Question 1

Twin-lead feeder (antenna to TV receiver) — typical inductance per metre What is the approximate inductance per unit length for a common twin feeder used to connect an antenna to a TV receiver?

Accepted Answer

Introduction / Context: Balanced twin-lead (e.g., 300 Ω twin-lead) was historically used to feed TV antennas. Its per-metre inductance L and capacitance C determine the characteristic impedance and propagation velocity. This question asks for a typical L value for such twin feeders. Given Data / Assumptions: Typical consumer twin-lead geometry and polyethylene dielectric. Order-of-magnitude engineering estimate rather than an exact spec. Concept / Approach: For transmission lines, Z0 ≈ sqrt(L/C). Common twin-lead has Z0 around 300 Ω with C around 60–70 pF/m. Rearranging gives L ≈ Z0^2 * C. Using C ≈ 67 pF/m and Z0 ≈ 300 Ω yields L on the order of 0.6–0.7 μH/m, matching handbook values. Step-by-Step Solution: 1) Adopt representative values: Z0 ≈ 300 Ω, C ≈ 67 pF/m.2) Compute L ≈ Z0^2 * C ≈ (300^2) * 67e-12 H/m.3) 300^2 = 90000; multiply → 90000 * 67e-12 ≈ 6.03e-6 H/m ≈ 0.60 μH/m. Verification / Alternative check: Transmission-line tables list L near 0.6–0.7 μH/m for 300 Ω twin-lead, consistent with the back-of-the-envelope calculation. Why Other Options Are Wrong: 0.2 μH/m: too low for typical 300 Ω twin-lead parameters. 1.6 μH/m or 10 μH/m: too high; would imply very different C or geometry. Common Pitfalls: Confusing twin-lead with 50 Ω or 75 Ω coax values; mixing units of pF/m and μH/m. Final Answer: 0.6 μH/m

Question 2

Transmission-line theory — infinite (endless) line behavior Statement: In an infinite line, the input impedance at every point equals the characteristic impedance Z0. Choose the correct truth value.

Accepted Answer

Introduction / Context: The concept of an ‘‘infinite’’ transmission line is a theoretical construct where the line extends without termination, so no reflection ever returns. Understanding its input impedance clarifies why a finite line perfectly terminated in Z0 appears the same to the source. Given Data / Assumptions: Uniform, lossless or low-loss line with characteristic impedance Z0. No discontinuities or terminations to cause reflections. Concept / Approach: With no reflection, the voltage and current form a pure traveling wave. The ratio V/I remains constant everywhere and equals Z0. Thus, the input impedance looking into any point of an infinite line is Z0, independent of distance along the line. Step-by-Step Reasoning: 1) Infinite length → no reflection from any end.2) Pure traveling wave → constant V/I ratio.3) Therefore Z_in = V/I = Z0 at every point. Verification / Alternative check: Equivalently, a finite line terminated in Z0 has Γ = 0 and presents Z0 at its input, mimicking the infinite-line condition. Why Other Options Are Wrong: False: would imply some position-dependent impedance despite the absence of reflections, which contradicts traveling-wave theory. Common Pitfalls: Confusing standing-wave patterns (which require reflections) with the reflection-free traveling-wave case. Final Answer: True

Question 3

Rectangular waveguide — identify the dominant (lowest-cutoff) mode Which mode is the dominant mode in a standard rectangular metallic waveguide (i.e., has the lowest cutoff frequency)?

Accepted Answer

Introduction / Context: Rectangular waveguides support discrete TE and TM modes. The dominant mode is the one with the lowest cutoff frequency and thus the first to propagate as frequency increases. Correctly identifying the dominant mode is fundamental for design and interpretation of waveguide behavior. Given Data / Assumptions: Rectangular guide with broad dimension a and narrow dimension b. Perfectly conducting walls (idealized). Concept / Approach: Cutoff frequencies are proportional to mode indices: for TE_mn, f_c ∝ sqrt((m/a)^2 + (n/b)^2). The TE10 mode sets m = 1, n = 0, producing the smallest cutoff because it leverages the largest dimension a and requires no field variation along b. Hence TE10 is dominant in rectangular waveguides. Step-by-Step Reasoning: 1) Compute relative ordering: TE10 uses only the broad wall dimension → lowest f_c.2) TE01 varies along the narrow dimension b → higher cutoff than TE10.3) Modes with higher indices (TE02, TM11) have even higher cutoff. Verification / Alternative check: Standard waveguide charts list TE10 as the dominant mode; practical guides are sized such that operating frequency is above TE10 cutoff and below the next higher-order mode to avoid multimoding. Why Other Options Are Wrong: TE01/TE02/TM11: all have higher cutoff than TE10 in a rectangular guide of the same dimensions. Common Pitfalls: Confusing circular (TE11 dominant) with rectangular (TE10 dominant) guides; misreading indices. Final Answer: TE10

