Microwave Engineering (EC-702 (A)) Unit 5 Notes

I. TRANSMISSION LINES AND WAVEGUIDES

A. Microstrip Lines

Structure: Conductor strip (width $W$, thickness $t$) on dielectric substrate (height $h$, permittivity $$\displaystyle \epsilon_r $$) with ground plane.

DiagramCANVAS: Cross-section showing top strip, dielectric, and ground plane.
Effective dielectric constant ($$\displaystyle \epsilon_{eff} $$): Accounts for fringing fields; part in air, part in dielectric.

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

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

For $$\displaystyle W/h < 1 $$, use modified formula.

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

For $W/h \leq 1$:

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

For $W/h \geq 1$:

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

[!TIP] $$\displaystyle \epsilon_{eff} $$ lies between $1$ and $$\displaystyle \epsilon_r $$. $$\displaystyle Z_0 $$ decreases as $W$ increases or $$\displaystyle \epsilon_r $$ increases.

B. Slotted Line (Slotted Waveguide)

Construction: Waveguide with a longitudinal slot on broad wall; probe detects field strength at slot.
Working: Probe moved along slot to locate voltage maxima/minima.
Measurements:
- VSWR: $$\displaystyle \text{VSWR} = \frac{V_{\text{max}}}{V_{\text{min}}} = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$
- Guided wavelength ($$\displaystyle \lambda_g $$): Distance between two minima $$\displaystyle = \lambda_g/2 $$, so $$\displaystyle \lambda_g = 2 \Delta z_{\text{min}} $$.
- Impedance: From VSWR ($|\Gamma|$) and distance $$\displaystyle d_{\text{min}} $$ from load to first minimum, compute phase $$\displaystyle \theta = -2\beta d_{\text{min}} $$, then $$\displaystyle Z_L = Z_0 \frac{1+\Gamma}{1-\Gamma} $$.
[!TIP] Ensure probe does not disturb field; use calibrated detector.

C. Rectangular Waveguides

Modes: TE$$\displaystyle _{mn} $$ ($$\displaystyle E_z=0 $$), TM$$\displaystyle _{mn} $$ ($$\displaystyle H_z=0 $$); $$\displaystyle m,n=0,1,2,\dots $$ but for TE, $m$ and $n$ cannot both be zero; for TM, neither can be zero.
Dominant mode: TE$$\displaystyle _{10} $$ (lowest cutoff).
Cutoff wavelength ($$\displaystyle \lambda_c $$):

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

For TE$$\displaystyle _{10} $$: $$\displaystyle \lambda_c = 2a $$.

Cutoff frequency ($$\displaystyle f_c $$): $$\displaystyle f_c = \frac{c}{\lambda_c} = \frac{c}{2a} $$ for TE$$\displaystyle _{10} $$ in air.
Guided wavelength ($$\displaystyle \lambda_g $$):

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

Phase velocity ($$\displaystyle v_p $$):

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

Wave impedance:
- TE modes: $$\displaystyle Z_{\text{TE}} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$
- TM modes: $$\displaystyle Z_{\text{TM}} = \eta \sqrt{1 - (f_c/f)^2}} $$
where $$\displaystyle \eta = \sqrt{\mu/\epsilon} $$ is intrinsic impedance of filling medium.

[!TIP] For TM$$\displaystyle _{11} $$, compute $$\displaystyle f_c $$ using $a,b$, then $$\displaystyle Z_{\text{TM}} $$.

D. Circular Waveguides

Dominant mode: TE$$\displaystyle _{11} $$ (lowest cutoff).
Cutoff wavelength for TE$$\displaystyle _{mn} $$: $$\displaystyle \lambda_c = \frac{2\pi a}{x'_{mn}} $$, where $$\displaystyle x'_{mn} $$ is $m$-th root of $$\displaystyle J_n'(x)=0 $$. For TE$$\displaystyle _{11} $$: $$\displaystyle x'_{11} \approx 1.841 $$, so $$\displaystyle \lambda_c \approx 3.41a $$.
Cutoff frequency: $$\displaystyle f_c = \frac{c}{\lambda_c} \approx \frac{0.293 c}{a} $$ (air-filled).
Guided wavelength: $$\displaystyle \lambda_g = \lambda / \sqrt{1 - (f_c/f)^2} $$.
Mode analysis: For given $f$ and radius $a$, compute $$\displaystyle f_c $$ for each mode; propagation if $$\displaystyle f > f_c $$.

