Microwave Engineering (EC-702 (A)) - Unit 3 Short Notes

Unit 3: Microwave Engineering - Comprehensive Short Notes

0. Introduction and Applications of Microwaves

• Frequency Range: 300 MHz to 300 GHz (wavelength 1 m to 1 mm).

• Key Characteristics: Short wavelengths allow compact systems, high directivity with small antennas, wide bandwidth for high data rates, and significant effects from parasitic elements and skin depth.

• Major Applications:

  • Radar: Aircraft navigation, weather forecasting, speed detection.

  • Satellite Communications: TV broadcasting, GPS, global telephony.

  • Microwave Ovens: Dielectric heating of water molecules in food.

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

  • Wireless Networks: Wi-Fi (2.4/5 GHz), 5G (mmWave bands).

  • Radio Astronomy: Studying celestial objects via microwave emissions.

[!TIP] Exam Focus: Be prepared to list 4-5 applications and briefly explain the underlying principle for at least two (e.g., dielectric heating in ovens, skin effect in heating).

1. Transmission Lines and Waveguides

1.1 Transmission Line Fundamentals

• Primary Constants (per unit length):

  • R: Series resistance (Ω/m) – conductor loss.

  • L: Series inductance (H/m) – magnetic energy storage.

  • G: Shunt conductance (S/m) – dielectric loss.

  • C: Shunt capacitance (F/m) – electric energy storage.

• Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta $$

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

  • $\beta$ (rad/m): Phase constant (phase change per unit length).

• Phase Velocity ($$\displaystyle v_p $$): Speed of a constant phase point. $$\displaystyle v_p = \omega/\beta $$.

• Group Velocity ($$\displaystyle v_g $$): Speed of energy/information transfer. $$\displaystyle v_g = d\omega/d\beta $$. For lossless lines: $$\displaystyle v_p \cdot v_g = c^2 / \varepsilon_r $$.

• Characteristic Impedance ($$\displaystyle Z_0 $$): Input impedance of an infinitely long line.

  • Lossy: $$\displaystyle Z_0 = \sqrt{(R + j\omega L)/(G + j\omega C)} $$

  • Lossless ($$\displaystyle R=G=0 $$): $$\displaystyle Z_0 = \sqrt{L/C} $$

• Reflection Coefficient ($\Gamma$): Ratio of reflected to incident voltage wave at a load discontinuity.

$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$$

• Voltage Standing Wave Ratio (VSWR): Ratio of max to min voltage on a standing wave pattern.

  Derivation: $$\displaystyle |V_{max}| = |V^+|(1+|\Gamma|) $$, $$\displaystyle |V_{min}| = |V^+|(1-|\Gamma|) $$

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

1.2 Planar Transmission Lines

1.2.1 Microstrip Line

• Structure: Conductor strip on a dielectric substrate with a ground plane.

• Effective Dielectric Constant ($$\displaystyle \varepsilon_{eff} $$): Accounts for fringing fields in air and dielectric.

  • For $W/d \geq 1$:

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

*   Significance: Determines electrical length and phase velocity ($$\displaystyle v_p = c/\sqrt{\varepsilon_{eff}} $$).

• Characteristic Impedance ($$\displaystyle Z_0 $$): Derived using quasi-TEM approximation (valid for $h \ll \lambda$).

  • For $W/d \leq 1$: $$\displaystyle Z_0 \approx \frac{60}{\sqrt{\varepsilon_{eff}}} \ln\left(\frac{8d}{W} + \frac{W}{4d}\right) $$

  • For $W/d \geq 1$: $$\displaystyle Z_0 \approx \frac{120\pi}{\sqrt{\varepsilon_{eff}} \left( W/d + 1.393 + 0.667\ln(W/d + 1.444) \right)} $$

• Advantages: Easy to fabricate, integrate active devices, low cost.

• Limitations: Radiation loss, dispersion ($$\displaystyle \varepsilon_{eff} $$ and $$\displaystyle Z_0 $$ vary with frequency), limited power handling.

