receives echo. Time delay → range; Doppler shift → velocity.

   • Uses: Air traffic control, weather, military.

2. Communication:

   • Satellite links, cellular backhaul, point-to-point.

   • Advantages: Large bandwidth, small antennas, directional beams.

3. Heating: Microwave ovens (2.45 GHz) heat water molecules.

D. Special Topics

• MASER (Microwave Amplification by Stimulated Emission of Radiation):

  • Uses stimulated emission in ammonia or ruby.

  • Extremely low noise (quantum limited).

  • Applications: Radio astronomy, low-noise amplifiers.

• TWT Amplifier: High power (kW), wide bandwidth (octave), used in satellite transponders, radar.

• Microwave Resonators: Cavity (high Q), dielectric (compact), YIG (tunable).

VI. COMPARISON AND ADVANCED TOPICS

A. Comparison of Transmission Lines

B. Solid-State Microwave Sources

[!TIP] Modern systems use solid-state (Gunn, IMPATT, FET) for compactness; vacuum tubes (TWT, klystron) for high power/bandwidth.

Key Formulas Summary:

• $$\displaystyle Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}} $$

• $$\displaystyle \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} $$

• $$\displaystyle S = \frac{1+|\Gamma|}{1-|\Gamma|} $$

• $$\displaystyle \lambda_c = \frac{2}{\sqrt{(m/a)^2+(n/b)^2}} $$ (rect waveguide)

• $$\displaystyle Z_{\text{TM}} = \eta \sqrt{1 - (f_c/f)^2} $$

• $$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}}{1 - S_{22}\Gamma_L} $$

• Oscillation: $$\displaystyle |\Gamma_{in} \Gamma_f| \geq 1 $$, $$\displaystyle \angle(\Gamma_{in} \Gamma_f) = 0 $$

Common Pitfalls:

• Confusing TE/TM mode indices.

• Forgetting effective $\epsilon$ in microstrip.

• Misapplying VSWR formula for complex $\Gamma$.

• Ignoring that S-parameters require matched ports for definition.

• In oscillator design, forgetting phase condition.