Question 4

Spectrum planning (PM/FM style): Approximately how many phase-modulated (PM) radio channels can be accommodated in a total usable spectrum of 300 MHz, assuming a typical allocated channel spacing of about 200 kHz (as used for wideband FM/PM broadcas…

Accepted Answer

Introduction: This problem tests spectrum-allocation reasoning for phase modulation (PM) channels that are similar in occupied bandwidth to wideband FM broadcasting. The key idea is to convert the total usable spectrum into an integer count of channels using typical per-channel spacing with practical guard bands already implicit in that spacing. Given Data / Assumptions: Total available spectrum = 300 MHz. Typical wideband PM/FM channel spacing ≈ 200 kHz per channel (includes guard for adjacent-channel interference under standard allocations). Uniform channelization across the band. Concept / Approach: Channel count is total bandwidth divided by per-channel spacing. For broadcast-style PM (akin to wideband FM), practical planning commonly adopts 200 kHz spacing. This spacing folds in Carson’s-rule considerations (peak deviation and modulating frequency) and adjacent-channel protections. Step-by-Step Solution: 1) Convert 300 MHz to kHz: 300 MHz = 300,000 kHz.2) Adopt spacing = 200 kHz per channel.3) Channel count = 300,000 kHz / 200 kHz = 1500 channels.4) Round to an integer (already exact here): 1500. Verification / Alternative check: If one chose a slightly more conservative spacing (for example 250 kHz), the count becomes 1200; if the band plan is explicitly WBFM-like, 200 kHz is standard and yields 1500. Hence 1500 is the reasonable textbook answer. Why Other Options Are Wrong: 10/100/500: Far too small; they imply per-channel spacings in multi-MHz or ~0.6 MHz ranges, inconsistent with WBFM/PM planning. 1200: Matches 250 kHz spacing, which is more conservative than the common 200 kHz broadcast plan. Common Pitfalls: Confusing Carson’s occupied bandwidth with regulatory channel spacing. The spacing already bakes in adjacent-channel protection and planning rules. Final Answer: 1500

Question 5

In a klystron amplifier, what does the RF field in the buncher gap do to the electron beam as a whole?

Accepted Answer

Introduction: Klystron amplifiers rely on velocity modulation of an electron beam. The RF field across the buncher gap is time-varying; electrons reaching the gap at different RF phases experience different forces, which is fundamental to beam bunching and subsequent energy transfer to the output cavity. Given Data / Assumptions: Input RF drives the buncher cavity of a klystron amplifier. Electrons traverse the buncher gap at various RF phases. Drift space converts velocity modulation into density (current) modulation. Concept / Approach: Because the electric field reverses with RF phase, electrons arriving near the positive half-cycle are accelerated, while those arriving near the negative half-cycle are decelerated. This mixed acceleration/retardation produces velocity spread that evolves into tight bunches downstream, enabling efficient power extraction in the catcher cavity. Step-by-Step Solution: 1) Apply RF to the buncher → alternating electric field across the gap.2) Electrons see different instantaneous field polarities depending on arrival time.3) Some gain kinetic energy (accelerated), some lose it (retarded).4) In the drift region, faster electrons overtake slower electrons → bunching. Verification / Alternative check: Small-signal klystron theory explicitly models the induced velocity modulation as proportional to the RF gap voltage, predicting downstream current modulation at the same frequency. Why Other Options Are Wrong: All accelerated or all retarded: contradicts the phase-dependent interaction. No change: would preclude bunching and amplification. Common Pitfalls: Assuming uniform acceleration because of the DC beam voltage; the superimposed RF field is what creates the alternating acceleration/retardation necessary for gain. Final Answer: causes acceleration of some electrons and retardation of others

Question 6

Transmission lines with finite attenuation: Under what condition is power transmission to the load most efficient?