1.2.2 Stripline

• Structure: Central conductor sandwiched between two ground planes (dielectric on both sides). TEM mode is the dominant mode.

• Characteristic Impedance: $$\displaystyle Z_0 \propto 1/(\text{strip width}) $$, lower than microstrip for same width due to higher capacitance.

• Loss Considerations: Lower radiation loss than microstrip, but higher dielectric loss due to field confinement in dielectric.

1.2.3 Slot Line

• Structure: Slot in a ground plane with a narrow strip on the opposite side of the substrate. Fields are predominantly electric (E-field across slot).

• Comparison with Microstrip: Slot line has higher loss, higher characteristic impedance for same dimensions. Useful for series connections and non-reciprocal devices with ferrites.

1.2.4 Comparison of Planar Transmission Lines

1.3 Waveguides

1.3.1 Rectangular Waveguide (a x b, a > b)

• TE/TM Modes: Solutions to wave equation with boundary conditions. Denoted TE_mn/TM_mn.

• Cutoff Wavelength ($$\displaystyle \lambda_c $$): Wavelength above which mode propagates.

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

• Dominant Mode: TE₁₀ (lowest cutoff, $$\displaystyle m=1, n=0 $$). $$\displaystyle \lambda_{c10} = 2a $$.

• Guide Wavelength ($$\displaystyle \lambda_g $$): Wavelength of wave inside guide.

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

• Phase Velocity ($$\displaystyle v_p $$): $$\displaystyle v_p = f\lambda_g = c/\sqrt{1-(f_c/f)^2} > c $$.

• Group Velocity ($$\displaystyle v_g $$): $$\displaystyle v_g = c\sqrt{1-(f_c/f)^2} < c $$. $$\displaystyle v_p v_g = c^2 $$.

• Wave Impedance:

  • TE: $$\displaystyle Z_{TE} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$ (higher than in free space)

  • TM: $$\displaystyle Z_{TM} = \eta \sqrt{1 - (f_c/f)^2} $$ (lower than in free space)

  where $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$ is the intrinsic impedance of the filling medium.

• Example (TM₁₁): For a=3cm, b=2cm, f=10GHz, $$\displaystyle \varepsilon_r=1 $$:

  $$\displaystyle \lambda_0 = 3 $$ cm. $$\displaystyle \lambda_{c11} = 2/\sqrt{(1/0.03)^2 + (1/0.02)^2} \approx 1.09 $$ cm. $$\displaystyle f_c = c/\lambda_{c11} \approx 27.5 $$ GHz. Since $$\displaystyle f < f_c $$, TM₁₁ does NOT propagate. (Common pitfall: always check $$\displaystyle f > f_c $$).

1.3.2 Circular Waveguide (radius a)

• TE/TM Modes: Solutions involve Bessel functions. Modes: TE_0n, TE_mn, TM_0n, TM_mn (m=azimuthal index, n=radial index).

• Cutoff Wavelength ($$\displaystyle \lambda_c $$):

$$\lambda_c = \frac{2\pi a}{p'_{mn}}$$

where $$\displaystyle p'_{mn} $$ is the n-th root of derivative of Bessel function $$\displaystyle J_m $$. For dominant TE<sub>11</sub>, $$\displaystyle p'_{11} \approx 1.841 $$.

• Dominant Mode: TE₁₁ (lowest cutoff).

• Example (Determine all propagating modes): Given $$\displaystyle a=2 $$ cm, $$\displaystyle f=10 $$ GHz.

  1. Calculate $$\displaystyle f_c $$ for each mode: $$\displaystyle f_c = \frac{p'_{mn} c}{2\pi a} $$.

  2. TE₁₁: $$\displaystyle f_c \approx (1.841 * 3e8)/(2π*0.02) \approx 4.39 $$ GHz → Propagates.

  3. TE₂₁: $$\displaystyle p'_{21} \approx 3.054 $$ → $$\displaystyle f_c \approx 7.28 $$ GHz → Propagates.

  4. TE₀₁: $$\displaystyle p'_{01} \approx 3.832 $$ → $$\displaystyle f_c \approx 9.14 $$ GHz → Propagates.

  5. TM₁₁: $$\displaystyle p_{11} \approx 1.841 $$ → $$\displaystyle f_c \approx 4.39 $$ GHz → Propagates.

  6. Check higher modes (TE₃₁, TM₀₁, etc.) – find cutoff > 10 GHz → Cutoff.

  Result: Propagating modes: TE₁₁, TE₂₁, TE₀₁, TM₁₁.

