UNIT 1: Microwave Engineering - Comprehensive Short Notes

I. Introduction to Microwaves

Definition: Microwaves are electromagnetic waves with frequencies ranging from 300 MHz to 300 GHz, corresponding to wavelengths from 1 m to 1 mm.

Key Properties:

• Short wavelength → enables compact components and antennas.

• Quasi-optical behavior → can be guided by waveguides or propagate in free space; exhibit reflection, refraction, polarization.

• Line-of-sight propagation → limited diffraction, requires repeater stations for long distances.

• High bandwidth → supports wideband communication and high data rates.

• Low latency → due to high propagation speed (speed of light in medium).

Major Applications:

[!TIP] Exam Focus: Remember the exact frequency range (300 MHz – 300 GHz) and be prepared to list at least 4 applications with brief explanations.

II. Transmission Line Theory

Telegrapher’s Equations

For a distributed transmission line with primary constants R, L, G, C (per unit length):

$$\frac{\partial V(x,t)}{\partial x} = -R I(x,t) - L \frac{\partial I(x,t)}{\partial t}$$

$$\frac{\partial I(x,t)}{\partial x} = -G V(x,t) - C \frac{\partial V(x,t)}{\partial t}$$

Secondary Constants

Lossless Transmission Line ($$\displaystyle R=0, G=0 $$)

• $$\displaystyle \gamma = j\omega\sqrt{LC} = j\beta $$ → $$\displaystyle \alpha = 0 $$

• $$\displaystyle Z_0 = \sqrt{L/C} $$ (purely real)

• $$\displaystyle v_p = 1/\sqrt{LC} $$

• Wave equation: $$\displaystyle V(x) = V^+ e^{-j\beta x} + V^- e^{j\beta x} $$

Standing Waves & VSWR

• Reflection coefficient at load: $$\displaystyle \Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0} $$

• Voltage Standing Wave Ratio (VSWR):

$$S = \frac{1 + |\Gamma_L|}{1 - |\Gamma_L|}$$

[!TIP] Derivation is frequently asked. Start from total voltage $$\displaystyle V(x) = V^+(e^{-j\beta x} + \Gamma_L e^{j\beta x}) $$, find $$\displaystyle |V|_{max/min} $$, then ratio.

III. Transmission Line Structures

A. Microstrip Line

Structure: Conductor strip on dielectric substrate with ground plane.

Effective Dielectric Constant $$\displaystyle \varepsilon_{eff} $$

$$\varepsilon_{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12h/W}} \quad \text{(for } W/h \geq 1\text{)}$$

For $$\displaystyle W/h < 1 $$, use:

$$\varepsilon_{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \left[ \frac{1}{\sqrt{1+12h/W}} + 0.04\left(1 - \frac{W}{h}\right)^2 \right]$$

Characteristic Impedance $$\displaystyle Z_0 $$

For $W/h \geq 1$:

$$Z_0 = \frac{60}{\sqrt{\varepsilon_{eff}}} \ln\left(\frac{8h}{W} + \frac{W}{4h}\right)$$

For $$\displaystyle W/h < 1 $$:

$$Z_0 = \frac{120\pi}{\sqrt{\varepsilon_{eff}} \left[ \frac{W}{h} + 1.393 + 0.667\ln\left(\frac{W}{h} + 1.444\right) \right]}$$

Advantages: Easy to fabricate, integrate active devices, low cost. Disadvantages: Radiation loss, dispersion, limited power handling. Applications: MICs (Microwave Integrated Circuits), antennas, filters.

B. Stripline

Structure: Center conductor sandwiched between two ground planes in dielectric. Dominant Mode: TEM (no cutoff frequency). Characteristic Impedance:

$$Z_0 = \frac{30\pi}{\sqrt{\varepsilon_r}} \frac{1 - \frac{b}{W}}{1 + \frac{b}{W}} \quad \text{(for } b/W \geq 0.35\text{)}$$

where $b$ = spacing between ground planes, $W$ = strip width. Types: Sandwich (centered), bilateral (offset), unilateral (one side only).