Accepted Answer

Introduction: Standing waves on a transmission line are a signature of mismatch. In practical lines with loss (finite attenuation), reflections increase total loss and reduce the net power delivered to the load. Therefore, understanding the connection between standing waves (VSWR) and efficiency is central to RF design. Given Data / Assumptions: Finite-loss transmission line (α > 0). Load may or may not be matched to Z0. Single-frequency steady-state operation. Concept / Approach: Reflections caused by mismatch create forward and reflected waves that form standing waves. In a lossy line, reflected power makes a second pass back and forth, suffering attenuation each time and never fully reaching the load. Matching the load (VSWR = 1) eliminates reflections and maximizes power transfer to the load for the given line loss. Step-by-Step Solution: 1) For matched load: reflection coefficient Γ = 0 → VSWR = 1.2) No reflected wave → only one forward traversal → minimal dissipative loss for delivered power.3) For mismatch: |Γ| > 0 → multiple traversals → extra attenuation → reduced delivered power. Verification / Alternative check: Power at the load with mismatch can be written as a function of |Γ| and line loss; differentiating shows maximum at |Γ| = 0 (match). Empirically, system efficiency curves peak at VSWR = 1. Why Other Options Are Wrong: Least efficient at VSWR = 1: opposite of reality. Depends only on phase constant: ignores mismatch and attenuation. Standing waves always exist: false—matching removes them. Very large VSWR: large reflections waste power. Common Pitfalls: Confusing the effect of line loss with matching; even with some loss, eliminating reflections still maximizes delivered power. Final Answer: power transmission is most efficient when there are no standing waves on the line

Question 7

If the load impedance is much larger than the characteristic impedance (ZL » Z0), what is the approximate relationship between ZL, Z0, and VSWR?

Accepted Answer

Introduction: This item checks your facility with the relationship between load mismatch and standing wave ratio (VSWR) on a line terminated in a real load that is much larger than the line impedance. In that limit, the VSWR connects directly to the impedance ratio. Given Data / Assumptions: Purely resistive load ZL » Z0. Lossless or low-loss line for simplicity. Standard definitions: Γ = (ZL − Z0) / (ZL + Z0), VSWR = (1 + |Γ|) / (1 − |Γ|). Concept / Approach: When ZL » Z0, |Γ| ≈ (ZL − Z0) / (ZL + Z0) → (ZL / Z0 − 1) / (ZL / Z0 + 1) ≈ 1 − 2 Z0 / ZL. The corresponding VSWR ≈ (1 + |Γ|) / (1 − |Γ|) ≈ (2 − 2 Z0 / ZL) / (2 Z0 / ZL) ≈ ZL / Z0. Hence, ZL ≈ Z0 * VSWR. Step-by-Step Solution: 1) Start from Γ = (ZL − Z0) / (ZL + Z0).2) Assume ZL / Z0 ≫ 1 ⇒ |Γ| ≈ 1 − 2 Z0 / ZL.3) Compute VSWR = (1 + |Γ|) / (1 − |Γ|) ≈ (2 − 2 Z0 / ZL) / (2 Z0 / ZL) = (ZL / Z0) − 1 ≈ ZL / Z0.4) Rearranged: ZL ≈ Z0 * VSWR. Verification / Alternative check: Numerical example: Z0 = 50 Ω, ZL = 1000 Ω ⇒ exact VSWR ≈ 1000/50 = 20 (very close), confirming the approximation. Why Other Options Are Wrong: Proportionalities using VSWR − 1 or inverses do not match the high-ratio limit. Equations setting Z0 = ZL(VSWR) invert the relationship incorrectly. Common Pitfalls: Applying the approximation outside its range; for modest mismatches, use exact formulas rather than the ZL/Z0 ≈ VSWR shortcut. Final Answer: ZL ≈ Z0 * VSWR (for ZL » Z0)

Question 8

Semiconductor choice for varactor (varicap) diodes: which materials are commonly used?

Accepted Answer

Introduction: Varactor (varicap) diodes exploit the voltage-dependent junction capacitance of a reverse-biased p–n junction for tuning oscillators, filters, and frequency multipliers. Material choice affects Q factor, capacitance ratio (Cmax/Cmin), breakdown, and frequency performance. Given Data / Assumptions: Reverse-biased junction operated in the depletion region. Goal is tunable capacitance with low series resistance and adequate breakdown voltage. High-frequency (RF to microwave) operation is typical. Concept / Approach: Silicon varactors are widely used for RF tuning because Si technology offers mature processing, good Q, and cost efficiency. Gallium arsenide (GaAs) varactors are preferred at higher microwave/millimeter-wave frequencies due to superior electron mobility and lower series resistance, enabling higher Q and lower loss at very high frequencies. Both materials appear in commercial varactor families depending on the target band and performance trade-offs. Step-by-Step Solution: 1) Identify the function: change capacitance with reverse bias.2) Map performance needs to material properties: mobility, breakdown, series resistance, junction profile.3) Recognize both Si and GaAs are commonly used in practice. Verification / Alternative check: Microwave components catalogs list Si hyperabrupt varactors for VCOs and GaAs planar varactors for mm-wave multipliers—evidence that both materials are standard choices. Why Other Options Are Wrong: Only Si or only GaAs: too restrictive. Neither Si nor GaAs / only Ge: inconsistent with mainstream device technology. Common Pitfalls: Assuming one material universally dominates; selection is application-specific and frequency-dependent. Final Answer: either silicon or GaAs

Question 9

In a traveling-wave tube (TWT) using a helix slow-wave structure, what is the direction of the RF electric field associated with the applied signal that interacts with the electron beam?

