Q: Ideal transformer voltage step-up: With a turns ratio of 20 (secondary-to-primary) and a primary AC voltage of 12 V, what is the secondary voltage produced?

Introduction / Context: This question tests the fundamental transformer relationship between turns ratio and voltage ratio in an ideal transformer. Understanding this relationship is essential for sizing transformers and predicting output voltages in power and electronics applications. Given Data / Assumptions: Primary (input) voltage, Vp = 12 V AC. Turns ratio r = Ns/Np = 20 (secondary turns to primary turns). Ideal transformer (no losses), so voltage scales directly with turns. Concept / Approach: For an ideal transformer, the voltage ratio equals the turns ratio. Hence Vs/Vp = Ns/Np. If Ns/Np > 1, the transformer steps the voltage up. If Ns/Np < 1, it steps the voltage down. Step-by-Step Solution: Vs/Vp = Ns/NpVs = Vp * (Ns/Np)Vs = 12 * 20Vs = 240 V Verification / Alternative check: Because the secondary has twenty times as many turns as the primary, the secondary voltage must be twenty times the primary voltage. 12 V * 20 = 240 V confirms the result. Why Other Options Are Wrong: 12 V: No step-up; contradicts the stated turns ratio. 120 V: Only a 10:1 step, not 20:1. 2,400 V: Off by a factor of 10; that would be a 200:1 ratio. Common Pitfalls: Mixing up the ratio as Np/Ns instead of Ns/Np, or assuming transformers change DC (they require AC to function properly). Final Answer: 240 V

Q: Coefficient of coupling (k): If 4% of the primary flux does not link the secondary, what is the coupling coefficient of the transformer?

Introduction / Context: The coefficient of coupling, k, measures how effectively magnetic flux produced by a primary winding links to the secondary winding. It ranges from 0 (no coupling) to 1 (perfect coupling). Practical transformers seek high k for efficient energy transfer. Given Data / Assumptions: Flux not linked to secondary = 4% of total primary flux. Thus, flux that does link = 96% of total. Assume standard definition: k equals the fraction of total primary flux linking the secondary in magnitude. Concept / Approach: By definition, k ≈ linked flux / total primary flux for tightly coupled windings. If 4% is lost (leakage flux), 96% couples into the secondary. Step-by-Step Solution: Leakage flux = 4% ⇒ Coupled fraction = 100% - 4% = 96%k = 0.96 Verification / Alternative check: A well-designed transformer typically has k close to 1 but not exactly 1 due to leakage inductance; 0.96 is realistic and consistent with the statement of 4% uncoupled flux. Why Other Options Are Wrong: 0.4 and 9.6: Outside the valid 0–1 range for k (9.6 is impossible; 0.4 would imply 60% leakage). 4: Dimensionally and numerically incorrect; k must be ≤ 1. Common Pitfalls: Confusing percent with fractional values (e.g., taking k = 96 instead of 0.96) or forgetting that k cannot exceed 1. Final Answer: 0.96

Q: Mutual inductance from coupling coefficient: Two coils have L1 = 75 mH and L2 = 105 mH with coefficient of coupling k = 0.45. What is the mutual inductance M between the coils?

Introduction / Context: Mutual inductance quantifies how effectively a changing current in one coil induces voltage in another via shared magnetic flux. In coupled-coil problems, the coupling coefficient k (0 ≤ k ≤ 1) summarizes how much of the flux links both coils. Computing M from L1, L2, and k is a common task in transformer design and filter analysis. Given Data / Assumptions: L1 = 75 mH. L2 = 105 mH. Coefficient of coupling k = 0.45. Linear magnetic operation (no saturation) and sinusoidal excitation. Concept / Approach: The relationship between mutual inductance and the self-inductances is M = k * sqrt(L1 * L2). This follows from defining k as the fraction of flux linkage shared between the coils. Units must be consistent (use H, not mH) during calculations, then convert the result to mH for the final answer. Step-by-Step Solution: Convert to henry: L1 = 75 mH = 0.075 H; L2 = 105 mH = 0.105 H.Compute product: L1 * L2 = 0.075 * 0.105 = 0.007875 H^2.Take square root: sqrt(0.007875) ≈ 0.088877 H.Multiply by k: M = 0.45 * 0.088877 ≈ 0.03999 H.Convert to mH: 0.03999 H ≈ 39.9 mH. Verification / Alternative check: Since k < 1, M must be less than sqrt(L1 * L2) ≈ 88.9 mH; 39.9 mH satisfies this bound. Also, M must be greater than 0; the computed value is reasonable given moderate coupling. Why Other Options Are Wrong: 3.54 mH / 7.88 mH: Too small; would imply an unrealistically tiny k for the given L1 and L2. 189.3 mH / 450 mH: Exceed sqrt(L1 * L2) or even L values, impossible for k ≤ 1. Common Pitfalls: Mixing mH and H units, leading to errors by factors of 1000. Using M = k * (L1 + L2) incorrectly; the correct relation uses the geometric mean. Final Answer: 39.9 mH

Q: Impedance matching with an ideal transformer: What turns ratio (primary:secondary) is required to match a 1 kΩ source resistance to a 160 Ω load?

