Microwave Engineering (EC-702 (A)) - Unit 4 Short Notes

UNIT 4: Microwave Engineering

1. Introduction to Microwaves and Applications

Microwaves are electromagnetic waves with frequencies ranging from 300 MHz to 300 GHz (wavelengths 1 m to 1 mm). They occupy the upper end of the radio frequency spectrum.

Key Applications:

• Radar Systems: Object detection, speed measurement, weather monitoring.

• Satellite Communication: Uplink/downlink, TV broadcasting, global positioning.

• Microwave Ovens: Dielectric heating of food (typically 2.45 GHz).

• Medical Therapy: Diathermy for deep tissue heating, cancer treatment (hyperthermia).

• Remote Sensing: Atmospheric profiling, earth resource mapping, military surveillance.

[!TIP]

Exam Focus: Be prepared to list applications and briefly explain two with frequency bands and principles (e.g., radar uses pulse modulation, satellite uses geostationary orbits).

2. Transmission Lines and Guided Wave Structures

2.1 General Transmission Lines

Primary Constants (per unit length):

• $R$: Series resistance (Ω/m)

• $L$: Series inductance (H/m)

• $G$: Shunt conductance (S/m)

• $C$: Shunt capacitance (F/m)

Secondary Constants:

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

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

• Attenuation Constant: $$\displaystyle \alpha = \text{Re}\{\gamma\} $$ (Np/m)

• Phase Constant: $$\displaystyle \beta = \text{Im}\{\gamma\} $$ (rad/m)

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

For low-loss lines ($R \ll \omega L$, $G \ll \omega C$):

$$Z_0 \approx \sqrt{\frac{L}{C}}, \quad \gamma \approx j\omega\sqrt{LC}, \quad \alpha \approx \frac{R}{2Z_0} + \frac{GZ_0}{2}, \quad v_p \approx \frac{1}{\sqrt{LC}}$$

[!TIP]

Common Pitfall: Forgetting that $$\displaystyle Z_0 $$ is complex when losses are significant. In exams, if $R$ and $G$ are given, compute full expression.

2.2 Microstrip Lines

Structure: Conductor strip on dielectric substrate with ground plane.

Effective Dielectric Constant ($$\displaystyle \varepsilon_{\text{eff}} $$):

Accounts for fringing fields in air and dielectric.

For $W/h \leq 1$:

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

For $W/h \geq 1$:

$$\varepsilon_{\text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2\sqrt{1 + 12h/W}}$$

Characteristic Impedance ($$\displaystyle Z_0 $$):

For $W/h \leq 1$:

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

For $W/h \geq 1$:

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

Factors Affecting:

• Higher $$\displaystyle \varepsilon_r $$ → lower $$\displaystyle Z_0 $$, lower $$\displaystyle v_p $$.

• Larger $W/h$ → lower $$\displaystyle Z_0 $$.

• Frequency dependence due to dispersion.

[!TIP]

Exam Derivation: Show how $$\displaystyle \varepsilon_{\text{eff}} $$ is derived from capacitance equivalence (air + dielectric). Memorize formulas for $W/h \leq 1$ and $\geq 1$.

2.3 Strip Lines

Structure: Two parallel ground planes with a central conductor strip sandwiched between dielectric layers.

Dominant Mode: Quasi-TEM (transverse electromagnetic, no cutoff frequency).

Higher-Order Modes: TE, TM modes with cutoff frequencies. First higher-order mode is usually TE₀₁ or TM₁₀ depending on dimensions.

Field Distribution: Concentrated between conductors; minimal radiation.

Impedance Characteristics: $$\displaystyle Z_0 $$ depends on strip width $W$, dielectric thickness $b$, and $$\displaystyle \varepsilon_r $$. Lower $$\displaystyle Z_0 $$ compared to microstrip for same $W$ due to better field confinement.

[!TIP]

Comparison: Strip line has lower loss and better shielding than microstrip but harder to fabricate (requires drilling for connections).

