UNIT 1: MICROWAVE ENGINEERING
I. TRANSMISSION LINES & WAVEGUIDES (Fundamentals)
A. Transmission Line Theory
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Primary Constants:
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R (Ω/m): Resistance per unit length (conductor loss).
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L (H/m): Inductance per unit length (magnetic energy storage).
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C (F/m): Capacitance per unit length (electric energy storage).
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G (S/m): Conductance per unit length (dielectric loss).
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Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta $$, where:
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$\alpha$ (Np/m): Attenuation constant.
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$\beta$ (rad/m): Phase constant.
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Characteristic Impedance: $$\displaystyle Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} $$ (Ω).
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Phase Velocity: $$\displaystyle v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}} $$ (for low-loss lines).
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Voltage Standing Wave Ratio (VSWR) and Reflection Coefficient (Γ):
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$$\displaystyle \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$.
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$$\displaystyle \text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$.
[!TIP] VSWR is always ≥ 1. For matched load ($$\displaystyle Z_L = Z_0 $$), Γ = 0, VSWR = 1.
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B. Planar Transmission Lines
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Microstrip Line:
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Structure: Conductor strip on dielectric substrate with ground plane.
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Effective Dielectric Constant (for $W/d \leq 1$):
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$$\varepsilon_{\text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12d/W}}$$
- Characteristic Impedance (for $W/d \leq 1$):
$$Z_0 = \frac{60}{\sqrt{\varepsilon_{\text{eff}}}} \ln\left(\frac{8d}{W} + \frac{W}{4d}\right)$$
[!CAUTION] Microstrip is quasi-TEM; dispersion increases with frequency.
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Slot Line: Inverted microstrip (slot in ground plane). Higher loss, used for series connections.
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Strip Line: Sandwiched between ground planes. Supports pure TEM mode as dominant; higher-order modes (TE/TM) exist at higher frequencies.
C. Waveguides
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Rectangular Waveguide ($a \times b$, $$\displaystyle a > b $$):
- Cutoff Wavelength for TE/TM$$\displaystyle _{mn} $$:
$$\lambda_c = \frac{2}{\sqrt{(m/a)^2 + (n/b)^2}}$$
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Dominant Mode: TE$$\displaystyle _{10} $$ ($$\displaystyle \lambda_c = 2a $$).
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Guided Wavelength: $$\displaystyle \lambda_g = \frac{\lambda_0}{\sqrt{1 - (\lambda_0/\lambda_c)^2}} $$.
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Wave Impedance (TM modes):
$$Z_{\text{TM}} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}}$$
where $$\displaystyle \eta = \sqrt{\mu_0/\varepsilon_0} $$ (intrinsic impedance of free space).
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Circular Waveguide (radius $r$):
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Dominant Mode: TE$$\displaystyle _{11} $$.
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Cutoff Wavelength: $$\displaystyle \lambda_c = \frac{2\pi r}{1.841} $$ (first zero of J$$\displaystyle _1' $$).
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For given $f$ and $r$, propagating modes satisfy $$\displaystyle f > f_c $$.
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Wave Impedance vs. Intrinsic Impedance: In waveguides, wave impedance is frequency-dependent and differs from $\eta$; for TE modes, $$\displaystyle Z_{\text{TE}} > \eta $$, for TM modes, $$\displaystyle Z_{\text{TM}} < \eta $$.
II. SCATTERING PARAMETERS (S-PARAMETERS)
A. Definition & Need
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Used at microwave frequencies because:
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Voltages/currents not well-defined due to distributed nature.
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Measurement of Z/Y parameters requires open/short circuits, impractical at high f (parasitics, resonances).
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S-parameters use incident/reflected waves, measured with matched terminations.
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General S-Matrix 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.
B. Derivation & Properties
- Reciprocal Two-Port (material symmetric):
$$S_{12} = S_{21}$$
- Lossless Two-Port (no power dissipation):
$$|S_{11}|^2 + |S_{12}|^2 = 1, \quad |S_{22}|^2 + |S_{21}|^2 = 1$$
- Symmetry for reciprocal networks: $$\displaystyle [S] = [S]^T $$.
C. Applications & Analysis
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Oscillator Design: Oscillation condition: $$\displaystyle |\Gamma_{in} \Gamma_s| \geq 1 $$ and $$\displaystyle \angle(\Gamma_{in} \Gamma_s) = 0 $$, where $$\displaystyle \Gamma_{in} $$ = input reflection coefficient of active device, $$\displaystyle \Gamma_s $$ = reflection coefficient of feedback network.
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Stability Factor (Rollett): $$\displaystyle K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2|S_{12}S_{21}|} $$, where $$\displaystyle \Delta = S_{11}S_{22} - S_{12}S_{21} $$. Network unconditionally stable if $$\displaystyle K > 1 $$ and $$\displaystyle |\Delta| < 1 $$.
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Return Loss (RL) when port 2 terminated in $$\displaystyle \Gamma_L $$:
$$\text{RL} = -20 \log_{10} |S_{11} + \frac{S_{12}S_{21}\Gamma_L}{1 - S_{22}\Gamma_L}|$$
III. PASSIVE MICROWAVE COMPONENTS
A. Matching Networks
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Need: Maximize power transfer, minimize reflections.