E. Strip Lines

Structure: Two parallel ground planes with a center conductor embedded in homogeneous dielectric.

DiagramCANVAS: Cross-section: top and bottom ground planes, dielectric slab, center strip.
Comparison with microstrip:

| Feature | Strip Line | Microstrip | |---------|------------|------------| | Shielding | Excellent (enclosed) | Poor (open) | | Radiation | Low | High | | Dispersion | Low | High | | Fabrication | Difficult (drilling for connectors) | Easy |
Dominant mode: Quasi-TEM (due to conductor presence, but near-TEM).
Higher-order modes: TE, TM appear at higher frequencies.

F. TEM Mode

Definition: Transverse Electromagnetic mode; both $\vec{E}$ and $\vec{H}$ perpendicular to propagation direction.
Conditions: Exists only in two-conductor transmission lines (coaxial, two-wire) with uniform cross-section and homogeneous medium.
Properties: No cutoff frequency (propagates down to DC); $$\displaystyle v_p = 1/\sqrt{LC} $$; $$\displaystyle Z_0 = \sqrt{L/C} $$.
Relevance: Used in low-frequency and broadband applications; not supported in single-conductor waveguides (rectangular, circular).

II. MICROWAVE VACUUM TUBES (ELECTRON DEVICES)

A. Klystron Amplifiers

1. Reflex Klystron

Construction: Cathode → Anode (accelerating) → Cavity gap → Drift space → Repeller (reflector) → back to cavity.

DiagramCANVAS: Schematic showing cathode, anode, cavity, drift tube, repeller.
Working:
1. Electrons emitted, accelerated by anode voltage.
2. Pass through cavity gap; RF field modulates velocity (velocity modulation).
3. Electrons travel to repeller; repeller voltage reflects them back.
4. Reflected electrons bunch in drift space; bunched electrons pass cavity again, transferring energy to RF field → oscillation.
Mode curve: Output power vs. repeller voltage. Shows periodic peaks; each peak corresponds to a mode (different bunching conditions). Highest peak is used.
Applications: Low-power oscillator (mW range), local oscillators, signal sources.

[!TIP] Mode spacing $$\displaystyle \Delta V_r \approx V_0 / (2N) $$, where $N$ is number of cycles in drift space.

2. Two-Cavity Klystron Amplifier

Construction: Two resonant cavities (input and output) separated by drift tube.
Velocity modulation: RF signal in input cavity creates voltage across gap, modulating electron velocity.
Bunching: Faster electrons catch slower ones, forming electron bunches in drift tube.
Amplification: Bunched electrons pass through output cavity gap, induce current, amplify RF signal.
Gain: $$\displaystyle G \propto (V_0)^2 $$ (beam voltage squared), depends on cavity coupling.

B. Traveling Wave Tube (TWT)

Construction: Electron gun → Helix (slow-wave structure) → Collector.

DiagramCANVAS: TWT with electron gun, helix, collector, magnetic focusing.
Interaction mechanism:
- Electron beam travels along helix with velocity $$\displaystyle v_e $$.
- RF wave travels along helix with phase velocity $$\displaystyle v_p \approx c $$ (but helix slows it to $$\displaystyle v_p \approx v_e $$).
- Continuous interaction: electrons gain energy from RF wave (if $$\displaystyle v_e > v_p $$) or lose energy (if $$\displaystyle v_e < v_p $$), leading to amplification.
Amplification: Broadband (due to helix), high gain (40–50 dB), low noise.

[!TIP] Helix provides wide bandwidth but limited power; coupled-cavity TWT for higher power.