2.4 Waveguide Analysis

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

• TM₁₁ Mode:

  • Cutoff wavelength: $$\displaystyle \lambda_c = \frac{2}{\sqrt{(1/a)^2 + (1/b)^2}} $$

  • Wave Impedance: $$\displaystyle Z_{\text{TM}} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$, where $$\displaystyle \eta = \sqrt{\mu_0/\varepsilon_0} $$

  • Field Patterns: $$\displaystyle E_z \neq 0 $$, $$\displaystyle H_z = 0 $$; $$\displaystyle E_x $$, $$\displaystyle E_y $$, $$\displaystyle H_z $$ vary sinusoidally across cross-section.

Circular Waveguide (radius $r$):

• TEₘₙ Modes:

  • Cutoff wavelength: $$\displaystyle \lambda_c = \frac{2\pi r}{\chi'_{mn}} $$ (first root of derivative of Bessel function)

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

  • Cutoff frequency: $$\displaystyle f_c = \frac{\chi'_{mn}}{2\pi r \sqrt{\mu\varepsilon}} $$

• TMₘₙ Modes:

  • $$\displaystyle \lambda_c = \frac{2\pi r}{\chi_{mn}} $$ (roots of Bessel function $$\displaystyle J_m $$)

• Dominant Mode: TE₁₁ (lowest cutoff).

[!TIP]

Exam Problem: Given $a$, $b$, $f$, find $$\displaystyle Z_{\text{TM11}} $$ by computing $$\displaystyle f_c $$, then $$\displaystyle Z_{\text{TM}} $$. For circular, identify possible modes by checking $$\displaystyle f > f_c $$.

2.5 Transmission Line Modes

TEM Mode (Transverse Electromagnetic):

• $$\displaystyle E_z = H_z = 0 $$; fields entirely transverse.

• Propagation constant $$\displaystyle \gamma = j\beta $$, no cutoff frequency ($$\displaystyle f_c = 0 $$).

• Occurs in two-wire lines, coaxial cables.

• Phase velocity $$\displaystyle v_p = 1/\sqrt{LC} $$, independent of frequency (non-dispersive).

• Significance: Simplifies analysis; used in low-frequency RF.

[!TIP]

Key Point: Waveguides (rectangular, circular) cannot support TEM; they support TE/TM only.

2.6 Microstrip vs. Slot Line

[!TIP]

Exam Comparison: Slot line is the complementary structure; $$\displaystyle Z_{0,\text{slot}} \approx \frac{\eta^2}{4Z_{0,\text{microstrip}}} $$ for same dimensions.

3. Active Microwave Devices

3.1 Vacuum Tubes

3.1.1 Reflex Klystron

• Construction: Cathode → resonant cavity (buncher & catcher) → repeller → anode.

• Velocity Modulation: Electron beam accelerated by voltage $$\displaystyle V_a $$, passes through buncher cavity gap where RF field modulates velocity.

• Bunching: Electrons with higher velocity catch up with slower ones, forming dense bunches at catcher cavity.

• Mode Curve: Plot of output power vs. repeller voltage $$\displaystyle V_r $$. Peaks correspond to electron flight time = integer multiple of RF period.

• Applications: Low-power local oscillator (1–50 GHz, ~10 mW).

[!TIP]

Diagram: Show electron paths, buncher/catcher gaps, repeller voltage negative relative to cathode.

3.1.2 Traveling Wave Tube (TWT)

• Structure: Electron gun → helix (slow-wave structure) → collector.

• Interaction Mechanism: RF signal travels along helix at $$\displaystyle v_p \approx c $$; electron beam travels faster ($$\displaystyle v_e > v_p $$). Axial electric field from helix velocity-modulates beam, forming bunches. Bunched beam induces current on helix, amplifying RF.

• Amplifier vs. Oscillator: With feedback (e.g., internal reflections) → oscillator.

• Advantages: Wide bandwidth (octave), high gain (40–50 dB), moderate power (watts to kilowatts).

[!TIP]

Key Formula: Synchronous condition $$\displaystyle v_e \approx v_p $$; helix reduces $$\displaystyle v_p $$ to match electron velocity.

3.1.3 Magnetron

• Types: Cylindrical (common), coaxial, inverted coaxial.

• Oscillation Mechanism:

  1. Electrons emitted from cathode attracted to anode (positive voltage).

  2. Magnetic field $B$ applied perpendicular; at critical field (electron-cyclotron resonance), electrons follow curved paths.