C. Slot Line

Structure: Slot in ground plane on dielectric substrate; opposite side has transmission line. Field: $$\displaystyle E_z $$ and $$\displaystyle H_x $$, $$\displaystyle H_y $$ dominant (quasi-TE). Comparison with Microstrip:

D. Other Structures (Brief)

• Coaxial cable: TEM mode, shielded, low loss, used up to ~20 GHz.

• Two-wire line: Balanced, TEM, used for HF/VHF.

• Coplanar Waveguide (CPW): Conductor on top, ground planes on sides; easy for active device mounting.

IV. Waveguides

A. Rectangular Waveguide (a × b, a > b)

Modes: TE$$\displaystyle _{mn} $$ ($$\displaystyle E_z=0 $$), TM$$\displaystyle _{mn} $$ ($$\displaystyle H_z=0 $$). No TEM mode.

Cutoff Wavelength & Frequency

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

$$f_c = \frac{c}{2\sqrt{\varepsilon_r}} \sqrt{(m/a)^2 + (n/b)^2}$$

Dominant mode: TE$$\displaystyle _{10} $$ (lowest $$\displaystyle f_c $$).

Phase & Group Velocity

$$\lambda_g = \frac{\lambda_0}{\sqrt{1 - (f_c/f)^2}}$$

$$v_p = \frac{c}{\sqrt{\varepsilon_r} \sqrt{1 - (f_c/f)^2}} = f \lambda_g$$

$$v_g = \frac{c}{\sqrt{\varepsilon_r}} \sqrt{1 - (f_c/f)^2}$$

$$v_p v_g = \frac{c^2}{\varepsilon_r}$$

Wave Impedance

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

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

where $$\displaystyle \eta = \sqrt{\mu_0/\varepsilon_0} \approx 377\ \Omega $$ (free space).

Example (TM$$\displaystyle _{11} $$ mode): For a × b = 3 cm × 2 cm, $$\displaystyle f=10 $$ GHz: $$\displaystyle f_c = \frac{3\times10^8}{2\sqrt{\varepsilon_r}} \sqrt{(1/0.03)^2 + (1/0.02)^2} \approx 9.15 $$ GHz (air-filled). $$\displaystyle Z_{TM11} = \eta \sqrt{1 - (9.15/10)^2} \approx 377 \times 0.44 \approx 166\ \Omega $$.

B. Circular Waveguide (radius r)

Modes: TE$$\displaystyle _{0n} $$, TM$$\displaystyle _{0n} $$, hybrid modes. Dominant mode: TE$$\displaystyle _{11} $$ (lowest cutoff).

TE$$\displaystyle _{11} $$ Mode

Cutoff wavelength: $$\displaystyle \lambda_c = \frac{2\pi r}{1.841} $$ (first root of J$$\displaystyle _1' $$)

Cutoff frequency: $$\displaystyle f_c = \frac{1.841c}{2\pi r \sqrt{\varepsilon_r}} $$

Guided wavelength: $$\displaystyle \lambda_g = \frac{\lambda_0}{\sqrt{1 - (f_c/f)^2}} $$

Finding all possible modes (Example): r = 2 cm, f = 10 GHz (air-filled).

• TE$$\displaystyle _{mn} $$: $$\displaystyle f_{cmn} = \frac{X'_{mn}}{2\pi r} c $$ (X' = roots of J$$\displaystyle _m' $$)

• TM$$\displaystyle _{mn} $$: $$\displaystyle f_{cmn} = \frac{X_{mn}}{2\pi r} c $$ (X = roots of J$$\displaystyle _m $$)

Calculate $$\displaystyle f_c $$ for m,n = 0,1,2,... and list modes with $$\displaystyle f_c < 10 $$ GHz.

C. Waveguide Components (Brief)

• Bends: E-plane (90°), H-plane (90°); cause higher-order mode excitation.