C. Magnetron

Types: Cylindrical (multicavity), annular, strapped (for mode separation).
Oscillation mechanism:
1. Electrons emitted from cathode, accelerated toward anode (with cavities) by $$\displaystyle V_a $$.
2. Axial magnetic field $B$ perpendicular to $E$; electrons follow curved paths.
3. At critical field ($$\displaystyle B_c $$), electrons just reach anode.
4. Space-charge focusing: RF fields in cavities cause electrons to form rotating spokes; spokes transfer energy to cavities → oscillation.
Role of cavities: Each cavity acts as an LC resonator; frequency $$\displaystyle f \approx \frac{1}{2\pi\sqrt{LC}} \propto 1/\text{cavity dimensions} $$.
Output coupling: Loop or probe in one cavity.

[!TIP] Strapping (connecting alternate cavities) prevents mode competition.

III. SOLID-STATE MICROWAVE DEVICES

A. Gunn Diode

Principle: Transferred electron effect in GaAs (or InP). GaAs has two conduction band valleys:
- Lower valley: high mobility ($$\displaystyle \mu_1 $$), low energy.
- Upper valley: low mobility ($$\displaystyle \mu_2 $$), high energy.
At high electric fields ($$\displaystyle E > E_{\text{th}} $$), electrons gain energy and transfer to upper valley → negative differential resistance ($$\displaystyle dI/dE < 0 $$).
Domain formation: High-field domain (space charge) forms near cathode, travels to anode, collapses; repeats.
Domains of operation:

| Domain | Bias | Domain behavior | Application | |--------|------|----------------|-------------| | Stable amplification | Just above threshold | Single domain travels steadily | Low-noise amplifiers | | Relaxation oscillations | Moderate | Domain forms, collapses, reforms | Low-power oscillators | | LSA (Limited Space-charge Accumulation) | High | Multiple domains, high velocity | High-frequency oscillators (≥100 GHz) |
Gunn effect: Current instabilities due to domain propagation.

[!TIP] Gunn diodes require $n$-type GaAs with $$\displaystyle n \approx 10^{21}–10^{22} \text{ m}^{-3} $$.

B. Schottky Barrier Diode

Structure: Metal (e.g., Pt, Au) on $n$-type semiconductor (e.g., GaAs). Forms Schottky barrier.

DiagramCANVAS: Metal-semiconductor junction with depletion region.
I-V characteristics: $$\displaystyle I = I_s (e^{qV/nkT} - 1) $$, similar to PN but with lower forward voltage ($\sim 0.2–0.3$ V) and no minority carrier storage.
Advantages over PN diode: Fast switching (no charge storage), suitable for high frequencies (up to 100+ GHz).
Applications:
- Mixer: Nonlinear I-V used for frequency conversion (RF + LO → IF). Typically in diode ring mixer for balanced operation.
- Detector: Operates in square-law region ($V \ll kT/q$), output $$\displaystyle \propto V_{\text{RF}}^2 $$ → envelope detection.
[!TIP] Schottky diodes have lower capacitance than PN diodes, hence higher $$\displaystyle f_{\text{max}} $$.

C. IMPATT Diode

Principle: Impact ionization and transit-time delay → negative resistance.
- Structure: PN junction or PIN with high reverse bias near breakdown.
- Avalanche generates carriers; carriers drift across depletion region.
- Phase delay between AC current and voltage gives negative resistance.
TRAPATT mode: Trapped plasma avalanche transit time. High-current plasma forms and propagates, giving higher efficiency but higher noise.
Comparison:

| Device | Power | Noise | Frequency | |--------|-------|-------|-----------| | IMPATT | High (W) | High | Up to 100 GHz | | Gunn | Moderate | Low | Up to 100 GHz | | Schottky | Low | Moderate | Up to 100+ GHz |

[!TIP] IMPATT requires high bias voltage ($\sim 100$ V) and has high noise figure.