  3. Cavity resonators in anode: RF fields cause π-mode (alternating phase) oscillation.

  4. Rotating spokes of electrons couple energy to cavities.

• π-mode: Adjacent cavities 180° out of phase; maximum power output.

• Applications: Radar transmitters, microwave ovens (high power, low cost).

[!TIP]

Mode Identification: For $N$ cavities, π-mode has $N/2$ wavelength variation around circumference.

3.2 Solid-State Diodes

3.2.1 Gunn Diode

• Principle: Transferred electron effect in GaAs, InP. At high $E$-field, electrons transfer from lower to upper valley (higher effective mass → lower mobility).

• Domain Formation: High-field domain (space-charge) forms near cathode, travels to anode, extinguishes, then reforms.

• Domains of Operation:

  • Stable Amplification ($$\displaystyle V < V_{th} $$): Linear region.

  • Oscillation ($$\displaystyle V > V_{th} $$): Domain transit time sets frequency.

  • LSA Mode (Limited Space-Charge Accumulation): High-frequency, high-efficiency oscillation.

• Applications: Gunn oscillators (X-band), amplifiers, sensors.

[!TIP]

Gunn Effect: Negative differential resistance due to valley transfer; frequency $$\displaystyle f \approx v_{\text{sat}}/L $$.

3.2.2 Schottky Barrier Diode

• Structure: Metal-semiconductor junction (e.g., Au-GaAs).

• I-V Characteristics: $$\displaystyle I = I_s (e^{qV/nkT} - 1) $$; no minority carrier storage → fast response.

• As Mixer:

  • Nonlinear I-V produces sum/difference frequencies.

  • Conversion Loss: 4–8 dB (due to mismatch, diode resistance, noise).

  • Used in double-balanced mixers for image rejection.

• As Detector:

  • Operates in square-law region ($V \ll kT/q$): $$\displaystyle I \approx \alpha + \beta V^2 $$.

  • Output current $\propto$ RF power → envelope detection.

• Applications: Mixers, detectors, frequency converters.

[!TIP]

Advantage over PN: No charge storage → higher speed, lower noise.

3.2.3 IMPATT and TRAPATT Diodes

IMPATT (Impact Ionization Avalanche Transit Time):

• Structure: $$\displaystyle p^+ $$-$n$-$$\displaystyle n^+ $$ or $p$-$i$-$n$.

• Principle:

  1. High reverse bias → avalanche multiplication (impact ionization).

  2. Space charge from carriers causes phase delay between current and voltage.

  3. Negative resistance due to carrier drift time (transit time).

• Frequency: $$\displaystyle f \approx \frac{1}{2\pi \tau_{\text{drift}}} $$.

• Applications: High-power oscillators/amplifiers (W-band).

TRAPATT (Trapped Plasma Avalanche Transit Time):

• Principle: Plasma (highly conductive region) forms and gets "trapped" near junction; avalanche propagates slowly → higher efficiency.

• Efficiency: ~30% (vs. IMPATT ~10–15%).

• Applications: Medium-power, high-efficiency sources.

[!TIP]

Comparison: IMPATT has higher power but lower efficiency; TRAPATT used where DC power is limited.

3.2.4 BARITT Diode

• Principle: Barrier Injection and Transit Time. Uses $$\displaystyle p^+ $$-$n$-$$\displaystyle n^+ $$ or $p$-$i$-$n$ with low doping in $i$-layer.

• Carriers injected over barrier (thermionic emission) → drift through $i$-layer.

• Negative resistance due to phase shift between injection current and voltage.

• Advantages: Lower noise than IMPATT, moderate power.

• Limitations: Lower power and frequency than IMPATT.

• Applications: Low-power oscillators (<10 GHz).

3.2.5 Tunnel Diode (Esaki Diode)

• Structure: Heavily doped $p$-$n$ junction → narrow depletion region.

• I-V Curve: Negative resistance region due to quantum tunneling.

• Modes of Operation:

  • Resistive: Used as switch.

  • Capacitive: Reverse bias, junction capacitance varies with voltage → varactor.

  • Inductive: Negative resistance with parasitic inductance → microwave oscillator.

• Applications: High-speed switches, low-noise amplifiers, oscillators (up to 100 GHz).

[!TIP]

Key: Tunneling dominates at forward bias; peak current $$\displaystyle I_p $$, valley current $$\displaystyle I_v $$, peak-to-valley ratio determines quality.