• Twists: Rotate polarization; gradual to avoid radiation.

• Tapers: Impedance matching between different waveguide sizes.

• Windows: Seals for pressurized systems; thin dielectric sheets.

V. Network Analysis (S-parameters)

Need for S-parameters

At microwave frequencies:

• Voltage & current not well-defined (distributed, standing waves).

• Open/short circuits impractical (parasitics, resonances).

• S-parameters use incident & reflected waves → measured with matched terminations.

Definition for N-port

$$b_i = \sum_{j=1}^{N} S_{ij} a_j$$

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

S-matrix Properties

1. Reciprocity: $$\displaystyle S_{ij} = S_{ji} $$ (for reciprocal networks).

2. Losslessness: $$\displaystyle \mathbf{S}^H \mathbf{S} = \mathbf{I} $$ (unitary matrix).

3. Symmetry: For reciprocal networks, S is symmetric.

Two-Port Reciprocal & Lossless Network

Given $$\displaystyle S_{11} = S_{22} $$ (reciprocal) and $$\displaystyle |S_{11}|^2 + |S_{21}|^2 = 1 $$ (lossless).

General form:

$$\mathbf{S} = \begin{bmatrix} S_{11} & S_{12} \\ S_{12} & S_{11} \end{bmatrix}$$

with $$\displaystyle |S_{11}|^2 + |S_{12}|^2 = 1 $$.

Example (Dec 2024 Q13): Given $$\displaystyle S_{11}=0.2\angle0^\circ $$, $$\displaystyle S_{22}=0.1\angle0^\circ $$, $$\displaystyle S_{12}=S_{21}=0.6\angle90^\circ $$.

• Reciprocal? Yes ($$\displaystyle S_{12}=S_{21} $$).

• Lossless? Check $$\displaystyle |S_{11}|^2+|S_{12}|^2 = 0.04+0.36=0.4 \neq 1 $$ → not lossless.

Measurement

• Use network analyzer with calibrated standards (SOLT: Short, Open, Load, Thru).

• Measure $$\displaystyle S_{11} $$, $$\displaystyle S_{21} $$, $$\displaystyle S_{12} $$, $$\displaystyle S_{22} $$ at each frequency.

Applications

• Design of matching networks, amplifiers, oscillators.

• Stability analysis (K-factor).

• Component characterization.

VI. Impedance Matching

Need for Matching

• Maximum power transfer: $$\displaystyle Z_L = Z_0^* $$.

• Minimize reflection → reduce VSWR, standing waves, losses.

• Improve signal-to-noise ratio and system linearity.

A. Matching Networks

L-section (Lumped)

Two reactive elements (L or C) in L-configuration.

Design: Given $$\displaystyle Z_L = R_L + jX_L $$, choose series/shunt element to transform to $$\displaystyle Z_0 $$.

[!TIP] Two solutions: (1) Series element to cancel $$\displaystyle X_L $$, shunt to adjust $$\displaystyle R_L $$; (2) Shunt first, then series.

Single-stub Matching

Use shorted or opened stub at distance $d$ from load.

1. Calculate $$\displaystyle \Gamma_L $$, plot on Smith Chart.

2. Move towards generator to $$\displaystyle |\Gamma| = |\Gamma_{in}| $$.

3. Add stub to cancel susceptance/susceptance.

B. Impedance Transformers

Quarter-wave Transformer

$$Z_{01} = \sqrt{Z_0 Z_L}$$

Only perfect at $$\displaystyle f_0 $$; bandwidth limited by:

$$\frac{\Delta f}{f_0} \approx \frac{4}{\pi} \sin^{-1}\left(\frac{|Z_L - Z_0|}{Z_L + Z_0}\right)$$

Multi-section Transformers

• Binomial: Maximally flat response; $$\displaystyle Z_{0i} $$ from binomial coefficients.

• Chebyshev: Equiripple passband; better bandwidth for same number of sections. Bandwidth Enhancement: More sections → wider bandwidth; Chebyshev gives sharper cutoff.