D. Microwave Bipolar Junction Transistor (BJT)

Working: NPN/PNP; minority carrier injection from emitter to base, amplification via base current control.
Frequency limitations:
- Carrier transit time across base: $$\displaystyle \tau_b = W_b^2/(2D_n) $$ or $$\displaystyle W_b/v_{\text{sat}} $$.
- Base charging time: Time to charge base-emitter capacitance.
- Junction capacitances: $$\displaystyle C_{be} $$, $$\displaystyle C_{bc} $$ (Miller effect).
- $$\displaystyle f_T $$ (transition frequency): $$\displaystyle f_T = \frac{1}{2\pi (\tau_e + \tau_b + \tau_c)} $$, where $\tau$ are delay times.
Applications: Amplifiers up to few GHz; limited above 5 GHz due to transit time.

[!TIP] Reduce base width $$\displaystyle W_b $$ to increase $$\displaystyle f_T $$ (e.g., HBTs).

E. Field-Effect Transistor (FET)

Structure: MESFET (Metal-Semiconductor FET) on GaAs: gate (Schottky), source, drain.

DiagramCANVAS: MESFET cross-section with gate, source, drain on GaAs.
Operation: Gate voltage controls depletion width, hence channel conductivity → voltage-controlled current.
Advantages over BJT:
- Higher input impedance (gate is reverse-biased Schottky).
- No minority carrier storage → faster.
- Better high-frequency performance ($$\displaystyle f_{\text{max}} > 100 $$ GHz).
Applications: Low-noise amplifiers, power amplifiers, oscillators (X-band and above).

F. Other Semiconductor Devices

BARITT Diode: Barrier Injection and Transit Time. Uses barrier (e.g., p-i-n) with injection delay; negative resistance from carrier drift. Lower efficiency than IMPATT, but lower noise.
MASER (Microwave Amplification by Stimulated Emission of Radiation):
- Principle: Population inversion in paramagnetic ions (e.g., Cr$$\displaystyle ^{3+} $$ in ruby) pumped by microwave/optical energy. Stimulated emission amplifies weak signals.
- Applications: Ultra-low-noise amplifiers (e.g., radio astronomy), frequency standards.
[!TIP] MASER operates at cryogenic temperatures; precursor to laser.

IV. NETWORK ANALYSIS WITH S-PARAMETERS

A. Scattering Matrix (S-Matrix)

Definition: Relates incident ($$\displaystyle a_i $$) and reflected ($$\displaystyle b_i $$) waves at ports:

$$ \mathbf{b} = \mathbf{S} \mathbf{a} $$

where $$\displaystyle S_{ij} = \frac{b_i}{a_j}\bigg|_{a_k=0 \ (k \neq j)} $$.

Need at microwave: Cannot measure true voltage/current due to standing waves, connectors, frequency dependence. S-parameters use traveling waves, measurable with network analyzer.
$$\displaystyle S_{ij} $$ meaning:
- $$\displaystyle S_{ii} $$: Reflection coefficient at port $i$ when all other ports matched.
- $$\displaystyle S_{ij} $$ ($i \neq j$): Transmission coefficient from port $j$ to port $i$.

B. Challenges with Traditional Parameters (Z, Y, h, ABCD)

True open/short impossible: At microwave, open has fringing capacitance, short has inductance.
Connector issues: Parasitics, repeatability.
Frequency-dependent effects: Skin effect, dielectric loss make parameters complex and hard to separate.
S-parameters avoid these by using wave-based definitions.

C. Properties of S-Parameters

Reciprocity: For reciprocal network, $$\displaystyle S_{ij} = S_{ji} $$ (all $i,j$).
Losslessness: For lossless network, $$\displaystyle \mathbf{S}^\dagger \mathbf{S} = \mathbf{I} $$ (unitary matrix).

Equivalently, for each row $i$: $$\displaystyle \sum_{j=1}^N |S_{ij}|^2 = 1 $$, and rows are orthogonal.

[!TIP] Lossless implies energy conservation; check row sums of $$\displaystyle |S_{ij}|^2 $$.

D. Two-Port Reciprocal and Lossless Network

General S-matrix:

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

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

$$ |S_{11}|^2 + |S_{12}|^2 = 1 \quad \text{(row 1)} $$

$$ |S_{21}|^2 + |S_{22}|^2 = 1 \quad \text{(row 2)} $$

and orthogonality: $$\displaystyle S_{11}S_{11}^* + S_{12}S_{12}^* = 1 $$, etc.

