Nano Electronics (EC-702 (C)) - Unit 3 Short Notes

UNIT 3: Microwave Engineering

1. Transmission Lines and Waveguides

Microstrip Lines

A microstrip line consists of a conducting strip on a dielectric substrate with a ground plane. Due to fringing fields, the effective dielectric constant $$\displaystyle \epsilon_{eff} $$ lies between air (1) and substrate $$\displaystyle \epsilon_r $$.

• Effective Dielectric Constant:

  For $W/h \leq 1$:

$$ \epsilon_{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \left(1 + 12\frac{h}{W}\right)^{-1/2} $$

For $W/h \geq 1$:

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

\boxed{\epsilon_{eff} \text{ increases with } \epsilon_r \text{ and decreases with } W/h}

• Characteristic Impedance:

  For $W/h \leq 1$:

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

For $W/h \geq 1$:

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

\boxed{Z_0 \text{ decreases as } W/h \text{ increases}}

[!TIP]

For exams, remember $$\displaystyle \epsilon_{eff} $$ is always between 1 and $$\displaystyle \epsilon_r $$, and $$\displaystyle Z_0 $$ is inversely related to $W/h$.

Stripline and Slot Line

• Higher-Order Modes: Both support TE/TM modes at high frequencies; cutoff depends on dimensions.

Slotted Line

• Construction: Coaxial line with a longitudinal slot in outer conductor; a probe moves along the slot to sample electric field.

• Working Principle: The slot allows measurement of voltage standing wave pattern inside the coaxial line.

• Measurements:

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

  2. Wavelength: $$\displaystyle \lambda_g = 2 \times $$ distance between consecutive minima.

  3. Impedance: From first minimum distance $$\displaystyle d_{min} $$ from load, $$\displaystyle \Gamma = \frac{VSWR-1}{VSWR+1} e^{-j2\beta d_{min}} $$, then $$\displaystyle Z_L = Z_0 \frac{1+\Gamma}{1-\Gamma} $$.

Transmission Line Parameters

Given primary constants $R, L, C, G$ per unit length:

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

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

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

[!EXAMPLE]

For $$\displaystyle R=40\ \Omega/km $$, $$\displaystyle L=2.5\ mH/km $$, $$\displaystyle C=0.009\ \mu F/km $$, $$\displaystyle G=0.29\ \mu mho/km $$, $$\displaystyle f=1\ kHz $$:

$$\displaystyle \omega = 2\pi \times 10^3 $$, compute $R+j\omega L$ and $G+j\omega C$, then $\gamma$ and $$\displaystyle Z_0 $$.

Rectangular Waveguide

• Dimensions $a \times b$ ($$\displaystyle a > b $$).

• TE/TM Modes: $$\displaystyle TE_{mn} $$ ($$\displaystyle E_z=0 $$), $$\displaystyle TM_{mn} $$ ($$\displaystyle H_z=0 $$).

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

• Cutoff Frequency: $$\displaystyle f_c = \frac{c}{2} \sqrt{(m/a)^2 + (n/b)^2} $$ (air-filled).

• Dominant Mode: $$\displaystyle TE_{10} $$.

• Characteristic Wave Impedance:

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

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

  For $$\displaystyle TM_{11} $$: $$\displaystyle f_c = \frac{c}{2} \sqrt{(1/a)^2 + (1/b)^2} $$, then $$\displaystyle Z_{TM11} = \eta \sqrt{1 - (f_{c11}/f)^2} $$.

Circular Waveguide

• Radius $a$.

• Dominant Mode: $$\displaystyle TE_{11} $$ ($$\displaystyle \lambda_c \approx 3.41 a $$).

• Cutoff Frequency: $$\displaystyle f_c = \frac{1.841 c}{2\pi a} $$.

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

• TE/TM Modes for Transmission: All modes with $$\displaystyle f > f_c $$ propagate. For $$\displaystyle a=2\ cm $$, $$\displaystyle f=10\ GHz $$, compute $$\displaystyle f_c $$ for $$\displaystyle TE_{mn} $$ and $$\displaystyle TM_{mn} $$ using Bessel function roots.