C. Hybrid Tee (Matched Hybrid Tee / Magic Tee)

Construction: Combination of E-plane tee (series) and H-plane tee (shunt) with matched ports.

S-matrix Derivation (Dec 2025 Q11)

Properties:

• Isolation: Port 1–2, 3–4 isolated → $$\displaystyle S_{12}=S_{21}=S_{34}=S_{43}=0 $$.

• Equality: $$\displaystyle |S_{13}| = |S_{14}| $$, $$\displaystyle |S_{23}| = |S_{24}| $$.

• Sum/difference: Signals at ports 3,4 add/subtract at ports 1,2.

Ideal S-matrix:

$$\mathbf{S} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 \\ 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{bmatrix}$$

Applications: Mixers, couplers, power dividers, antenna arrays.

VII. Measurement Techniques

A. Slotted Line

Construction: Coaxial or waveguide with axial slot; movable probe detects electric field.

Working: Probe samples $|V|$ along line; records standing wave pattern.

Measurements:

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

2. Guided wavelength $$\displaystyle \lambda_g $$: Distance between two $$\displaystyle V_{max} $$ (or $$\displaystyle V_{min} $$) points.

3. Load impedance:

   • Find $$\displaystyle \Gamma_L = \frac{S-1}{S+1} e^{-j2\beta d} $$ (d = distance from load to first min/max).

   • Use Smith Chart or calculate $$\displaystyle Z_L = Z_0 \frac{1+\Gamma_L}{1-\Gamma_L} $$.

B. VSWR & Reflection Coefficient

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

C. Power Measurement

VIII. Microwave Components

A. Directional Coupler

Four-port: Input (1), Through (2), Coupled (3), Isolated (4).

• Coupling factor (C): $$\displaystyle C = -20\log|S_{21}| $$ (dB) or $$\displaystyle -20\log|S_{31}| $$.

• Directivity (D): $$\displaystyle D = 20\log\left|\frac{S_{31}}{S_{41}}\right| $$ (dB).

Ideal S-matrix (symmetric coupler):

$$\mathbf{S} = \begin{bmatrix} 0 & \tau & j\kappa & 0 \\ \tau & 0 & 0 & j\kappa \\ j\kappa & 0 & 0 & \tau \\ 0 & j\kappa & \tau & 0 \end{bmatrix}$$

with $$\displaystyle |\tau|^2 + |\kappa|^2 = 1 $$.

Types: Waveguide (aperture coupling), microstrip (edge/coupled lines), branch-line (quadrature hybrid).

B. Circulator

Symbol: Triangle with arrows showing direction (1→2, 2→3, 3→1). S-matrix (3-port):

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

Working (using magic tee + phase shifter): Combine E-plane and H-plane tees with 90° phase shift to achieve non-reciprocal rotation. Applications: Duplexers (transmit/receive), isolators, mixer LO distribution.

C. Isolator

Principle: Ferrite material in magnetic field → non-reciprocal rotation (Faraday rotation). Signal passes forward, absorbed in termination for reverse. Types:

• Resonance absorption: Ferrite rod at waveguide wall; resonance at specific frequency.

• Field displacement: Ferrite slab shifts field to lossy wall for reverse direction.

D. Phase Shifters

Diode phase shifter: Uses varactor (voltage-controlled capacitance) or PIN diode (switches transmission line lengths).

• Broadband: Switched-line (digital) or loaded-line (analog).

• Tuned: Resonant circuits (narrowband).

E. YIG Resonator

Structure: Yttrium Iron Garnet (YIG) sphere; coupling loops; DC magnetic bias.

Working: Magnetic bias sets resonance frequency $$\displaystyle f_0 \propto B_0 $$. Input RF excites spin precession; energy transferred to output if $$\displaystyle f_{RF} = f_0 $$. Tuning: Wide tuning range (2–20 GHz) by varying $$\displaystyle B_0 $$; high Q (~1000–10000). Applications: Tunable filters, oscillators, frequency discriminators.