Example: Lossless transmission line of length $l$:

$$\displaystyle S_{11}=0 $$, $$\displaystyle S_{21}=S_{12}=e^{-j\beta l} $$, $$\displaystyle S_{22}=0 $$.

E. S-Parameter Applications

1. Oscillator Design

Oscillation condition: $$\displaystyle |\Gamma_{\text{in}}| \geq 1 $$, where $$\displaystyle \Gamma_{\text{in}} $$ is input reflection coefficient with output terminated in feedback network.
Derivation:

For a two-port active device with S-parameters and a feedback network with reflection $$\displaystyle \Gamma_f $$ at output:

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

Oscillation requires $$\displaystyle |\Gamma_{\text{in}}| \geq 1 $$ and phase condition (total phase shift $$\displaystyle 0^\circ $$ or $$\displaystyle 360^\circ $$).

Stability factor: $$\displaystyle K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2|S_{12}S_{21}|} $$, where $$\displaystyle \Delta = S_{11}S_{22} - S_{12}S_{21} $$. $$\displaystyle K < 1 $$ indicates unconditional instability (potential oscillator).

[!TIP] For oscillator design, choose $$\displaystyle \Gamma_f $$ such that $$\displaystyle \Gamma_{\text{in}} $$ lies outside unit circle.

2. Mixer Analysis

S-parameters model conversion loss, isolation, and port matching.
Conversion loss: $$\displaystyle CL = \frac{P_{\text{RF}}}{P_{\text{IF}}} $$ (linear) or $$\displaystyle CL_{\text{dB}} = 10 \log_{10}(P_{\text{RF}}/P_{\text{IF}}) $$.
Isolation between ports given by $$\displaystyle |S_{ij}| $$ (e.g., LO-RF isolation $$\displaystyle |S_{23}| $$).

F. Problem-Solving with S-Parameters

Prove reciprocity: Show $$\displaystyle S_{ij} = S_{ji} $$.
Prove losslessness: Verify $$\displaystyle \mathbf{S}^\dagger \mathbf{S} = \mathbf{I} $$ (row sums $$\displaystyle =1 $$, orthogonal).
Return loss (RL): $$\displaystyle RL = -20 \log_{10} |S_{11}| $$ (dB).
Insertion loss (IL): $$\displaystyle IL = -20 \log_{10} |S_{21}| $$ (for matched ports).
Gain: $$\displaystyle G = |S_{21}|^2 $$ (linear) or $$\displaystyle G_{\text{dB}} = 20 \log_{10} |S_{21}| $$.
S-parameters with port 2 shorted:

$$\displaystyle a_2 = -b_2 $$ (short circuit). Solve:

$$\displaystyle b_2 = S_{21}a_1 + S_{22}a_2 = S_{21}a_1 - S_{22}b_2 $$

$$\displaystyle \Rightarrow b_2 = \frac{S_{21}}{1 + S_{22}} a_1 $$

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

So effective $$\displaystyle S_{11}^{\text{(short)}} = S_{11} - \frac{S_{12}S_{21}}{1 + S_{22}} $$.

V. PASSIVE COMPONENTS AND IMPEDANCE MATCHING

A. Impedance Matching Networks

Purpose: Maximum power transfer, minimize reflections (VSWR), improve isolation.
L-network: Two reactive elements (L or C) in L-shape.
- Topology: Series-shunt or shunt-series depending on $$\displaystyle Z_L $$ vs $$\displaystyle Z_0 $$.
- Design: $$\displaystyle Q = \sqrt{\max(Z_L/Z_0, Z_0/Z_L) - 1} $$.
Single-stub matching: Shunt stub (open/short) at distance $d$ from load.

Steps:
1. Normalize $$\displaystyle z_L = Z_L/Z_0 $$, plot on Smith chart.
2. Move toward generator to circle $$\displaystyle g = 1 $$ (conductance circle).
3. At intersection, add stub to cancel susceptance.
Two matching techniques:
1. Quarter-wave transformer: Single $\lambda/4$ line with $$\displaystyle Z_{0}' = \sqrt{Z_L Z_0} $$. Narrowband ($\sim 10\%$ bandwidth).
2. Stub matching: Single-stub (narrowband) or double-stub (wider, but limited by forbidden region).