1.3.3 TEM Mode in Transmission Lines

• Definition: Transverse ElectroMagnetic mode. Both E and H fields are entirely transverse to the direction of propagation (no $$\displaystyle E_z $$ or $$\displaystyle H_z $$).

• Condition for TEM: Requires two or more separate conductors (e.g., coaxial, two-wire, stripline). Cannot exist in a single-conductor waveguide (like rectangular/circular) – they support only TE/TM.

• Relevance: All planar lines (microstrip, stripline, slot) support a quasi-TEM mode (small longitudinal fields due to dispersion) at lower microwave frequencies, allowing use of TEM-based equations (like $$\displaystyle Z_0 = \sqrt{L/C} $$) with corrections.

2. Microwave Sources: Vacuum Tubes

2.1 Reflex Klystron

• Construction: Electron gun → Buncher cavity → Drift space → Repeller (reflects electrons) → Catcher cavity → Collector.

• Working (Velocity Modulation & Bunching):

  1. Electrons accelerated by voltage $$\displaystyle V_a $$ pass through buncher cavity gap.

  2. RF voltage in buncher gap causes velocity modulation: electrons crossing during positive half are accelerated, during negative half are decelerated.

  3. In drift space, faster electrons catch up with slower ones, forming electron bunches.

  4. Bunches induce RF voltage in catcher cavity (output) and are repelled by negative repeller voltage back to anode.

• Mode Curve: Plot of output power vs. repeller voltage ($$\displaystyle V_r $$). Shows modes (regions of oscillation) separated by mode jumps. Used for local oscillators due to frequency stability in a mode.

• Applications: Low-power (10-500 mW) local oscillators in receivers, pumps for parametric amplifiers.

2.2 Two-Cavity Klystron Amplifier

• Structure: Input cavity (buncher) → Drift space → Output cavity (catcher) → Collector.

• Working: Similar velocity modulation. RF input signal in buncher gap creates velocity-modulated beam. Bunched beam induces amplified RF signal in catcher cavity.

• Power Gain & Efficiency: Moderate gain (10-20 dB), efficiency ~10-20%. Used as medium-power amplifiers (watts to kW).

2.3 Traveling Wave Tube (TWT)

• Construction: Electron gun → Slow-wave structure (helix or coupled-cavity) → Collector. Helix is most common for wideband.

• Interaction Mechanism:

  1. Electron beam travels along axis near helix.

  2. RF signal fed into helix propagates at $$\displaystyle v_p \approx v_e $$ (electron velocity) due to slow-wave structure.

  3. Continuous interaction: RF electric field on helix accelerates/decelerates electrons, causing velocity modulation.

  4. Bunched electrons induce more energy into RF wave → amplification.

• Gain: Very high gain (40-60 dB), wide bandwidth (octave). Used as wideband amplifiers in satellite comms, EW systems.

2.4 Magnetron

• Types: Pulsed (high peak power, radar), Continuous-Wave (CW, lower power, microwave ovens).

• Oscillation Mechanism (π-mode):

  1. Electrons from cathode emitted into crossed E (radial) and B (axial) fields → Cycloidal motion.

  2. Electrons interact with resonant cavities in anode block.

  3. In π-mode (alternating cavities 180° out of phase), electron cloud forms and rotates, delivering energy to RF field.

  4. Strapping (rings connecting cavities) suppresses competing modes.

• Applications: Radar transmitters (pulsed), microwave ovens (CW, 2.45 GHz).

3. Microwave Sources: Solid-State Devices

3.1 Tunnel Diode

• Structure: Heavily doped PN junction (tunnel junction).