F. Microwave Resonators (General)

• Cavity resonators: Rectangular/circular; high Q; used in oscillators, filters.

• Dielectric resonators: Low loss ceramic; high Q; compact.

• Q-factor: $$\displaystyle Q = \frac{f_0}{\Delta f} = \frac{\text{stored energy}}{\text{power loss per cycle}} $$.

• Tuning: Mechanical (screw), dielectric tuning, varactor.

IX. Microwave Tubes

A. Klystron Amplifiers

Two-Cavity Klystron

Construction: Electron gun → buncher cavity → catcher cavity → collector. Velocity Modulation: RF signal in buncher modulates electron velocity → velocity modulated beam → density modulation (bunching) → energy extracted in catcher. Output: Amplified RF at catcher frequency.

Reflex Klystron (Dec 2025 Q2)

Construction: Cathode → repeller (negatively biased) → single cavity → collector. Working: Electrons accelerated into cavity; reflected by repeller; return to cavity in phase → oscillations. Mode Curve: Power vs. repeller voltage; multiple modes (n, n+1/2, n+1). Mode chosen by voltage.

B. Traveling Wave Tube (TWT) (Dec 2025 Q3)

Construction: Electron gun → helix slow-wave structure → collector. Interaction: Continuous velocity modulation → electron bunching → RF energy transfer to helix via continuous interaction. Gain: Broadband (octave bandwidth); high gain (40–50 dB). Types: Helix TWT (wideband), coupled-cavity TWT (high power). Applications: Satellite amplifiers, radar transmitters, EW systems.

C. Magnetron

Types: Cylindrical, coaxial, rising-sun. Oscillation Mechanism: π-mode (alternating anode vanes in-phase). Electrons interact with RF fields → bunching → oscillations. Construction: Cathode, anode block (resonant cavities), magnets (permanent/electro). Applications: Radar transmitters, microwave ovens.

D. MASER

Principle: Microwave Amplification by Stimulated Emission of Radiation. Uses population inversion in ammonia or ruby; stimulated emission → amplification. Applications: Ultra-low-noise amplification (radio astronomy, deep-space comms).

X. Solid-State Microwave Devices

A. Gunn Diode (Dec 2025 Q4, Nov 2023 Q11)

Principle: Transferred electron effect in GaAs/InP. Two valleys in conduction band: low-field (high mobility), high-field (low mobility). Above threshold, domain forms and propagates. Domains:

1. Below threshold: Stable amplification (small signal).

2. Above threshold: Stable oscillation (Gunn oscillator).

3. LSA mode: Limited space-charge accumulation; high power, high frequency. Applications: Oscillators (10–100 GHz), amplifiers.

B. IMPATT Diode (Dec 2025 Q16, Dec 2024 Q11, Nov 2023 Q18)

Principle: Impact ionization and transit time → negative resistance. Avalanche breakdown creates carriers; they drift through high-field region → current lags voltage. I-V: High breakdown voltage; negative resistance region. Types: Single-drift (Si), double-drift (GaAs, higher power). Applications: High-power oscillators/amplifiers (mm-wave), frequency multipliers.

C. TRAPATT Diode (Dec 2024 Q11)

Principle: Trapping and avalanche transit time. Carriers trapped in high-field region → avalanche → plasma formation → high efficiency. Efficiency: Higher than IMPATT (20–30% vs. 10–15%). Applications: High-efficiency oscillators.

D. BARITT Diode (Dec 2024 Q14)

Principle: Barrier injection and transit time. Uses p-n-p or n-p-n structure; injection limited by barrier → gradual carrier injection → less noise. Noise: Lower than IMPATT; power lower. Applications: Low-noise oscillators.