B. Impedance Transformers

Single-section transformer:

$$ Z_{0i} = \sqrt{Z_{0,i-1} \cdot Z_{0,i+1}} $$

for each $\lambda/4$ section. Bandwidth limited because reflection coefficient $$\displaystyle \Gamma = 0 $$ only at center frequency.

Multi-section transformer:
- Binomial: Maximally flat response; impedance profile from binomial expansion.
- Chebyshev: Equal ripple; wider bandwidth for same number of sections.
- Bandwidth increase: Stepped impedance profile smooths reflection coefficient vs frequency, extending bandwidth.
[!TIP] Multi-section transformers trade bandwidth for increased complexity and loss.

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

Construction: Combination of E-plane tee and H-plane tee with matched junction. Four ports:
- Ports 1,2: H-plane (collinear).
- Ports 3,4: E-plane (collinear).
DiagramCANVAS: Magic tee with four ports; E-plane and H-plane tees combined.
Working principle:
- E-plane tee: Signal into port 3 divides equally into ports 1 and 2 in phase.
- H-plane tee: Signal into port 1 divides equally into ports 3 and 4 with $$\displaystyle 180^\circ $$ phase difference.
- Combined: Magic tee properties:
  - Ports 1 and 2 isolated ($$\displaystyle S_{12}=0 $$).
  - Ports 3 and 4 isolated ($$\displaystyle S_{34}=0 $$).
  - All ports matched ($$\displaystyle S_{ii}=0 $$).
S-matrix derivation: From symmetry and isolation:

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

Applications:
- Sum/difference: Two signals in phase at H-ports add at port 3 (sum), out of phase add at port 4 (difference).
- Power divider/coupler.

D. Directional Coupler

Coupling factor ($C$): $$\displaystyle C = -20 \log_{10} |S_{31}| $$ (if port 1 input, port 3 coupled).
Directivity ($D$): $$\displaystyle D = -20 \log_{10} \left| \frac{S_{31}}{S_{41}} \right| $$ (isolation between coupled and isolated ports).
S-matrix for matched four-port directional coupler:

Assuming ports: 1=input, 2=through, 3=coupled, 4=isolated:

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

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

[!TIP] Coupling factor $$\displaystyle C = -20 \log_{10} |\kappa| $$; through loss $$\displaystyle = -20 \log_{10} |\tau| $$.

E. Circulator

Symbol: Triangle with arrows indicating circulation direction.
Schematic: Often using ferrite and magnetic field; can be built with two magic tees and a phase shifter.

DiagramCANVAS: Circulator using two magic tees connected by a $$\displaystyle 90^\circ $$ phase shifter.
Working: Non-reciprocal; signal from port 1 → port 2, port 2 → port 3, port 3 → port 1.
Simplified S-matrix (3-port):

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

F. Isolator

Principle: Uses ferrite in magnetic field to allow propagation in one direction only (non-reciprocal).

DiagramCANVAS: Isolator with ferrite slab and magnetic field.
Use: Protect source (e.g., klystron, TWT) from reflected power; placed between source and load.

G. YIG (Yttrium Iron Garnet) Resonator

Structure: YIG sphere (ferrimagnetic) placed in static magnetic field $$\displaystyle H_0 $$; coupling loops around sphere.

DiagramCANVAS: YIG sphere with bias magnet and coupling loops.
Working principle: Ferrimagnetic resonance. When microwave frequency $f$ matches Larmor frequency $$\displaystyle f_0 = \frac{\gamma}{2\pi} H_0 $$ ($\gamma$ = gyromagnetic ratio), YIG absorbs strongly → resonance.
Frequency tuning: $$\displaystyle f \propto H_0 $$; vary $$\displaystyle H_0 $$ by changing electromagnet current → continuous tuning over wide range (e.g., 2–18 GHz).
Applications: Tunable filters, oscillators, frequency references, YIG-tuned amplifiers.