2. S-Parameters

Definition and Necessity

At microwave frequencies, Z/Y parameters are impractical due to parasitic effects and lack of ideal terminations. S-parameters relate incident ($$\displaystyle a_i $$) and reflected ($$\displaystyle b_i $$) waves:

$$ \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} $$

Properties

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

• Losslessness: $$\displaystyle S^H S = I $$, i.e., $$\displaystyle \sum_k S_{ki}^* S_{kj} = \delta_{ij} $$.

• Symmetry: For symmetric networks, $$\displaystyle S_{11}=S_{22} $$, $$\displaystyle S_{12}=S_{21} $$.

Scattering Matrix for Reciprocal and Lossless Two-Port

From losslessness:

$$ |S_{11}|^2 + |S_{21}|^2 = 1, \quad |S_{22}|^2 + |S_{12}|^2 = 1, \quad S_{11} S_{21}^* + S_{12} S_{22}^* = 0 $$

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

Let $$\displaystyle S_{11} = |S_{11}| e^{j\theta_1} $$, $$\displaystyle S_{22} = |S_{11}| e^{j\theta_2} $$, $$\displaystyle S_{12} = |S_{12}| e^{j\theta_{12}} $$.

Then $$\displaystyle \theta_1 + \theta_2 = 2\theta_{12} + \pi $$ and $$\displaystyle |S_{12}| = \sqrt{1 - |S_{11}|^2} $$.

Thus:

$$ S = \begin{bmatrix} |S_{11}| e^{j\theta_1} & \sqrt{1-|S_{11}|^2} e^{j\theta_{12}} \\ \sqrt{1-|S_{11}|^2} e^{j\theta_{12}} & |S_{11}| e^{j\theta_2} \end{bmatrix} $$

If $$\displaystyle S_{12} $$ is real and positive ($$\displaystyle \theta_{12}=0 $$), then $$\displaystyle S_{22} = -S_{11}^* $$:

$$ S = \begin{bmatrix} S_{11} & \sqrt{1-|S_{11}|^2} \\ \sqrt{1-|S_{11}|^2} & -S_{11}^* \end{bmatrix} $$

Applications

• Oscillator Design: Oscillation condition: $$\displaystyle |\Gamma_{in} \Gamma_{out}| \geq 1 $$ and $$\displaystyle \angle(\Gamma_{in} \Gamma_{out}) = 0^\circ $$, where $$\displaystyle \Gamma_{in} $$ and $$\displaystyle \Gamma_{out} $$ are input/output reflection coefficients with feedback.

• Amplifier Analysis: Gain, stability circles using S-parameters.

Problem Solving

• Network Classification:

  • Reciprocal if $$\displaystyle S_{ij} = S_{ji} $$.

  • Lossless if $$\displaystyle S^H S = I $$ (check $$\displaystyle |S_{11}|^2+|S_{21}|^2=1 $$, etc.).

• Return Loss:

  With port 2 matched, $$\displaystyle RL = -20 \log_{10} |S_{11}| $$.

  With port 2 terminated in $$\displaystyle \Gamma_L $$:

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

Then $$\displaystyle RL = -20 \log_{10} |\Gamma_{in}| $$.

[!EXAMPLE]

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? $$\displaystyle |S_{11}|^2+|S_{21}|^2=0.04+0.36=0.4 \neq 1 $$, so no.

Return loss at port 1 with port 2 shorted ($$\displaystyle \Gamma_L=-1 $$):

$$\displaystyle \Gamma_{in} = 0.2 + \frac{0.6\angle90^\circ \times 0.6\angle90^\circ \times (-1)}{1 - 0.1\angle0^\circ \times (-1)} = 0.2 - \frac{0.36}{1.1} = -0.1273 $$,

$$\displaystyle RL = -20 \log_{10}(0.1273) \approx 19.9\ dB $$.