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Topologies: L-section, Π-section, T-section. Use Smith Chart for graphical design.
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Example: For $$\displaystyle Z_L = R_L + jX_L $$, normalize to $$\displaystyle z_L = Z_L/Z_0 $$, find matching point on Smith Chart, add series/shunt elements.
B. Impedance Transformers
- Quarter-Wave Transformer ($$\displaystyle \lambda_g/4 $$):
$$Z_{in} = \frac{Z_0^2}{Z_L}$$
Design: $$\displaystyle Z_0' = \sqrt{Z_0 Z_L} $$.
[!CAUTION] Bandwidth narrow; works only at center frequency $$\displaystyle f_0 $$ where $$\displaystyle \lambda_g/4 $$ is exact.
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Multi-Section Transformers:
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Binomial: Maximally flat response; $$\displaystyle Z_{0i} $$ from binomial expansion.
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Chebyshev: Equiripple passband; better bandwidth for same number of sections.
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Bandwidth ∝ number of sections.
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C. Directional Couplers & Hybrids
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Four-Port Directional Coupler:
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Coupling Factor (C): $$\displaystyle C = -20 \log_{10}|S_{21}| $$ (dB), power coupled to port 3.
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Directivity (D): $$\displaystyle D = C - I $$, where $I$ = isolation (dB) between ports 1–4.
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Ideal S-Matrix:
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$$[S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & j & 0 \\ 1 & 0 & 0 & j \\ j & 0 & 0 & 1 \\ 0 & j & 1 & 0 \end{bmatrix}$$
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Matched Hybrid Tee (Magic Tee):
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Combines E-plane and H-plane tees.
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Properties:
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Ports 1–2, 3–4 isolated.
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Equal power division (3 dB) from port 1 to ports 3 & 4, with 90° phase difference.
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S-Matrix:
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$$[S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \end{bmatrix}$$
D. Circulators & Isolators
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Circulator: Non-reciprocal 3-port; signal flows 1→2→3→1.
- Simplified S-Matrix:
$$[S] = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$
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Implementation: Magic Tee + phase shifter (90° in one arm).
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Isolator: Circulator with port 3 terminated in matched load; allows 1→2, blocks 2→1.
E. Slotted Line
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Construction: Coaxial/rectangular waveguide with movable short and probe detector.
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Measurements:
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VSWR: From voltage minima/maxima along line.
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Guided Wavelength $$\displaystyle \lambda_g $$: Distance between two minima.
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Impedance: Using Smith Chart with VSWR and distance to first minimum.
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IV. MICROWAVE TUBES (Vacuum Electron Devices)
A. Klystrons
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Two-Cavity Klystron Amplifier:
- Velocity Modulation: RF signal in input cavity modulates electron beam velocity → bunching → output cavity induces amplified RF.
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Reflex Klystron Oscillator:
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Electron beam passes through cavity, reflects off repeller, returns in phase.
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Mode Curve: Power vs. repeller voltage; modes correspond to different transit times.
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Used as low-power oscillator.
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B. Traveling Wave Tube (TWT)
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Slow-Wave Structure: Helix or coupled cavities to reduce phase velocity to match electron beam.
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Interaction: Continuous velocity modulation; RF wave and beam interact over entire length → amplification.
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Oscillator Configuration: With feedback (e.g., delayed line).
C. Magnetron
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Crossed Fields: $\vec{E} \perp \vec{B}$; electrons oscillate, bunch in cavities.
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Strapping: Reduces mode competition; ensures π-mode oscillation.
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Applications: Radar transmitters, microwave ovens.
V. SOLID-STATE MICROWAVE DEVICES
A. Gunn Diode
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Principle: Transferred Electron Effect (Gunn effect) in GaAs/InP.
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Domain Formation:
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Accumulation: High-field domain forms near cathode.
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Depletion: Domain propagates to anode.
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Quenching: Domain disappears at anode, cycle repeats.
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Modes: LSA (high power), transit-time, bias-circuit oscillations.
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Applications: X-band oscillators, amplifiers.
B. IMPATT & TRAPATT Diodes
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IMPATT: Impact ionization → avalanche multiplication → negative resistance. High power, high noise.
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TRAPATT: Trapped plasma avalanche triggered transit; higher efficiency than IMPATT.
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Frequency Limit: Set by avalanche buildup time and transit time.
C. Schottky Barrier Diode
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Metal-Semiconductor Junction: Barrier potential $$\displaystyle \phi_B $$.
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I-V Characteristic (thermionic emission): $$\displaystyle I = I_s (e^{qV/nkT} - 1) $$.
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Applications:
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Mixer: Nonlinear I-V for frequency conversion.
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Detector: Square-law region ($V \ll kT/q$) for power detection.
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D. Microwave BJT
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Structure: Emitter (heavily doped), thin base, collector.
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Frequency Limitations:
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Base transit time $$\displaystyle \tau_b = \frac{W_b^2}{2D_n} $$.
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Charge storage (diffusion capacitance).