E. Schottky Barrier Diode (Dec 2025 Q5, Nov 2023 Q10)

Structure: Metal-semiconductor junction (e.g., Au-GaAs). Characteristics: No minority carrier storage → fast response, low capacitance. Uses:

• Mixer: Nonlinear I-V for frequency conversion (RF + LO → IF).

• Detector: Envelope detection (zero-bias operation possible).

F. Microwave BJT (Dec 2025 Q6, Nov 2023 Q9)

Structure: Emitter-base-collector; vertical or lateral. Frequency Limitations:

• Base resistance $$\displaystyle r_b $$ and capacitances ($$\displaystyle C_{be} $$, $$\displaystyle C_{bc} $$) → RC time constant.

• Transit time across base → $$\displaystyle f_T = \frac{1}{2\pi \tau_{ec}} $$. Applications: Low-noise amplifiers (LNA), oscillators up to ~10 GHz.

G. Microwave FET

Small-signal model: Gate-source capacitance $$\displaystyle C_{gs} $$, gate-drain capacitance $$\displaystyle C_{gd} $$ (Miller effect), channel resistance. Gain: $$\displaystyle G = \frac{g_m}{2\pi f (C_{gs} + C_{gd}(1+g_m R_D))} $$. Types: MESFET (GaAs), HEMT (AlGaAs/GaAs) → high $$\displaystyle f_T $$ (>100 GHz). Advantages over BJT: High input impedance, better high-frequency performance, easier integration.

H. Tunnel Diode (Nov 2023 Q11)

Modes:

1. Oscillation: Bias in negative resistance region ($$\displaystyle V_p < V < V_v $$).

2. Amplification: Bias near peak point ($$\displaystyle V \approx V_p $$).

3. Switching: Bias beyond valley point ($$\displaystyle V > V_v $$). Applications: High-speed oscillators, amplifiers, switching circuits.

XI. Mixers and Detectors

A. Microwave Mixer

Working: Nonlinear device (diode, FET) multiplies RF and LO → sum/difference frequencies. Signals:

• RF: $$\displaystyle f_{RF} $$ (input).

• LO: $$\displaystyle f_{LO} $$ (local oscillator, high power).

• IF: $$\displaystyle |f_{RF} \pm f_{LO}| $$ (intermediate frequency, filtered output). Conversion loss: $$\displaystyle L_c = \frac{P_{RF}}{P_{IF}} $$ (typically 6–9 dB for diode mixer). Types: Single-ended (simple, high loss), balanced (LO-RF isolation), image-reject (filters image).

B. Detectors

Principle: Rectification using diode nonlinearity.

• Broadband: Zero-bias Schottky diode; no tuned circuit → wide bandwidth, low sensitivity.

• Tuned: Diode with LC tank → narrowband, high sensitivity; used in receivers.

XII. Oscillators and Amplifiers

A. Oscillators Using S-parameters

Oscillation Condition (Kurokawa):

$$|\Gamma_{in} \Gamma_s| \geq 1, \quad \angle \Gamma_{in} + \angle \Gamma_s = 0^\circ$$

where $$\displaystyle \Gamma_s $$ = source reflection coefficient, $$\displaystyle \Gamma_{in} $$ = input reflection coefficient of active device. Design:

1. Choose frequency where active device has $$\displaystyle |S_{11}| > 1 $$ (unstable).

2. Add feedback network (e.g., resonator) to satisfy condition.

3. Verify with stability factor $$\displaystyle K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2|S_{12}S_{21}|} < 1 $$ for unconditional stability; oscillator requires $$\displaystyle K < 1 $$ and $$\displaystyle |S_{11}S_{22} - \Delta| < 1 $$.

Examples: Gunn diode oscillator (parallel feedback), IMPATT oscillator (cavity or transmission line feedback).