H. Microwave Resonators (General)

Types:
- Cavity resonators: Metallic enclosures (rectangular, cylindrical); high $Q$ ($$\displaystyle 10^3–10^5 $$).
- Dielectric resonators: Low-loss ceramic (e.g., TiO$$\displaystyle _2 $$); high $Q$, compact.
- YIG resonators: Tunable via magnetic field.
Quality factor ($Q$):

$$ Q = \frac{\omega_0 \cdot \text{Stored energy}}{\text{Power loss}} = \frac{f_0}{\Delta f} $$

Measures energy storage vs loss.

Applications: Filters, oscillators, frequency stabilization, power combining.

VI. SYSTEM COMPONENTS AND MEASUREMENTS

A. Microwave Mixer (Frequency Converter)

Principle: Nonlinear device (diode) multiplies RF and LO signals:

$$\displaystyle i(t) \propto v_{\text{RF}}(t) \cdot v_{\text{LO}}(t) $$ → produces sum ($$\displaystyle f_{\text{RF}}+f_{\text{LO}} $$) and difference ($$\displaystyle |f_{\text{RF}}-f_{\text{LO}}| $$) frequencies.
Signals:
- RF (Radio Frequency): Input signal.
- LO (Local Oscillator): High-power reference.
- IF (Intermediate Frequency): Desired output (usually difference frequency).
Conversion loss (CL):

$$ CL = \frac{P_{\text{RF}}}{P_{\text{IF}}} \quad (\text{linear}) \quad \text{or} \quad CL_{\text{dB}} = 10 \log_{10} \frac{P_{\text{RF}}}{P_{\text{IF}}} $$

Typical 5–10 dB. Caused by mismatch, conversion loss, image rejection.

Block diagram (Schottky diode ring mixer):

DiagramCANVAS: Four Schottky diodes in ring configuration; RF, LO, IF ports.
- Balanced design suppresses even harmonics and noise.

B. Frequency Multipliers

Derivation: For nonlinear $$\displaystyle i = a_0 + a_1 v + a_2 v^2 + \cdots $$, with $$\displaystyle v = V \cos \omega t $$:

$$\displaystyle v^2 = V^2 \cos^2 \omega t = \frac{V^2}{2} (1 + \cos 2\omega t) $$ → generates $2\omega$.

In general, $$\displaystyle v^n $$ generates $n\omega$.
Operating principle:
1. Apply input $f$ to nonlinear device (diode, transistor).
2. Output contains harmonics $nf$.
3. Bandpass filter selects desired harmonic $nf$.
Applications: Generate higher frequencies from lower source (e.g., 2×, 4× multipliers in source modules).

C. Phase Shifters

Diode phase shifter:
- Varactor diode: Voltage-controlled capacitance; phase shift $\phi \propto \sqrt{C(V)}$.
- Switched-line: Digital phase shift; switches select different line lengths.
Broadband vs. tuned:
- Broadband: Switched-line or loaded-line; constant phase shift over wide band.
- Tuned: Varactor in resonant circuit; narrowband, continuous tuning.

D. Detectors

Broadband detectors:
- Operate in square-law region ($$\displaystyle V_{\text{RF}} \ll kT/q $$).
- Output $$\displaystyle V_{\text{out}} \propto P_{\text{RF}} $$ (power).
- Wide bandwidth, low sensitivity.
Tuned detectors:
- Use resonant circuit (LC) at IF.
- Narrowband, high sensitivity.
- Used in receivers with IF amplification.

E. Power Measurement

Bolometer bridge:
- Sensor: Thermistor or barretter (resistance changes with temperature).
- Bridge circuit: Balanced initially; RF power heats sensor, unbalances bridge.
- Measurement: Bridge voltage proportional to absorbed power.
- Types: Barretter (wire), thermistor (semiconductor).
[!TIP] Bolometer measures average power; requires calibration.