• I-V Characteristics: Exhibits negative differential resistance region due to quantum tunneling.

  • Regions: Cut-in, negative resistance, saturation.

• Modes: Oscillator (with tank circuit), amplifier, switch.

• Frequency Limitations: Capacitance limits to ~10 GHz. Used in low-power, high-speed applications.

3.2 Gunn Diode

• Gunn Effect: In GaAs/InP, due to two-valley energy band structure. High-field domain (space charge) forms and travels from cathode to anode.

• Domains of Operation:

  • Gunn Mode: Transit-time oscillation (most common).

  • LSA Mode: Limited Space Charge Accumulation (higher frequency, lower noise).

  • Limited Space Charge Mode: Low-field operation.

• Applications: Oscillators (X/Ku band, 1-100 mW), amplifiers (Gunn amplifiers).

3.3 IMPATT and TRAPATT Diodes

• IMPATT (Impact Ionization Avalanche Transit Time):

  • Principle: Reverse-biased PN junction. Avalanche multiplication creates plasma. Carrier drift through depletion region causes current phase lag → negative resistance.

  • High power (watts), high efficiency (15-30%), high noise.

• TRAPATT (Trapped Plasma Avalanche Transit Time):

  • Principle: Similar to IMPATT but plasma is trapped near junction, causing faster current rise → higher efficiency (30-50%).

  • Used in high-power pulsed oscillators/amplifiers.

3.4 BARITT Diode

• Principle: Barrier Injection and Transit Time. Uses drift region between two barriers (e.g., p-i-n, metal-semiconductor-metal). Carriers injected over barrier, drift slowly → negative resistance.

• Comparison with IMPATT: Lower noise, lower power, lower frequency. Used in low-noise, low-power oscillators.

3.5 Schottky Barrier Diode

• Structure: Metal (e.g., Au, Pt) on n-type semiconductor (e.g., GaAs, Si). No depletion region like PN junction.

• I-V Characteristics: Similar to PN but no minority carrier storage → faster response, lower capacitance.

• Advantages over PN: Lower capacitance, faster switching, no reverse recovery, simpler fabrication.

• Applications: Microwave mixers (downconversion), detectors (square-law region for power detection), varactors (phase shifters).

3.6 Microwave BJT

• Structure: NPN or PNP with very small base region.

• Frequency Limitations:

  • $$\displaystyle f_T $$ (Transition Frequency): Current gain = 1. $$\displaystyle f_T = \beta_0 f_\beta $$.

  • $$\displaystyle f_{max} $$ (Maximum Oscillation Frequency): Power gain = 1. $$\displaystyle f_{max} \approx f_T / (2\sqrt{R_g C_c}) $$.

  • Limitations: Base resistance, collector-base capacitance, carrier transit time.

• Applications: Low-noise amplifiers (LNAs), oscillators up to ~10 GHz (SiGe HBT extends to ~30 GHz).

3.7 Microwave FET (MESFET, HEMT)

• MESFET (Metal-Semiconductor FET): Schottky gate on GaAs. Voltage-controlled resistor.

• HEMT (High Electron Mobility Transistor): Heterojunction (AlGaAs/GaAs) creates high-mobility 2DEG channel. Highest $$\displaystyle f_T $$ and $$\displaystyle f_{max} $$ (>100 GHz).

• Advantages: High input impedance, low noise, good power gain, inherently stable.

• Applications: LNAs, power amplifiers, switches in MMICs, phased arrays.

4. Network Analysis and S-Parameters

4.1 Scattering Matrix (S-Matrix)

• Definition: For an N-port network, relates incident ($$\displaystyle a_i $$) and reflected ($$\displaystyle b_i $$) voltage waves at each port (with reference impedance $$\displaystyle Z_0 $$).

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

$$\displaystyle S_{ij} = b_i/a_j $$ (with all other ports matched to $$\displaystyle Z_0 $$).

• Key Properties:

  • Linearity: Superposition holds.