3.3 Microwave Transistors

3.3.1 Microwave BJT

• Structure: Emitter-base-collector with narrow base (sub-micron).

• Working: Minority carrier injection from emitter to base, diffusion to collector.

• Frequency Limitations:

  • Base transit time $$\displaystyle \tau_b = \frac{W_b^2}{2D_n} $$ → minimize $$\displaystyle W_b $$.

  • Base resistance $$\displaystyle r_{bb'} $$ → RC time constant.

  • Collector depletion capacitance $$\displaystyle C_{\mu} $$ → Miller effect.

• Applications: Amplifiers (low-noise, power), switches (up to ~10 GHz).

3.3.2 Microwave FET (MESFET/HEMT)

• Operation: Gate voltage controls channel conductivity (Schottky gate in MESFET; heterojunction in HEMT).

• Advantages:

  • High input impedance (voltage-controlled).

  • High $$\displaystyle f_T $$ (transition frequency) → HEMT up to 300 GHz.

  • Low noise, high gain.

• Applications: Low-noise amplifiers, mixers, power amplifiers (MMICs).

[!TIP]

HEMT vs MESFET: HEMT uses AlGaAs/GaAs heterojunction → higher electron mobility, lower noise.

3.4 Other Active Devices

MASER (Microwave Amplification by Stimulated Emission of Radiation):

• Principle: Stimulated emission from inverted population (e.g., ammonia beam, ruby crystal).

• Low-Noise Amplification: Quantum mechanical; noise temperature near 0 K.

• Applications: Deep-space communication, radio astronomy (precursor to laser).

4. Network Analysis and S-parameters

4.1 Scattering Matrix (S-parameters)

• Definition: Relates incident ($$\displaystyle a_n $$) and reflected ($$\displaystyle b_n $$) waves at ports: $$\displaystyle \mathbf{b} = \mathbf{S}\mathbf{a} $$.

• Necessity: At microwaves, $Z$, $Y$, $h$, $ABCD$ parameters are hard to measure due to parasitics and termination difficulties. S-parameters use wave concepts with matched terminations.

• Properties:

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

  • Losslessness: $$\displaystyle \mathbf{S}^\dagger \mathbf{S} = \mathbf{I} $$ (unitary).

  • Symmetry: For symmetric networks, $$\displaystyle S_{11} = S_{22} $$, etc.

  • Passivity: $$\displaystyle |\det(\mathbf{I} - \mathbf{S}\mathbf{S}^\dagger)| \geq 0 $$.

4.2 Derivation of S-matrix for Reciprocal & Lossless Two-Port

For a two-port:

$$b_1 = S_{11}a_1 + S_{12}a_2$$

$$b_2 = S_{21}a_1 + S_{22}a_2$$

Reciprocity: $$\displaystyle S_{12} = S_{21} $$.

Losslessness (no power dissipation):

$$|a_1|^2 - |b_1|^2 + |a_2|^2 - |b_2|^2 = 0$$

Substituting and equating coefficients yields:

$$|S_{11}|^2 + |S_{21}|^2 = 1, \quad |S_{22}|^2 + |S_{12}|^2 = 1, \quad S_{11}S_{11}^* + S_{12}S_{12}^* = 1$$

and $$\displaystyle S_{11}S_{21}^* + S_{12}S_{22}^* = 0 $$.

For symmetric lossless two-port: $$\displaystyle S_{11} = S_{22} $$.

4.3 S-parameters in Oscillator Design

• Oscillation Condition: Feedback network must provide positive feedback.

• In terms of reflection coefficients at ports 1 and 2 (with other ports terminated):

$$\left| \Gamma_1 \Gamma_2 \right| \geq 1 \quad \text{and} \quad \angle(\Gamma_1 \Gamma_2) = 0^\circ \ (\text{or } 2\pi n)$$

• Design: Choose active device (e.g., Gunn diode) with $$\displaystyle \Gamma_1 > 1 $$ (negative resistance) and feedback network (e.g., cavity) with $$\displaystyle \Gamma_2 $$ such that product satisfies condition.

[!TIP]

Stability Circles: On Smith chart, plot stability circles for $$\displaystyle |Γ_{in}| < 1 $$; oscillator operates outside circle.