Accepted Answer

Introduction: In a TWT, an electron beam exchanges energy with an RF wave that is slowed to near the beam’s axial velocity by a slow-wave structure (often a helix). Understanding the field orientation is essential for grasping the beam–wave interaction that produces gain over wide bandwidths. Given Data / Assumptions: Helix slow-wave structure supporting a reduced phase velocity. Axial electron beam passing through the helix center. Small-signal operation with synchronism conditions approximately satisfied. Concept / Approach: The slow-wave helix supports an electromagnetic wave whose dominant electric field component is axial (parallel to the beam). This axial E field modulates the beam velocity, enabling continuous interaction along the tube. Although the full field has transverse components, the component that couples strongly to the beam for energy exchange is along the axis. Step-by-Step Solution: 1) Recognize that efficient energy exchange requires an axial electric field acting on the axial electron velocity.2) The helix geometry slows the wave so its axial phase velocity matches the beam speed.3) Therefore, the interacting RF E field is primarily axial. Verification / Alternative check: Small-signal TWT theory derives gain from the axial field component Ez interacting with beam current; without Ez the coupling would be negligible. Why Other Options Are Wrong: Pure radial, circumferential, or fixed oblique angles do not describe the dominant coupling field in a helix TWT. Common Pitfalls: Confusing field visualization around the helix with the component that performs work on the beam; it is the axial E field that governs velocity modulation. Final Answer: is directed along the helix axis

Question 10

A small circular loop antenna has radiation resistance Rr = 0.01 Ω for one turn. What is the radiation resistance for a 5-turn loop of the same size and frequency (assume tightly wound so area effectively scales with turns)?

Accepted Answer

Introduction: For electrically small loop antennas, radiation resistance is proportional to the square of the magnetic dipole moment, which itself scales with the number of turns N. Consequently, Rr scales as N^2 when the loop dimensions and frequency remain constant. Given Data / Assumptions: One-turn radiation resistance Rr1 = 0.01 Ω. Five-turn loop, same geometry and frequency. Tight winding so total effective magnetic moment scales linearly with N. Concept / Approach: For small loops, Rr ∝ (N * A)^2, where A is the loop area. Holding A fixed and multiplying turns by N gives RrN = N^2 * Rr1. Step-by-Step Solution: 1) Use scaling: RrN = N^2 * Rr1.2) Substitute N = 5 and Rr1 = 0.01 Ω.3) Rr5 = 5^2 * 0.01 = 25 * 0.01 = 0.25 Ω. Verification / Alternative check: General small-loop expression Rr ≈ k * (N * A / λ^2)^2 (k is a constant) also shows quadratic scaling with N; the ratio method is therefore exact for fixed geometry. Why Other Options Are Wrong: 0.002/0.01/0.05 Ω: underestimate due to ignoring the N^2 dependence. 0.5 Ω: would correspond to N ≈ 7.07 turns, not 5. Common Pitfalls: Assuming linear scaling with N; for small loops it is quadratic because radiated power depends on the square of current moment. Final Answer: 0.25 Ω

Question 11

In a klystron amplifier, what does the 'bunching effect' physically accomplish in the electron beam?

Accepted Answer

Introduction: Klystron action hinges on two stages: first, the buncher cavity imposes velocity modulation on the electron beam; second, the drift space converts that velocity modulation into density (current) modulation, known as bunching. Understanding this conversion is essential to explain how RF power is extracted downstream. Given Data / Assumptions: RF drive applied to the buncher cavity. Finite drift length for the beam to evolve. Output (catcher) cavity that extracts power from bunched electrons. Concept / Approach: Electrons accelerated more strongly arrive earlier in the drift space, while retarded electrons lag; over a suitable distance, faster electrons catch slower ones, compressing into bunches at the RF frequency. This spatial density modulation corresponds to an RF beam current that can efficiently excite the output cavity’s fields and yield gain. Step-by-Step Solution: 1) Buncher imposes velocity modulation via its RF gap voltage.2) In the drift region, different velocities convert to arrival-time differences.3) Electrons crowd into bunches → current (density) modulation appears.4) Catcher cavity interacts with this modulated current to extract RF power. Verification / Alternative check: Small-signal theory shows the RF component of beam current at the catcher peaks at an optimal drift length where bunching is strongest. Why Other Options Are Wrong: Current-to-velocity conversion reverses causality; klystrons are velocity-to-current converters. “Both” or “neither” contradict standard operation. Eliminating all velocity spread is unrealistic and not required for bunching. Common Pitfalls: Assuming bunching occurs inside the cavity; it primarily develops in the drift space following the buncher. Final Answer: converts velocity modulation into current (density) modulation of the beam