  • Reciprocity: $$\displaystyle S_{ij} = S_{ji} $$ for reciprocal networks (e.g., passive, linear).

  • Losslessness: $$\displaystyle \sum_{j=1}^{N} |S_{ij}|^2 = 1 $$ for all $i$ (conservation of energy).

  • Symmetry: $$\displaystyle [S] = [S]^T $$ for reciprocal networks.

  • Unitary: $$\displaystyle [S][S]^\dagger = [I] $$ for lossless networks.

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

• General Form (4 unknowns):

$$[S] = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix}$$

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

• Losslessness: For port 1 excited ($$\displaystyle a_1 \neq 0, a_2=0 $$): $$\displaystyle |S_{11}|^2 + |S_{21}|^2 = 1 $$. For port 2 excited: $$\displaystyle |S_{22}|^2 + |S_{12}|^2 = 1 $$.

• Combined: $$\displaystyle |S_{11}| = |S_{22}| $$, $$\displaystyle |S_{12}| = |S_{21}| = \sqrt{1 - |S_{11}|^2} $$.

• Final Form (with phase $\theta$):

$$\boxed{[S] = \begin{bmatrix} S_{11} & \sqrt{1 - |S_{11}|^2} e^{j\theta} \\ \sqrt{1 - |S_{11}|^2} e^{j\theta} & S_{11} \end{bmatrix}}$$

where $\theta$ is an arbitrary phase.

4.3 Why S-Parameters?

• Difficulties with Z/Y/h/ABCD at Microwaves:

  1. Insurmountable parasitic effects: Inductance/capacitance of test leads and connectors.

  2. Inaccessible internal voltages/currents: Cannot measure total voltage/current at ports.

  3. Unstable with frequency: Parameters vary wildly with frequency.

• Advantages of S-Parameters:

  1. Measurable at reference planes: Use VNA, define ports with $$\displaystyle Z_0 $$.

  2. Directly related to power: $$\displaystyle |S_{ij}|^2 $$ is power transmission.

  3. Easier cascade calculations (using ABCD conversion if needed).

4.4 Applications of S-Parameters

• Oscillator Design: Stability condition $$\displaystyle |S_{11}S_{22} - S_{12}S_{21}| \geq 1 $$ or $$\displaystyle \Gamma_{in}\Gamma_{source} \geq 1 $$.

• Amplifier Design: Gain ($$\displaystyle G = |S_{21}|^2/(1-|S_{11}|^2)(1-|S_{22}|^2) $$), stability circles (constant $$\displaystyle |\Gamma_{in}|=1 $$, $$\displaystyle |\Gamma_{out}|=1 $$).

• Network Synthesis: Design filters, matching networks from $$\displaystyle S_{11} $$ specifications.

4.5 S-Parameter Problems

• Example: Given $$\displaystyle S_{11}=0.2\angle0° $$, $$\displaystyle S_{22}=0.1\angle0° $$, $$\displaystyle S_{12}=0.6\angle90° $$, $$\displaystyle S_{21}=0.6\angle90° $$.

  1. Reciprocal? Yes, $$\displaystyle S_{12}=S_{21} $$.

  2. Lossless? Check $$\displaystyle \sum |S_{ij}|^2 $$ for each row:

     • Row 1: $$\displaystyle |0.2|^2 + |0.6|^2 = 0.04 + 0.36 = 0.4 \neq 1 $$ → Not lossless.

  3. Return Loss at port 1 with port 2 shorted:

     • Port 2 shorted → $$\displaystyle a_2 = -b_2 $$.

     • $$\displaystyle b_2 = S_{21}a_1 + S_{22}a_2 = S_{21}a_1 + S_{22}(-b_2) $$ → $$\displaystyle b_2(1 + S_{22}) = S_{21}a_1 $$ → $$\displaystyle b_2 = \frac{S_{21}}{1+S_{22}} a_1 $$.

     • $$\displaystyle b_1 = S_{11}a_1 + S_{12}a_2 = S_{11}a_1 + S_{12}(-b_2) = S_{11}a_1 - S_{12}\frac{S_{21}}{1+S_{22}} a_1 $$.