VII. SPECIAL TOPICS & SHORT NOTES (From Past Papers)

A. TEM Mode of Propagation

Definition: Mode with both $\vec{E}$ and $\vec{H}$ entirely transverse to direction of propagation.
Existence: Only in two-conductor transmission lines (coaxial, two-wire) with homogeneous dielectric.
Properties: No cutoff frequency ($$\displaystyle f_c = 0 $$); propagates down to DC; $$\displaystyle v_p = 1/\sqrt{LC} $$; $$\displaystyle Z_0 = \sqrt{L/C} $$.
Contrast with TE/TM: TE/TM modes in waveguides have cutoff frequencies and $$\displaystyle E_z $$ or $$\displaystyle H_z $$ nonzero.

B. Velocity Modulation

Principle: In klystron, RF field in cavity gap modulates electron velocities.
Process: Electrons entering gap when voltage is increasing gain velocity; those when decreasing lose velocity → velocity spread.
Result: Velocity modulation leads to bunching in drift space → energy transfer to RF field.

C. Interaction Mechanism in TWT

Continuous interaction: Electron beam and RF wave travel along slow-wave structure (helix) with nearly equal velocities.
Energy transfer: If $$\displaystyle v_e > v_p $$, electrons lose energy to wave (amplification); if $$\displaystyle v_e < v_p $$, electrons gain energy (attenuation).
Bandwidth: Determined by helix dispersion; typically octave bandwidth.

D. Domains of Operation in Gunn Diode

Stable amplification: Bias just above threshold; single high-field domain travels steadily → stable oscillations.
Relaxation oscillations: Domain forms, travels, collapses at anode, reforms → pulse train (lower frequency).
LSA mode: High bias; multiple domains exist simultaneously → higher frequency oscillations ($$\displaystyle > 100 $$ GHz).

E. Conversion Loss in Mixers

Definition: Ratio of available RF input power to delivered IF output power:

$$ CL = \frac{P_{\text{RF}}}{P_{\text{IF}}} \quad (\text{linear}) \quad \text{or} \quad CL_{\text{dB}} = 10 \log_{10} \frac{P_{\text{RF}}}{P_{\text{IF}}} $$

Factors:
- Diode resistance and capacitance.
- Mismatch at RF, LO, IF ports.
- Image frequency rejection (if not filtered).
- Conversion loss inherent in mixing process (theoretical minimum 3 dB for ideal square-law).
Typical values: 5–10 dB for Schottky diode mixers.

F. Mode Curve in Reflex Klystron

Plot: Output power vs. repeller voltage $$\displaystyle V_r $$.
Shape: Periodic peaks and valleys; each peak corresponds to a mode (different number of RF cycles in drift space).
Use: Select operating mode by biasing at a peak; mode spacing $$\displaystyle \Delta V_r \approx V_0/(2N) $$, where $N$ = number of cycles.
Mode number: $$\displaystyle N = \frac{2d}{n\lambda_g} + \frac{1}{2} $$, where $d$ = drift length, $$\displaystyle \lambda_g $$ = guided wavelength.

G. Frequency Tuning in YIG Resonator

Tuning mechanism: Resonance frequency $$\displaystyle f_0 = \frac{\gamma}{2\pi} H_0 $$, where $\gamma$ = gyromagnetic ratio ($\approx 2.8$ MHz/G), $$\displaystyle H_0 $$ = static magnetic field.
Method: Vary current in electromagnet → changes $$\displaystyle H_0 $$ → continuous tuning over wide range (e.g., 2–18 GHz).
Linearity: $$\displaystyle f_0 $$ vs. $$\displaystyle H_0 $$ is linear.
Applications: Tunable filters, YIG-tuned oscillators (YTO), spectrum analyzers.

H. Broadband and Tuned Detectors

Broadband detectors:
- Operate in square-law region ($$\displaystyle v_{\text{RF}} \ll kT/q $$).
- Output voltage $$\displaystyle V_{\text{out}} \propto P_{\text{RF}} $$ (power).
- Wide bandwidth (no tuning), low sensitivity, used for power monitoring.
Tuned detectors:
- Use resonant circuit (LC) at desired frequency.
- Narrow bandwidth, high sensitivity, used in receivers for specific channel detection.
- Can be crystal detectors (diode + tank circuit).