Question 12

Assertion–Reason (IMPATT Physics): Assertion (A): An IMPATT diode is an avalanche diode. Reason (R): Avalanche breakdown occurs when a p–n junction is reverse biased.

Accepted Answer

Introduction: IMPATT (Impact Ionization Avalanche Transit-Time) diodes operate by generating carriers through avalanche multiplication and then using transit-time delay to achieve negative resistance at microwave frequencies. Recognizing the avalanche origin is central to identifying the device class. Given Data / Assumptions: Reverse-biased p–n or p–i–n junction structure. High electric field sufficient for impact ionization. Microwave operation where transit time is comparable to an RF period. Concept / Approach: Under strong reverse fields, energetic carriers ionize lattice atoms, creating electron-hole pairs and initiating avalanche multiplication. The resultant phase delay between RF voltage and current, due both to avalanche buildup and carrier transit through the drift region, produces an effective negative resistance that sustains oscillation or provides gain. Step-by-Step Solution: 1) Reverse bias the junction into the avalanche regime → impact ionization occurs.2) Avalanche current builds up with a characteristic delay.3) Carriers traverse the drift region introducing additional transit-time phase shift.4) Net result: current leads/ lags sufficiently to yield negative resistance at microwave frequencies. Verification / Alternative check: IMPATT oscillators and amplifiers are textbook avalanche devices; their noise properties and breakdown voltages reflect avalanche physics. Why Other Options Are Wrong: R not explaining A: incorrect; R directly states the physical trigger for avalanche behavior that defines IMPATTs. Claims that A or R are wrong contradict device fundamentals. Common Pitfalls: Confusing Zener tunneling breakdown (dominant at lower fields and higher doping) with avalanche breakdown; IMPATT relies on avalanche and transit time. Final Answer: Both A and R are correct and R is correct explanation of A

Question 13

A TE10 rectangular waveguide must operate from 25 to 35 GHz. If the band center is 1.5 times the cutoff frequency, what is the required broad dimension a?

Accepted Answer

Introduction: Rectangular waveguides are band-pass structures whose lower edge is set by the TE10 cutoff frequency fc = c / (2a). Given an operating band and a design rule tying the band center to cutoff, we can determine the broad wall dimension a directly. Given Data / Assumptions: Operating band: 25–35 GHz → center f0 = (25 + 35) / 2 = 30 GHz. Design rule: f0 = 1.5 * fc. TE10 cutoff: fc = c / (2a), where c ≈ 3 * 10^8 m/s. Concept / Approach: Use the center-to-cutoff relationship to find fc, then solve for a from the TE10 cutoff formula. Ensuring f0 exceeds fc by a safe factor guarantees single-mode operation over most of the band while avoiding excessive proximity to cutoff. Step-by-Step Solution: 1) Compute fc from f0 = 1.5 * fc ⇒ fc = f0 / 1.5 = 30 GHz / 1.5 = 20 GHz.2) Use fc = c / (2a) ⇒ a = c / (2 * fc).3) Substitute: a = (3 * 10^8) / (2 * 20 * 10^9) = 3e8 / 4e10 = 7.5e−3 m.4) Convert to millimeters: a = 7.5 mm. Verification / Alternative check: A = 7.5 mm corresponds to an fc of 20 GHz; operating between 25–35 GHz keeps the guide well above cutoff, consistent with low dispersion near the middle of the band. Why Other Options Are Wrong: 10–15 mm: would lower fc below 20 GHz, not meeting the 1.5× rule. 9 mm: gives fc ≈ 16.7 GHz, again too low given the specified relation. 6 mm: fc ≈ 25 GHz, making the low edge equal to cutoff, which violates the given design constraint. Common Pitfalls: Using free-space wavelength at band edges instead of the TE10 cutoff formula; cutoff depends only on a for the dominant mode. Final Answer: 7.5 mm