     • $$\displaystyle \Gamma_{in} = b_1/a_1 = S_{11} - \frac{S_{12}S_{21}}{1+S_{22}} $$.

     • Plug values: $$\displaystyle \Gamma_{in} = 0.2 - \frac{(0.6\angle90°)(0.6\angle90°)}{1+0.1} = 0.2 - \frac{0.36\angle180°}{1.1} = 0.2 - (-0.3273) = 0.5273 $$.

     • Return Loss (RL) = $$\displaystyle -20\log_{10}|\Gamma_{in}| = -20\log_{10}(0.5273) \approx 5.56 $$ dB.

5. Passive Components and Matching Networks

5.1 Impedance Matching

• Purpose: Maximize power transfer, minimize reflections (VSWR), improve noise figure.

• Common Techniques:

  1. L-Section: Uses one series and one shunt reactance (L or C). Two configurations for $$\displaystyle Z_L > Z_0 $$ and $$\displaystyle Z_L < Z_0 $$.

  2. Quarter-Wave Transformer: $$\displaystyle Z_{T1} = \sqrt{Z_0 Z_L} $$. Narrowband (bandwidth $\propto 1/n$ for n-sections).

  3. Stub Matching: Use open/short shunt stub (single or double stub) to cancel susceptance. Computationally simple.

5.2 Single-Section Impedance Transformer

• Quarter-Wave: $$\displaystyle Z_{in} = Z_0^2 / Z_L $$ when $$\displaystyle l = \lambda_g/4 $$. $$\displaystyle Z_T = \sqrt{Z_0 Z_L} $$.

• Limitation: Bandwidth inversely proportional to number of sections. Single section is very narrowband.

5.3 Multi-Section Impedance Transformer

• Design: Multiple quarter-wave sections with intermediate impedances $$\displaystyle Z_1, Z_2, ... $$.

• Bandwidth Enhancement:

  • Binomial (Maximally Flat): Chebyshev polynomial of 1st kind. Maximizes bandwidth for given ripple.

  • Chebyshev (Equiripple): Allows specified ripple in passband for wider bandwidth.

• Trade-off: More sections → wider bandwidth, higher loss, more complex fabrication.

5.4 Directional Couplers

• Four-Port Device: Ports 1-4. Couples power from input (1) to through (2) and coupled (3). Isolated port (4).

• Coupling Factor (C): $$\displaystyle C = -20\log_{10}|S_{31}| $$ (dB). Power from port 1 to port 3.

• Directivity (D): $$\displaystyle D = C - I $$, where Isolation $$\displaystyle I = -20\log_{10}|S_{41}| $$. Measures ability to isolate coupled from isolated port.

• S-Matrix for Matched Four-Port Directional Coupler:

  • Conditions: All ports matched ($$\displaystyle S_{ii}=0 $$), reciprocal ($$\displaystyle S_{ij}=S_{ji} $$), symmetry.

  • Ideal 3 dB Coupler (Hybrid):

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

*   Properties: $$\displaystyle |S_{21}| = |S_{31}| = 1/\sqrt{2} $$, $$\displaystyle S_{41}=0 $$.

5.5 Hybrid Tee and Circulators

5.5.1 Matched Hybrid Tee (Magic Tee)

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

• Properties:

  • Sum Port (1): Signals in phase → add at port 2, cancel at port 3.

  • Difference Port (4): Signals 180° out of phase → cancel at port 2, add at port 3.

  • Isolation: Between port 1 & 4, and port 2 & 3.

• S-Matrix Derivation: From symmetry and matching, and assuming 3 dB coupling:

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

(Same as ideal 3 dB coupler).

5.5.2 Circulator

• Symbol: Triangle with arrows indicating circulation (1→2→3→1).

• Working Principle: Uses ferrite material in static magnetic field ($$\displaystyle H_0 $$) → non-reciprocal Faraday rotation. Signal entering port 1 is rotated and exits port 2, etc.