4.4 Challenges in Traditional Parameter Measurement

• Parasitic Effects: Inductances/capacitances of connectors, probes become significant.

• Termination Issues: Achieving perfect 50 Ω match at high frequencies is difficult.

• Directivity Errors: In directional couplers, finite isolation causes measurement errors.

• Frequency-Dependent Behavior: Parameters vary rapidly with frequency; calibration needed.

• S-parameters avoid these by using wave incident/reflected definitions with matched loads.

4.5 Applications of S-parameters

• Network Characterization: Gain, match, isolation.

• Stability Analysis: Stability circles for amplifier design.

• Impedance Matching: Using $$\displaystyle \Gamma = S_{11} $$ for input match.

• Cascading Networks: Using ABCD conversion or S-parameter multiplication.

• Device Modeling: Extract model parameters from measured S-parameters.

5. Passive Microwave Components

5.1 Matching Networks and Impedance Transformers

5.1.1 Matching Networks

• Purpose: Match load impedance $$\displaystyle Z_L $$ to line $$\displaystyle Z_0 $$ for maximum power transfer ($$\displaystyle Γ=0 $$).

• Techniques:

  1. L-section: Two reactive elements (series-shunt or shunt-series). Unique solution for given $$\displaystyle Z_L $$, $$\displaystyle Z_0 $$.

  2. Quarter-wave Transformer: $$\displaystyle Z_{0T} = \sqrt{Z_0 Z_L} $$; narrowband (bandwidth $$\displaystyle \approx \frac{4}{\pi} \sin^{-1}\left|\frac{Z_L - Z_0}{Z_L + Z_0}\right| $$).

  3. Single-Stub: Shunt or series stub at specific distance from load. Uses Smith chart.

  4. Double-Stub: Two stubs at fixed distance; more bandwidth, tolerant to frequency changes.

[!TIP]

Design Rule: For L-section, choose configuration based on $$\displaystyle Z_L > Z_0 $$ or $$\displaystyle Z_L < Z_0 $$.

5.1.2 Impedance Transformers

• Single-Section Quarter-Wave: $$\displaystyle Z_0(x) = \sqrt{Z_0 Z_L} $$ constant; bandwidth limited by frequency sensitivity of $\lambda/4$ condition.

• Multi-Section Transformer:

  • Binomial Response: Maximally flat (Butterworth) $$\displaystyle |Γ|^2 = |Γ_0|^2 \cos^{2N}(N\theta) $$; wider bandwidth, no ripple.

  • Chebyshev Response: Equiripple in passband; sharper cutoff, wider bandwidth for same ripple.

• Bandwidth Enhancement: Increase number of sections $N$; characteristic impedances $$\displaystyle Z_{0i} $$ from binomial/Chebyshev polynomials.

[!TIP]

Formulas: For binomial $N$-section, $$\displaystyle Z_{0i} = Z_0 \left[ \prod_{k=1}^{i} \frac{1 + \cos\left(\frac{(2i-N)\pi}{2N}\right)}{1 + \cos\left(\frac{(2k-N)\pi}{2N}\right)} \right]^{1/2} $$.

5.2 Power Dividers, Couplers, and Tees

5.2.1 Hybrid Tee (Magic Tee)

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

• Scattering Matrix (ports 1=input, 2=series, 3=shunt, 4=isolated):

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

• Properties:

  • Isolation: Ports 2 and 3 isolated ($$\displaystyle S_{32}=0 $$).

  • Coupling: Equal split between 1→2 and 1→3 (3 dB).

  • Phase: 1→2 and 1→3 in-phase; 1→4 out-of-phase with 2, in-phase with 3.

• Applications: Mixers, balanced amplifiers, phase shifters.

[!TIP]

Derivation: Use symmetry and orthogonality of E-plane/H-plane modes; enforce $$\displaystyle S_{ii}=0 $$ for match, $$\displaystyle S_{ij}=0 $$ for isolation.

5.2.2 Directional Coupler

• Definition: Four-port with power coupled from input (port 1) to coupled port (port 3) in one direction.

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

• Directivity ($D$): $$\displaystyle D = C - I $$, where $$\displaystyle I = -20\log|S_{41}| $$ (isolation between coupled and isolated ports).