Question 14

In radar engineering, the duty cycle of a pulsed transmitter is defined as the fraction of time the RF is ON during one pulse-repetition interval. If PRF is the pulse repetition frequency (in Hz) and pulse width is τ seconds, what is the correct exp…

Accepted Answer

Introduction / Context: In pulsed radar, the transmitter emits short bursts of RF energy separated by silent intervals. The duty cycle quantifies how “busy” the transmitter is within each repetition period and directly affects average power, thermal loading, and system range versus resolution trade-offs. Given Data / Assumptions: Pulsed operation with rectangular pulses for simplicity. PRF is the pulse repetition frequency in Hz (pulses per second). Pulse width τ is the duration of each pulse in seconds. Repetition period T = 1 / PRF. Concept / Approach: Duty cycle is the fraction of time the transmitter is ON in one PRF period. It is dimensionless and equals ON-time divided by total period. With ON-time = τ and period T = 1/PRF, duty cycle D = τ / T = τ * PRF. Step-by-Step Solution: 1) Define T = 1 / PRF.2) Duty cycle D = ON-time / period = τ / T.3) Substitute T: D = τ / (1/PRF) = τ * PRF.4) Check units: τ (s) * PRF (1/s) ⇒ dimensionless, which is correct. Verification / Alternative check: Average power P_avg equals peak power P_peak multiplied by duty cycle. If τ doubles at constant PRF, P_avg doubles, matching practical radar transmitter behavior. Why Other Options Are Wrong: PRF / pulse width: Yields units of 1/s^2; not dimensionless. Pulsed width / PRF: Equals τ / (1/PRF) only if inverted; this gives duty *period, not a fraction. (Pulse width) + PRF: Adds incompatible quantities. 1 / (PRF * pulse width): The reciprocal of the correct expression. Common Pitfalls: Confusing PRF (Hz) with PRI (s); mixing average and peak powers; forgetting that duty cycle must be dimensionless and typically far less than 1 in radar. Final Answer: Duty cycle = PRF * (pulse width)

Question 15

For a standard rectangular waveguide supporting TEm0 modes, which mode between TE20 and TE30 has the lower cutoff frequency, and why?

Accepted Answer

Introduction / Context: Rectangular waveguides support discrete transverse electric (TE) and transverse magnetic (TM) modes. Each mode begins to propagate only when the operating frequency exceeds its cutoff frequency. Understanding which modes cut on first is fundamental for single-mode operation and low-dispersion design. Given Data / Assumptions: Standard rectangular waveguide with dimensions a (broad wall) and b (narrow wall). Comparing TEm0 modes: TE20 and TE30. Same physical guide; material losses do not change cutoff ordering. Concept / Approach: For a rectangular guide, the cutoff frequency for TEmn is f_c = (c / 2) * sqrt( (m/a)^2 + (n/b)^2 ). For TEm0 specifically (n = 0), this simplifies to f_c = (m * c) / (2a). Thus cutoff scales linearly with m for these modes. Step-by-Step Solution: 1) TE20: m = 2 ⇒ f_c(TE20) = (2 * c) / (2a) = c / a.2) TE30: m = 3 ⇒ f_c(TE30) = (3 * c) / (2a) = 1.5 * c / a.3) Since 1.5 * c / a > c / a, TE30 cuts on at a higher frequency than TE20.4) Therefore TE20 has the lower cutoff and will propagate first as frequency increases. Verification / Alternative check: Waveguide mode charts and measured S21 vs frequency show TE10 first, followed by TE20, then TE30 for increasing frequency in a fixed guide, consistent with f_c ∝ m for TEm0. Why Other Options Are Wrong: Same cutoff: False because m differs (2 vs 3) for TEm0. TE30 lower: Opposite of the linear dependence. Material dependence: Material affects loss, not geometric cutoff order. Independent of indices: Contradicts the mode formula. Common Pitfalls: Confusing TE and TM formulas; forgetting that for TEm0, only the broad-wall dimension a sets cutoff; assuming dielectric filling changes ordering (it scales all cutoffs similarly). Final Answer: TE20 has lower cutoff frequency than TE30

Question 16

For achieving a specified high directivity, how does the required physical size (aperture) of an antenna vary as the operating wavelength decreases?