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Miller effect ($$\displaystyle C_{cb} $$ multiplied by gain).
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Figure of Merit: $$\displaystyle f_T = \frac{1}{2\pi \tau} $$, where $\tau$ includes $$\displaystyle \tau_b $$, $$\displaystyle \tau_e $$, $$\displaystyle \tau_c $$.
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Applications: Low-noise amplifiers up to ~10 GHz.
E. Other Devices
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BARITT: Barrier injection and transit time; lower noise than IMPATT, lower power.
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Tunnel Diode: Negative resistance due to tunneling; used in high-speed oscillators.
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MASER: Microwave amplification by stimulated emission of radiation; uses paramagnetic crystal in magnetic field; ultra-low noise amplifier.
VI. MICROWAVE SYSTEMS & APPLICATIONS
A. Mixers
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Working: Nonlinear device (diode) mixes RF and LO → sum/difference frequencies (IF).
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Conversion Loss: $$\displaystyle L_c = \frac{P_{RF}}{P_{IF}} $$ (typically 6–9 dB); includes mismatch, conversion, and noise losses.
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Topologies: Single-ended, balanced (improves isolation).
B. Oscillators
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S-Parameter Design: Active device with $$\displaystyle |S_{11}| > 1 $$ (negative resistance) coupled to resonator.
- Oscillation condition: $$\displaystyle |\Gamma_{in} \Gamma_{res}| \geq 1 $$, $$\displaystyle \angle(\Gamma_{in} \Gamma_{res}) = 0 $$.
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Feedback Principle: Positive feedback at $$\displaystyle f_0 $$; loop gain ≥ 1, phase shift 0°.
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Resonators: YIG, cavity, dielectric.
C. Frequency Converters & Multipliers
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Converter: Schottky diode mixer with LO; outputs IF.
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Multiplier: Nonlinear device generates harmonics; $$\displaystyle f_{out} = n f_{in} $$; efficiency decreases with $n$.
D. Phase Shifters
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Diode Phase Shifter:
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Switching-type: Binary (0°/180°); uses transmission line sections.
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Variable capacitance-type: Continuous phase shift; varactor diodes.
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Broadband: Switched-line or loaded-line designs.
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Tuned: Resonant circuits; narrowband.
VII. MICROWAVE RESONATORS & MEASUREMENTS
A. YIG Resonator
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Structure: Yttrium Iron Garnet sphere in static magnetic field $$\displaystyle B_0 $$.
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Working: Ferromagnetic resonance; RF field excites spin precession at $$\displaystyle f = \frac{\gamma}{2\pi} B_0 $$, where $\gamma$ = gyromagnetic ratio.
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Tuning: Vary $$\displaystyle B_0 $$ → change $f$.
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Applications: Tunable filters, oscillators, frequency meters.
B. Microwave Resonators
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Types: Cavry (high Q), dielectric (low loss), planar (microstrip).
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Q-Factor: $$\displaystyle Q = \frac{f_0}{\Delta f} $$; measures energy storage vs. loss.
C. Power Measurement
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Bridges: Bolometer (temperature-sensitive resistance), thermistor.
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Detectors:
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Tuned: Narrowband, high sensitivity.
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Broadband: Wideband, lower sensitivity.
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D. Measurement Challenges
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Z/Y/h/ABCD Parameters: Require precise open/short/load standards; parasitics and resonances cause errors at microwave f.
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S-Parameters: Use matched terminations; directly measurable with vector network analyzer (VNA).
VIII. ADDITIONAL TOPICS (Short Notes)
A. Solid-State Sources Overview
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Gunn/IMPATT: High power, moderate noise; radar, transmitters.
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Schottky: Mixers, detectors; low noise, high speed.
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Gunn: Low-cost oscillators; automotive radar, local oscillators.
B. Device Comparisons
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Microstrip vs. Slot Line:
| Feature | Microstrip | Slot Line | |---------|------------|-----------| | Loss | Lower | Higher | | Dispersion | Moderate | Higher | | Integration | Easy (planar) | Difficult | | Connection | Shunt elements | Series elements |
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Strip Line Modes:
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Dominant: TEM (no cutoff, dispersionless).
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Higher-Order: TE/TM (cutoff frequencies depend on substrate thickness).
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C. Specialized Devices
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TWT Amplifier: Broadband, high power (kW); used in satellite comms, radar.
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Microwave Resonators: High Q essential for filters/oscillators; cavity resonators common.
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Power Measurement Bridges: Calorimetric (bolometer) or resistive (thermistor) for accurate average power.
[!EXAM TIP]
- VSWR/Γ: Always derive relationship from voltage maxima/minima.
- S-parameters: Remember lossless condition $$\displaystyle |S_{11}|^2 + |S_{12}|^2 = 1 $$.
- Waveguides: TE$$\displaystyle _{10} $$ dominant in rectangular; TE$$\displaystyle _{11} $$ in circular.
- Gunn Diode: Explain domain formation clearly.
- Magic Tee: S-matrix shows 3 dB split and 90° phase difference.
- YIG: Tuning via magnetic field, not electrical.
- IMPATT: High noise due to avalanche process.