3. Microwave Passive Components

Impedance Matching Networks

• Purpose: Match load $$\displaystyle Z_L $$ to source $$\displaystyle Z_0 $$ for maximum power transfer.

• L-section: Two reactive elements in L-configuration.

  • For $$\displaystyle Z_L > Z_0 $$: series inductor then shunt capacitor, or vice versa.

  • For $$\displaystyle Z_L < Z_0 $$: series capacitor then shunt inductor.

• Quarter-Wave Transformer: $$\displaystyle Z_T = \sqrt{Z_0 Z_L} $$, length $$\displaystyle \lambda_g/4 $$, narrowband.

[!TIP]

L-section offers two possible configurations; choose based on component Q and realizability.

Impedance Transformers

• Single-Section: Quarter-wave transformer, narrow bandwidth.

• Multi-Section: Cascaded quarter-wave sections with intermediate impedances $$\displaystyle Z_0 < Z_1 < ... < Z_n < Z_L $$ to increase bandwidth.

• Bandwidth Enhancement: More sections give wider bandwidth but higher loss. Tapered designs (e.g., Klopfenstein) optimize.

Directional Couplers

• Coupling Factor ($C$): $$\displaystyle C = -20 \log_{10} |S_{14}| $$ (input at port 1, coupled at port 4).

• Directivity ($D$): $$\displaystyle D = C - I $$, where $$\displaystyle I = -20 \log_{10} |S_{13}| $$ (isolation).

• S-Matrix for Ideal Four-Port (all ports matched, perfect isolation):

$$ S = \begin{bmatrix} 0 & -j & 0 & -j \\ -j & 0 & -j & 0 \\ 0 & -j & 0 & -j \\ -j & 0 & -j & 0 \end{bmatrix} $$

Hybrid Tee (Magic Tee)

• Construction: Four ports: two collinear (1,2), two perpendicular (3,4). Combines E-plane and H-plane tee properties.

• Properties:

  • Power into port 1: equal split to ports 2 (in-phase) and 3 (out-of-phase), port 4 isolated.

  • Power into port 4: equal split to ports 2 and 3 with 90° phase difference.

• S-Matrix Derivation:

  Assume matched ports: $$\displaystyle S_{11}=S_{22}=S_{33}=S_{44}=0 $$.

  From properties: $$\displaystyle S_{21}=S_{12}=\alpha $$, $$\displaystyle S_{31}=S_{13}=-\alpha $$, $$\displaystyle S_{41}=S_{14}=0 $$, $$\displaystyle S_{23}=S_{32}=0 $$, $$\displaystyle S_{24}=S_{42}=\beta $$, $$\displaystyle S_{34}=S_{43}=\beta $$.

  Unitarity gives $$\displaystyle |\alpha|^2+|\beta|^2=1 $$, and for equal split, $$\displaystyle |\alpha|=|\beta|=1/\sqrt{2} $$.

  Choosing phases: $$\displaystyle \alpha=1/\sqrt{2} $$, $$\displaystyle \beta=1/\sqrt{2} $$ yields:

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

Circulators and Isolators

• Circulator: Non-reciprocal three or four-port; power flows sequentially (e.g., 1→2, 2→3, 3→1).

• Isolator: Two-port circulator with one port terminated.

• Working using Magic Tee and Phase Shifter:

  A four-port circulator can be realized by connecting a magic tee with a 90° phase shifter in one arm. The phase shifter breaks reciprocity, enabling circulation.

• S-Matrix for Ideal Three-Port Circulator:

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

Phase Shifters

• Diode Phase Shifters: Use PIN or varactor diodes.

  • PIN diode: Forward bias → resistive (short); reverse bias → capacitive (open). Switching changes electrical length.

  • Varactor: Voltage-controlled capacitance for continuous phase shift.

• Broadband vs Tuned Detectors:

  • Broadband: Respond over wide band (e.g., thermocouple).

  • Tuned: Resonant circuit for specific frequency (e.g., crystal detector with IF filter).