XIII. Special Topics

A. Frequency Multipliers

Derivation: Nonlinear device generates harmonics: $$\displaystyle i(t) = a_0 + a_1 v(t) + a_2 v^2(t) + ... $$

If $$\displaystyle v(t) = V_0 \cos \omega t $$, then $$\displaystyle v^2(t) \propto \cos 2\omega t $$ → doubler. Principle: Input at $$\displaystyle f_{in} $$ → filter selects $$\displaystyle n \cdot f_{in} $$ harmonic. Types: Doublers ($$\displaystyle n=2 $$), triplers ($$\displaystyle n=3 $$); use Schottky diodes or varactors.

B. Power Measurement Bridges

• Bolometer bridge: Bolometer (thin resistor) in bridge; RF power heats → resistance change → bridge imbalance.

• Thermistor bridge: Similar with thermistor; more robust.

• Calorimeter: RF power heats fluid; measure temperature rise → accurate DC-equivalent power.

C. Additional Short Notes (From Past Papers)

TEM Mode (Nov 2023 Q16)

• Transverse ElectroMagnetic: $$\displaystyle E_z = 0 $$, $$\displaystyle H_z = 0 $$; all fields transverse.

• Exists only in two-conductor systems (coax, two-wire, stripline).

• No cutoff frequency → propagates at DC.

• $$\displaystyle \varepsilon_{eff} = \varepsilon_r $$, $$\displaystyle Z_0 = \sqrt{L/C} $$.

S-parameters & Applications (Dec 2025 Q17)

• Applications: Design of amplifiers (gain, stability), oscillators (oscillation condition), filters (insertion loss), antenna matching (reflection coefficient), network characterization.

Diode Phase Shifters (Dec 2025 Q18)

• PIN diode phase shifter: Switchable transmission line sections; digital phase states.

• Varactor phase shifter: Voltage-controlled capacitance in transmission line → continuous phase shift.

• Broadband: Switched-line (multiple paths).

• Tuned: Resonant circuits (narrowband, high phase shift).

Microwave Resonators (Dec 2024 Q16)

• Cavity: Metallic enclosure; high Q; used in oscillators, filters.

• Dielectric: Ceramic puck; lower loss than air; compact.

• Q-factor: $$\displaystyle Q = \frac{\omega_0 \cdot \text{stored energy}}{\text{power loss}} $$.

• Tuning: Mechanical (screw), dielectric tuning, varactor.

Power Measurement Bridges (Dec 2024 Q16)

• Bolometer bridge: Most accurate; uses thermally sensitive resistor.

• Thermistor bridge: Similar; thermistor has negative temp coefficient.

• Calorimeter: Absolute measurement; wide bandwidth.

TWT Amplifier (Dec 2024 Q15)

• Broadband amplification (octave bandwidth).

• High gain (40–50 dB), moderate power (10s of watts to kW).

• Slow-wave structure (helix) reduces phase velocity to match electron beam.

• Applications: Satellite transponders, radar, electronic warfare.

Final Exam Strategy:

1. Derivations: Practice microstrip $$\displaystyle \varepsilon_{eff} $$, $$\displaystyle Z_0 $$; VSWR from $\Gamma$; S-matrix for magic tee; two-port reciprocal/lossless S-matrix.

2. Formulas: Memorize key formulas for $\gamma$, $$\displaystyle Z_0 $$, $$\displaystyle \lambda_g $$, $$\displaystyle f_c $$ (rect/circ waveguide), VSWR, coupling/directivity.

3. Diagrams: Draw microstrip cross-section, magic tee, slotted line, klystron, TWT, YIG resonator.

4. Comparisons: Microstrip vs. slot line; TE vs. TM; lumped vs. distributed matching; circulator vs. isolator.

5. Applications: Link each device/component to at least 2 real-world uses.

[!CAUTION] Common Pitfalls:

• Confusing phase velocity $$\displaystyle v_p $$ and group velocity $$\displaystyle v_g $$ in waveguides.

• Forgetting that TEM mode does not exist in hollow waveguides.

• Misapplying VSWR formula: $$\displaystyle S = (1+|\Gamma|)/(1-|\Gamma|) $$ always.

• Assuming S-parameters are measured with open/short loads (they use matched loads).