Accepted Answer

Introduction / Context: High-directivity antennas concentrate radiated energy into narrow beams. Dishes, horn antennas, and phased arrays are common examples. Designers often ask how physical dimensions scale when changing frequency (or wavelength) for a target directivity and beamwidth. Given Data / Assumptions: Comparing antennas of the same type and efficiency. Goal: maintain the same directivity while changing operating wavelength λ. Free-space conditions; mutual coupling and platform constraints are ignored. Concept / Approach: For an effective aperture antenna, directivity D is approximately D ≈ 4π * A_e / λ^2, where A_e is effective aperture (proportional to physical area A times efficiency). Holding D constant implies A_e ∝ λ^2. Therefore, as λ decreases (frequency rises), the required area drops with λ^2. Step-by-Step Solution: 1) Start from D ≈ 4π * A_e / λ^2.2) For fixed D and efficiency, A_e ∝ λ^2.3) Physical aperture area A scales like A_e, so A ∝ λ^2.4) Hence when λ decreases, A (and linear dimensions such as diameter) decrease proportionally (diameter ∝ λ for a given beamwidth). Verification / Alternative check: Practical systems demonstrate this scaling: a 1 m dish at X-band provides far greater directivity than at L-band; conversely, the same directivity at X-band needs a much smaller dish than at L-band. Why Other Options Are Wrong: Increases: Opposite of aperture scaling for fixed D. Unaffected: Ignores λ^2 dependence in aperture theory. Depends only on polarization: Polarization does not set required physical aperture for D. Oscillates with wavelength: No such periodic relation in basic aperture theory. Common Pitfalls: Mixing up gain scaling with frequency-dependent efficiency; comparing dissimilar antenna types; overlooking edge-taper and blockage effects which slightly modify constants but not the λ^2 law. Final Answer: It decreases as wavelength decreases

Question 17

When multiple resonant cavities are coupled together (e.g., in filters or klystron intermediate cavities), how many distinct resonant frequencies typically result?

Accepted Answer

Introduction / Context: Resonant cavities form the basis of many microwave components. When several cavities are coupled, the electromagnetic energy can distribute across them, producing collective normal modes whose frequencies are shifted relative to the uncoupled resonance. This is analogous to coupled mechanical oscillators. Given Data / Assumptions: n nominally identical (or similar) cavities. Weak to moderate coupling through irises, slots, or transmission lines. Losses are small enough that resonances remain well-defined. Concept / Approach: Coupling lifts degeneracy: each cavity no longer resonates independently. The system supports n normal modes, each with a distinct field distribution and resonant frequency. The frequency set spans a passband whose width increases with coupling strength, a principle central to bandpass cavity filters. Step-by-Step Solution: 1) Start with one cavity: one resonance at f0.2) Couple a second cavity: the original resonance splits into two (symmetric and antisymmetric modes).3) Generalize to n cavities: obtain n collective modes with frequencies f1, f2, …, fn.4) Adjust coupling (iris size) to control bandwidth and ripple characteristics of the resulting filter. Verification / Alternative check: Measured S21 of n-cavity filters shows n poles within the passband; circuit models using coupled resonator theory predict the same pole count and spacing determined by coupling coefficients. Why Other Options Are Wrong: One frequency only: Ignores mode splitting. Continuum: Requires distributed structures without discrete resonators or very strong losses. Conductivity-only dependence: Coupling, not just loss, sets mode count. Zero resonances: Non-physical for practical low-loss coupling. Common Pitfalls: Assuming identical frequencies remain unchanged with coupling; overlooking end-effects and loading which slightly detune edge cavities. Final Answer: n distinct resonant frequencies (mode splitting)

Question 18

At RF frequencies of about 10 MHz and higher, the effective impedance of a practical ground connection is primarily dominated by which behavior?

Accepted Answer

Introduction / Context: RF grounding is critical for stable circuits, low noise, and EMC compliance. However, a real “ground” is made of straps, vias, wires, chassis seams, and earth bonds that exhibit frequency-dependent impedance rather than an ideal zero-ohm reference. Given Data / Assumptions: Frequency regime: ≥ 10 MHz (HF into VHF and above). Typical ground realized by wires or straps with finite length and geometry. Nearby conductors introduce stray capacitances. Concept / Approach: The impedance of a conductor with length l and inductance L behaves as Z ≈ jωL at RF, often dominating over ohmic resistance R. Even a few centimeters of wire can have tens of nanohenries; at 10–100 MHz the reactive term jωL becomes large relative to R, making the ground connection predominantly inductive. Step-by-Step Solution: 1) Estimate inductance of a short strap or wire (order of 10–50 nH).2) Compute reactance: X_L = ωL = 2πfL; at 10 MHz and L = 20 nH, X_L ≈ 2π * 10^7 * 20 * 10^−9 ≈ 12.6 Ω.3) Compare with resistance (milliohms): the reactive term dominates, indicating inductive behavior.4) Recognize that wide, short ground planes reduce L and lower RF ground impedance. Verification / Alternative check: Vector network analyzer measurements of ground leads show rising inductive reactance with frequency. Using ground planes and via stitching significantly lowers impedance, confirming inductive dominance of thin leads. Why Other Options Are Wrong: Resistive: True at low frequency; not dominant at tens of MHz. Capacitive: Stray capacitance exists, but lead inductance dominates for simple wire grounds. Exactly half resistive/half capacitive: Not generally true; depends on geometry and frequency. Ideal ground: No physical conductor is ideal at RF. Common Pitfalls: Using long skinny ground leads; forgetting loop area; neglecting return-current paths. Prefer ground planes, multiple vias, and short connections to minimize inductance. Final Answer: Inductive