4. Microwave Active Devices: Solid-State Diodes

Gunn Diode

• Principle: Gunn effect in GaAs/InP due to transferred electron mechanism (negative differential resistance).

• Domains of Operation:

  1. Quenched Domain: Low field, no domain, current increases with voltage.

  2. Accumulation Layer: Domain forms at cathode.

  3. Transit Time: Domain moves to anode, current drops.

  4. Saturation: Domain absorbed, current rises.

• Applications: Microwave oscillators (X-band).

IMPATT and TRAPATT Diodes

• IMPATT (Impact Ionization Avalanche Transit Time):

  Avalanche multiplication + carrier transit time → negative resistance. Structure: p⁺-n⁻-n⁺ or p-i-n.

• TRAPATT (Trapped Plasma Avalanche Triggered Transit):

  Plasma formation and collapse; higher power than IMPATT.

Schottky Barrier Diode

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

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

• Characteristics: Low capacitance, low noise.

• Use as Mixer and Detector:

  • Mixer: Nonlinear I-V for frequency conversion.

  • Detector: Rectification of RF signal.

Tunnel Diode

• Structure: Heavily doped p-n junction.

• Modes: Forward bias exhibits negative differential resistance due to tunneling. Used in oscillators/amplifiers.

BARITT Diode

• Principle: Barrier Injection and Transit Time. Carriers injected over a barrier (thermionic emission), drift through intrinsic region → negative resistance. Lower noise than IMPATT.

[!TIP]

Gunn and IMPATT for high-power sources; Schottky for mixers/detectors due to speed.

5. Microwave Active Devices: Tubes and Transistors

Klystron Amplifiers

• Reflex Klystron:

  • Construction: Electron gun, resonant cavity (with reflector/repeller), collector.

  • Working: Velocity modulation → bunching → energy transfer to cavity.

  • Mode Curve: Output power vs frequency; peaks correspond to cavity resonances.

• Two-Cavity Klystron:

  • Velocity Modulation: RF in buncher cavity modulates electron velocity; drift space forms bunches; catcher cavity extracts energy.

Traveling Wave Tube (TWT)

• Interaction Mechanism: Electron beam interacts with RF wave on a slow-wave structure (helix/coupled cavities). Beam velocity synchronized to wave phase velocity → continuous energy transfer.

• Working: Electron gun → focusing → helix → collector.

• Applications: High-power broadband amplifiers (satellite comms, radar).

Magnetron

• Types: Pulsed (high peak power), CW (continuous wave).

• Oscillation Mechanism:

  Crossed E and B fields; electrons interact with cavities.

  π-mode: Adjacent cavities 180° out of phase, maximum output. Strapping prevents mode jumping.

Microwave Transistors

• BJT:

  • Working: Current amplification via base control.

  • Frequency Limitations: Base transit time, collector capacitance, $$\displaystyle f_T $$ (transition frequency).

  • Applications: Low-noise amplifiers up to few GHz.

• FET:

  • Basic Relations: $$\displaystyle I_D = f(V_{GS}, V_{DS}) $$, transconductance $$\displaystyle g_m $$, gate-source capacitance $$\displaystyle C_{gs} $$.

  • Characteristics: High input impedance, suitable for high frequency.

  • Applications: Amplifiers, mixers up to tens of GHz.

[!TIP]

TWT: broadband high power; Klystron: high gain, stable; Magnetron: high peak power for radar.

6. Microwave Measurements and Detectors

Voltage Standing Wave Ratio (VSWR)

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

• Derivation from Reflection Coefficient $\Gamma$:

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

Conversely, $$\displaystyle \Gamma = \frac{VSWR-1}{VSWR+1} e^{j\theta} $$.

Measurement Techniques (Slotted Line)

1. Connect slotted line to load.

2. Move probe to find $$\displaystyle V_{max} $$ and $$\displaystyle V_{min} $$ → VSWR.

3. $$\displaystyle \lambda_g = 2 \times $$ distance between minima.