• S-matrix for ideal matched coupler:

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

with $$\displaystyle |\tau|^2 + |\kappa|^2 = 1 $$, $$\displaystyle \tau = \sqrt{1-\kappa^2} $$ (through path), $\kappa$ (coupling).

5.2.3 Circulator and Isolator

• Symbol: Triangle with arrows (circulator); arrow one-way (isolator).

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

  • Two magic tees connected via two arms with 90° phase shift.

  • Signal entering port 1 → travels through tee1 → phase shifter → tee2 → port 2.

  • Reverse path destructive interference → isolation.

• Simplified S-matrix (3-port circulator):

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

• Isolator: Circulator with port 3 terminated → port 1→2 transmitted, 2→1 absorbed.

• Applications: Circulator for duplexing; isolator for protecting source.

5.3 Phase Shifters

5.3.1 Diode Phase Shifter

• Principle: Use switching diodes (PIN) or varactor diodes to change electrical length.

  • Switching Type: Diode ON (short) or OFF (open) → 0° or 180° shift.

  • Varactor Type: Capacitance controlled by bias → continuous phase shift.

• Broadband: Switching diodes (digital phase shifters).

• Tuned: Varactors (analog phase shifters, narrowband).

• Applications: Phased array antennas, beam steering.

6. Measurement Techniques

6.1 Slotted Line

• Construction: Section of waveguide/coaxial line with longitudinal slot; movable probe detects field.

• Working:

  1. Probe measures $|V(z)|$ along line.

  2. Plot standing wave pattern → find $$\displaystyle V_{\max} $$, $$\displaystyle V_{\min} $$, distance between them.

• Determination:

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

  • Wavelength: $$\displaystyle \lambda_g = 2 \times $$ distance between two successive minima.

  • Impedance: From $$\displaystyle Γ = |Γ|e^{j\theta} $$, $$\displaystyle |Γ| = (S-1)/(S+1) $$, $\theta$ from min position; then $$\displaystyle Z_{\text{in}} = Z_0 \frac{1+Γ}{1-Γ} $$.

[!TIP]

Precision: Use precision slotted line for low-VSWR measurements; probe should not disturb field.

6.2 Voltage Standing Wave Ratio (VSWR)

• Definition: $$\displaystyle S = \frac{V_{\max}}{V_{\min}} $$.

• Derivation:

  Voltage along line: $$\displaystyle V(z) = V^+ (e^{-j\beta z} + Γ e^{j\beta z}) $$.

  $$\displaystyle |V(z)| = |V^+| \left| 1 + Γ e^{j2\beta z} \right| $$.

  Max when $$\displaystyle e^{j2\beta z} $$ in phase with $Γ$: $$\displaystyle V_{\max} = |V^|(1+|Γ|) $$.

  Min when out of phase: $$\displaystyle V_{\min} = |V^+|(1-|Γ|) $$.

  Hence:

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

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

6.3 Power Measurement Bridges

• Bolometer Bridge:

  • Sensor: Bolometer (barretters) – resistance increases with temperature (from RF heating).

  • Bridge: Balanced with DC bias; RF power changes resistance → unbalance → measure power.

  • Calibration: Known RF power vs. DC substitution.

• Thermistor Bridge:

  • Sensor: Thermistor (semiconductor, negative $dR/dT$).

  • Bridge: Self-balancing; RF heating changes resistance → feedback adjusts DC to maintain balance.

  • Advantage: Faster response, higher sensitivity than bolometer.

[!TIP]

Key Difference: Bolometer uses metal (positive $dR/dT$), thermistor uses semiconductor (negative $dR/dT$).

7. Resonators and Oscillators

7.1 YIG Resonator

• Structure: Yttrium Iron Garnet (YIG) sphere (~0.5 mm) placed in static magnetic field $$\displaystyle B_0 $$, surrounded by RF coil.

• Frequency Tuning: Resonance frequency $$\displaystyle f \propto B_0 $$ (electron spin resonance). Tuning range: 1–20 GHz.

• Linearity: Good linearity between $f$ and $$\displaystyle B_0 $$.

• Applications:

  • Tunable filters (YIG filters).

  • Oscillators (YIG-tuned oscillators).

  • Dispersive delay lines.

• Advantages: High $Q$ (~10,000), wide tuning, low phase noise.