• Implementation with Magic Tee + Phase Shifter: Combine magic tee with 90° phase shifter in one arm to break reciprocity.

• Ideal S-Matrix (3-port):

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

• Applications: Isolate source from load reflections (with isolator), duplexers in transceivers.

5.5.3 Isolator

• Principle: Circulator with port 3 terminated in matched load. Only allows signal from port 1 to port 2, not reverse.

• Use: Protect sensitive sources (e.g., klystron, TWT) from reflected power.

5.6 Diode Phase Shifters

• Principle: Vary phase of RF signal using voltage-controlled reactance (varactor) or switched transmission line lengths.

• Types:

  • Varactor Phase Shifter: Continuous tuning, limited phase range, lossy.

  • Switched-Line (Digital): Switch between different line lengths (e.g., 0°, 90°, 180°, 270°). Low loss, discrete steps.

• Broadband vs. Tuned: Switched-line is broadband; varactor is tuned (narrower).

• Applications: Phased array antennas (beam steering), modulators.

6. Measurement Techniques

6.1 Slotted Line

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

• Working Principle: Measures voltage standing wave pattern. Probe output (detector current) $$\displaystyle \propto |V(z)|^2 $$.

• Measurements:

  1. VSWR: $$\displaystyle VSWR = V_{max}/V_{min} $$ from probe readings.

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

  3. Impedance: Locate voltage minimum, measure distance $d$ from reference plane to first minimum. Then $$\displaystyle \Gamma = |\Gamma|e^{-j2\beta d} $$, where $$\displaystyle |\Gamma| = (VSWR-1)/(VSWR+1) $$. Load impedance $$\displaystyle Z_L = Z_0(1+\Gamma)/(1-\Gamma) $$.

• Calibration: Requires known reference plane and matched termination. Accuracy limited by probe coupling and losses.

6.2 Voltage Standing Wave Ratio (VSWR)

• Definition: $$\displaystyle VSWR = |V_{max}|/|V_{min}| = (1+|\Gamma|)/(1-|\Gamma|) $$.

• Significance: Measures match quality. VSWR=1 (perfect match), VSWR=∞ (total reflection).

• Measurement: Slotted line, VNA (direct readout).

6.3 Power Measurement

• Bridges:

  • Boltzmann (Thermistor) Bridge: Uses thermistor in bridge circuit. Thermistor resistance changes with temperature (power dissipation). Self-balancing for accurate power measurement.

  • Wattmeter (Thermocouple): RF power heats thermocouple → DC voltage output.

• Detectors:

  • Thermistors: Negative temperature coefficient. Used in bridges for high accuracy.

  • Thermocouples: Direct RF-to-DC conversion. Simpler, less accurate.

• Calibration: Compare with known standard source (e.g., calorimeter).

7. System Components

7.1 Microwave Mixers

• Principle: Nonlinear device (diode, transistor) combines RF ($$\displaystyle f_{RF} $$) and LO ($$\displaystyle f_{LO} $$) signals → sum/difference frequencies ($$\displaystyle f_{RF} \pm f_{LO} $$). One is selected as IF.

• Signals:

  • RF: Input signal to be converted.

  • LO: High-power local oscillator.

  • IF: Intermediate frequency (usually lower).

• Conversion Loss (CL): Ratio of available IF power to available RF power (in dB). $$\displaystyle CL = P_{IF(dBm)} - P_{RF(dBm)} - G_{LO} $$? Typically: $$\displaystyle CL = P_{RF} - P_{IF} $$ (dB) for a given LO drive.

• Types:

  • Single-Ended: One diode. Simple, poor isolation, high conversion loss.

  • Balanced (Diode Ring): Two diodes in ring. Good RF/LO/IF isolation, lower noise.

  • Image-Reject: Uses filtering to reject image frequency.

• Schottky Diode Mixer: Most common. Fast switching, low capacitance. Used in receivers, spectrum analyzers.

7.2 Microwave Oscillators

• Design using S-Parameters (Reflection Oscillator):

  • Connect active device (with $$\displaystyle S_{11} $$) to a feedback network (with reflection $$\displaystyle \Gamma_f $$) at input.