4. Measure distance $$\displaystyle d_{min} $$ from load to first minimum → $$\displaystyle \Gamma = \frac{VSWR-1}{VSWR+1} e^{-j2\beta d_{min}} $$ → $$\displaystyle Z_L = Z_0 \frac{1+\Gamma}{1-\Gamma} $$.

Detectors

• Tuned Detectors: Resonant circuit selects specific frequency (e.g., crystal detector with IF amp).

• Broadband Detectors: Wideband response (e.g., thermistor, thermocouple for power).

Power Measurement

• Bridges: Bolometer bridge (temperature-sensitive resistor), thermocouple bridge.

7. Mixers, Oscillators, and Frequency Converters

Microwave Mixer

• Working Principle: Nonlinear device (diode) combines RF and LO → sum/difference frequencies.

• Signals:

  • RF: Input signal.

  • LO: High-power local oscillator.

  • IF: Output intermediate frequency.

• Conversion Loss: $$\displaystyle L_c = P_{RF}/P_{IF} $$ (typically 6–9 dB) due to image and conversion process.

Oscillator Design using S-Parameters

• Oscillation Condition:

  For a two-port with feedback,

$$ |\Gamma_{in} \Gamma_{out}| \geq 1 \quad \text{and} \quad \angle(\Gamma_{in} \Gamma_{out}) = 0^\circ $$

where

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

$$\displaystyle \Gamma_{out} = S_{22} + \frac{S_{12} S_{21} \Gamma_S}{1 - S_{11} \Gamma_S} $$.

For oscillator, $$\displaystyle \Gamma_S $$ and $$\displaystyle \Gamma_L $$ often chosen as open/short.

Frequency Multipliers

• Derivation: Nonlinear device generates harmonics; output at $$\displaystyle n f_{in} $$ for $$\displaystyle n^{th} $$ harmonic.

• Operating Principle: Varactor or step-recovery diode for efficient multiplication.

Frequency Converters

• Block Diagram with Schottky Diodes:

  RF and LO combined via hybrid/directional coupler → Schottky diode → IF filter → IF output.

  DiagramCANVAS: Block diagram: RF input → hybrid combiner → Schottky diode → IF filter → IF output; LO input to hybrid

8. Resonators and Frequency Control

YIG Resonator

• Structure: Yttrium Iron Garnet (YIG) sphere in magnetic field, coupled to microwave cavity.

• Working Principle: Magnetic resonance frequency tunable by external field: $$\displaystyle f = \frac{\gamma H}{2\pi} $$ ($\gamma$: gyromagnetic ratio).

• Frequency Tuning: Linear with magnetic field.

• Applications: Tunable filters, oscillators, frequency meters.

Microwave Resonators

• Types: Cavity (rectangular/circular), dielectric, planar (microstrip).

• Characteristics: High Q-factor, narrow bandwidth.

Phase Shifters (Revisited)

• Diode Phase Shifters: PIN/varactor diodes change electrical length. PIN: switching (digital); varactor: continuous (analog).

9. Applications and Special Topics

Applications of Microwaves

• Radar: Object detection, speed measurement.

• Communications: Satellite, cellular, Wi-Fi.

• Heating: Microwave ovens.

• Others: Medical therapy, industrial drying.

Solid-State Microwave Sources

• Practical Uses:

  • Mobile phones: GaAs FETs.

  • Satellite transponders: TWTs, klystrons.

  • Radar: Magnetrons, TWTs.

  • Local oscillators: Gunn diodes.

Special Devices and Concepts

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

  • Principle: Population inversion in ammonia/other media → stimulated emission at microwaves.

  • Applications: Low-noise amplifier in radio astronomy.

• TEM Mode:

  • Characteristics: Both E and H transverse to propagation; no cutoff frequency.

  • Relevance: Only mode in two-conductor lines (coaxial, parallel wire).

[!TIP]

MASER is precursor to laser; TEM mode is fundamental for transmission lines.