7.2 Microwave Resonators

• Types:

  • Cavity Resonators: Metallic enclosure (rectangular, cylindrical). $Q$ high (10⁴–10⁵); used in filters, oscillators.

  • Dielectric Resonators: High-$$\displaystyle \varepsilon_r $$ ceramic (e.g., TiO₂). $Q$ moderate (10³–10⁴); compact, used in MMICs.

  • Planar Resonators: Microstrip open/short stubs, ring resonators. Lower $Q$; integrated circuits.

• Quality Factor ($Q$):

$$Q = \frac{\omega_0 \cdot \text{Stored Energy}}{\text{Power Loss}} = \frac{f_0}{\Delta f}$$

• Applications: Filters, oscillators, frequency-selective networks.

8. Mixers, Detectors, and Frequency Multipliers

8.1 Microwave Mixers

• Principle: Nonlinear device (diode, transistor) multiplies RF and LO signals → sum/difference frequencies.

• Signals:

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

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

  • IF ($$\displaystyle f_{IF} = |f_{RF} - f_{LO}| $$): Intermediate frequency output.

• Conversion Loss ($$\displaystyle L_c $$):

$$L_c = \frac{P_{RF}}{P_{IF}} \text{ (linear)} \quad \text{or} \quad L_c(\text{dB}) = P_{RF}(\text{dBm}) - P_{IF}(\text{dBm})$$

Typical: 4–9 dB (due to diode resistance, mismatch, noise).

• Types: Single-ended, single-balanced, double-balanced (better isolation, lower spurious).

8.2 Frequency Multipliers

• Principle: Nonlinear device generates harmonics; output filtered at $$\displaystyle n f_{in} $$.

• Derivation: For input $$\displaystyle V = V_0 \cos \omega t $$, output current $$\displaystyle I = a_0 + a_1 V + a_2 V^2 + \cdots $$. $$\displaystyle V^2 $$ term produces $2\omega$ component.

$$V^2 = V_0^2 \cos^2 \omega t = \frac{V_0^2}{2} (1 + \cos 2\omega t)$$

Hence doubler; higher-order terms give $n\omega$.

• Applications: Frequency synthesis (e.g., generate 60 GHz from 20 GHz source).

8.3 Detectors

8.3.1 Tuned Detectors

• Principle: Preselector (tuned circuit) + detector (diode).

• Narrowband: Only signals within bandwidth of tuned circuit detected.

• Applications: Spectrum analyzers, signal identification.

8.3.2 Schottky Diode as Detector

• Operates in square-law region ($$\displaystyle V_{RF} \ll kT/q \approx 25 $$ mV at 300 K).

• Output DC: $$\displaystyle I_{DC} \propto V_{RF}^2 \propto P_{RF} $$.

• Used in power detectors, envelope detectors.

9. Additional and Specialized Topics

9.1 TWT Amplifier

• Configuration: TWT as power amplifier (input RF, output amplified RF).

• Gain: 40–50 dB.

• Bandwidth: Octave or more (helix TWT).

• Noise Figure: 6–10 dB (higher than low-noise amplifiers).

• Applications: Satellite transponders, radar transmitters, ECM systems.

9.2 Power Measurement Bridges

• Bolometer Bridge: See 6.3.

• Thermistor Bridge: See 6.3.

• Operation: Bridge balanced with DC; RF power unbalances → measure substitution power.

9.3 Brief Notes

MASER:

• Principle: Stimulated emission from inverted population (quantum).

• Low-noise amplification (noise temp ~ few K).

• Used in radio astronomy, deep-space comms.

BARITT Diode:

• Barrier injection and transit time.

• Lower noise than IMPATT, lower power/frequency.

• Used in low-power oscillators (<10 GHz).

Isolator vs Circulator:

• Circulator: 3-port; signal circulates 1→2→3→1.

• Isolator: Circulator with one port terminated → allows transmission only one way (1→2).

• Isolator protects source from reflections.

TEM Mode:

• Transverse electromagnetic; $$\displaystyle E_z = H_z = 0 $$.

• Exists in two-wire, coaxial lines.

• No cutoff frequency; dispersionless ($$\displaystyle v_p = 1/\sqrt{LC} $$).

• Not supported in waveguides.

\boxed{\text{End of Unit 4 Notes}}