  • Oscillation Condition (Rollett Stability Factor):

$$|\Gamma_{in}| = |S_{11} + \frac{S_{12}S_{21}\Gamma_f}{1 - S_{22}\Gamma_f}| \geq 1$$

*   **Simplified (for one-port negative resistance device like Gunn):** $$\displaystyle |\Gamma_d \cdot \Gamma_{ext}| \geq 1 $$, where $$\displaystyle \Gamma_d $$ is device reflection coefficient (from $$\displaystyle S_{11} $$), $$\displaystyle \Gamma_{ext} $$ is external circuit reflection.

• Stability: Ensure $$\displaystyle K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2|S_{12}S_{21}|} > 1 $$ for unconditional stability.

• Examples: Gunn diode oscillator (cavity or microstrip), YIG oscillator (tunable).

7.3 Frequency Multipliers

• Principle: Nonlinear device (diode, transistor) generates harmonics of input frequency. Output tuned to $$\displaystyle n \cdot f_{in} $$ (n=2,3,...).

• Design Considerations:

  • Input matching at $$\displaystyle f_{in} $$.

  • Output matching at $$\displaystyle n f_{in} $$.

  • Suppress fundamental and other harmonics.

  • Efficiency decreases with n.

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

7.4 Detectors

• Tuned Detectors: Narrowband (tuned circuit), high sensitivity, used for specific frequency monitoring.

• Broadband Detectors: Wideband (diode + resistor), lower sensitivity, used for power monitoring across band.

• Diode Detectors:

  • Square-Law Region: $$\displaystyle I \propto V^2 \propto P_{RF} $$. Used for low-power detection (power meters).

  • Linear Region: $I \propto V$. Used for AM detection.

• Applications: Power meters, receivers (envelope detection), modulation monitors.

8. Special Devices and Resonators

8.1 YIG Resonator

• Structure: Yttrium Iron Garnet (YIG) sphere (few mm) placed in a static magnetic field $$\displaystyle H_0 $$ and coupled to microwave energy via loops or antennas.

• Working Principle: Ferromagnetic Resonance (FMR). At frequency $$\displaystyle f = \frac{\gamma}{2\pi} H_0 $$ (γ = gyromagnetic ratio ≈ 2.8 MHz/Oe), YIG absorbs energy strongly.

• Frequency Tuning: Vary $$\displaystyle H_0 $$ (electromagnet) → changes resonant frequency. Wide tuning range (2-40 GHz typical).

• Applications:

  • YIG Filter: Tunable bandpass filter.

  • YIG Oscillator: Very low phase noise.

  • Tunable Resonator: In network analyzers, frequency synthesizers.

8.2 Microwave Resonators

• Types:

  • Cavity Resonators: Metallic enclosure (rectangular, cylindrical). High Q (10^4-10^6). Used in oscillators, filters.

  • Dielectric Resonators: Low-loss ceramic (e.g., TiO₂). High Q, small size. Used in oscillators, filters.

  • Planar Resonators: Microstrip open/short stubs, ring resonators. Lower Q (100-1000). Used in MMICs.

• Quality Factor (Q): $$\displaystyle Q = \frac{\text{Stored Energy}}{\text{Energy Dissipated per Cycle}} = \frac{f_0}{\Delta f} $$ (fractional bandwidth).

• Applications: Filters (bandpass, bandstop), oscillators (frequency stabilization), frequency-selective networks.

8.3 MASER (Microwave Amplification by Stimulated Emission of Radiation)

• Principle: Stimulated emission from inverted population in ammonia (NH₃) or other paramagnetic medium at cryogenic temperatures (liquid helium). Amplifies weak signals with extremely low noise (noise temperature ~ few K).

• Construction: Ammonia beam passes through resonant cavity in magnetic field. Pumping creates population inversion.

• Applications: Radio astronomy (receive weak cosmic signals), deep-space communication, low-noise front-ends.

END OF UNIT 3 NOTES