Question 19

For a varactor (varicap) diode used as a voltage-controlled capacitor, how should the applied reverse bias relate to the device’s avalanche breakdown voltage during normal operation?

Accepted Answer

Introduction / Context: Varactor diodes provide a voltage-dependent junction capacitance used in tunable filters, VCOs, and phase shifters. Safe biasing is essential to achieve wide tuning ranges without damaging the device or injecting excessive noise. Given Data / Assumptions: Operation in reverse bias to exploit depletion-region capacitance. Breakdown voltage V_BR is specified by the manufacturer. Goal: maintain reliable, low-noise capacitance control. Concept / Approach: The depletion width increases with reverse bias, reducing junction capacitance approximately as C_j ∝ (V_R + V_bi)^−n (with n depending on doping profile). However, exceeding V_BR causes avalanche breakdown with large currents, potential thermal damage, and severe noise, rendering the varactor unsuitable for linear tuning. Step-by-Step Solution: 1) Bias the diode in reverse (no forward conduction) to access the variable C_j(V_R).2) Choose a tuning range with V_R below V_BR, including margin for ripple and RF swing.3) Add series resistance or limiters if necessary to protect against transients.4) Verify linearity and Q across the intended bias range. Verification / Alternative check: Datasheets show recommended reverse-bias ranges safely below V_BR, with capacitance–voltage curves specified only in that region. Circuits violating V_BR suffer breakdown and device degradation. Why Other Options Are Wrong: Exceed breakdown: Destroys the device and increases noise. Above/below without effect: Reliability is strongly affected by breakdown. Much greater than breakdown: Unsafe and unnecessary. Alternating with forward conduction: Not how varactors are used; they are not switch-mode capacitors. Common Pitfalls: Ignoring RF swing on the junction, which adds to DC bias; failing to account for temperature variation of V_BR; omitting ESD protection. Final Answer: It must remain less than the avalanche breakdown voltage

Question 20

Skin depth δ in a good conductor varies with frequency as δ ∝ 1 / √f. Compared to 500 MHz, what is the skin depth at 1000 MHz (expressed as a ratio δ_1000MHz / δ_500MHz)?

Accepted Answer

Introduction / Context: Skin effect confines AC current to a thin layer near a conductor’s surface at high frequencies, increasing effective resistance and loss in RF components, transmission lines, and transformers. Quantifying how skin depth changes with frequency helps in conductor sizing and plating decisions. Given Data / Assumptions: Skin depth δ ∝ 1 / √f for a given material (constant conductivity and permeability). Two frequencies: f1 = 500 MHz, f2 = 1000 MHz. Material properties are unchanged and temperature effects are neglected. Concept / Approach: Since δ ∝ f^−1/2, the ratio of skin depths at two frequencies is δ2 / δ1 = √(f1 / f2). Doubling the frequency halves the argument inside the square root, reducing δ by a factor of √2 ≈ 1.414. Step-by-Step Solution: 1) Write the proportionality: δ ∝ 1 / √f.2) Form ratio: δ_1000 / δ_500 = √(500 / 1000).3) Compute: √(0.5) ≈ 0.7071.4) Therefore, skin depth at 1000 MHz is about 0.707 times the depth at 500 MHz. Verification / Alternative check: For copper at room temperature, δ ≈ 66 / √f(MHz) micrometers is a common rule of thumb, confirming that δ at 1000 MHz is approximately 66/√1000 ≈ 2.09 μm and at 500 MHz is ≈ 2.95 μm; their ratio ≈ 0.71. Why Other Options Are Wrong: 2.0 and 12.0: Imply larger skin depth at higher frequency, opposite of skin effect. 0.5 or 0.25: Would correspond to 4x or 16x frequency, not 2x. Common Pitfalls: Using linear instead of square-root scaling; forgetting that increased frequency increases surface resistance and insertion loss; ignoring changes in permeability for magnetic materials. Final Answer: 0.707

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