Introduction / Context: Transformers can match impedances between a source and a load to maximize power transfer or set a desired loading. The impedance seen at the primary is the secondary load scaled by the square of the turns ratio. Selecting the correct turns ratio prevents reflections, improves efficiency, and protects the source from excessive loading. Given Data / Assumptions: Source resistance R_s = 1 kΩ. Load resistance R_L = 160 Ω. Ideal transformer (lossless) for calculation. Concept / Approach: The impedance referred to the primary is Z_p = (N_p/N_s)^2 * Z_s, where Z_s is the secondary impedance (the load). For a perfect match, Z_p should equal the source resistance R_s. Solve for the ratio N_p/N_s using the equality (N_p/N_s)^2 * R_L = R_s. Step-by-Step Solution: Set up match: (N_p/N_s)^2 * 160 = 1000.Compute ratio squared: (N_p/N_s)^2 = 1000 / 160 = 6.25.Take square root: N_p/N_s = sqrt(6.25) = 2.5.Therefore, required turns ratio (primary:secondary) = 2.5:1. Verification / Alternative check: Check reflection: With 2.5:1, reflected load at primary is (2.5)^2 * 160 = 6.25 * 160 = 1000 Ω, exactly matching the source. Any other ratio would mis-match and change the power transfer conditions. Why Other Options Are Wrong: 0.4:1 or 1:2.5: These imply stepping down primary turns, reflecting the load to 25.6 Ω or 64 Ω equivalent, not 1 kΩ. 6.25:1 or 16:1: These ratios overshoot, reflecting loads to 6.25^2 * 160 or 16^2 * 160, far from 1 kΩ. Common Pitfalls: Using voltage ratio instead of the squared relationship for impedance. Inverting the ratio (secondary:primary) and obtaining the reciprocal result by mistake. Final Answer: 2.5:1

Question 1

DC on transformers again: A 6 V battery is connected across the primary of a transformer with turns ratio 8 (Np/Ns or 8:1). What steady-state secondary voltage results?

Accepted Answer

Introduction / Context: Transformers need time-varying magnetic fields to induce a secondary voltage. A DC battery, after a brief switching transient, produces no sustained change in flux, so no sustained induced secondary voltage appears. This question reinforces that concept. Given Data / Assumptions: Primary excitation is DC: 6 V battery. Turns ratio magnitude 8 (step-down or step-up irrelevant for DC steady state). Steady-state condition after initial connection transient. Concept / Approach: Induced EMF magnitude is proportional to dΦ/dt. With DC, dΦ/dt → 0 as current stabilizes; therefore, Vs → 0 V in steady state. (Practical note: continuous DC can overheat the primary and saturate the core; this is outside the scope of the numeric answer.) Step-by-Step Solution: Apply DC → only a momentary change in flux during switch-on.After settling, flux is constant → induced voltage is zero.Secondary steady-state voltage = 0 V. Verification / Alternative check: An oscilloscope would show a spike at connection then decay to zero volts on the secondary, confirming the reasoning. Why Other Options Are Wrong: 6 V, 48 V, 0.75 V: These apply AC ratio logic to DC, which is invalid at steady state. 24 V: Also misapplies the turns relation to DC. Common Pitfalls: Using the turns ratio directly on DC values. Ignoring transient versus steady-state distinction. Final Answer: 0 V

Question 2

Reflected resistance: A transformer has a 1:6 turns ratio (Np:Ns = 1:6) and a 470 Ω load on the secondary. What is the equivalent load resistance seen at the primary?

Accepted Answer

Introduction / Context: Transformers reflect impedances by the square of the turns ratio. This principle is pivotal in matching sources to loads, optimizing power transfer, and designing audio/RF interfaces. Given Data / Assumptions: Turns ratio Np:Ns = 1:6. Secondary (load-side) resistance RL = 470 Ω. Ideal transformer approximation. Concept / Approach: Reflected (equivalent) resistance at the primary is R_in = (Np/Ns)^2 * RL. Because Np < Ns, the resistance seen at the primary will be much smaller than RL (step-down of resistance by the squared ratio). Step-by-Step Solution: Compute the square of the turns ratio: (Np/Ns)^2 = (1/6)^2 = 1/36.R_in = (1/36) * 470 Ω ≈ 13.0556 Ω.Rounded to the nearest whole ohm, R_in ≈ 13 Ω. Verification / Alternative check: As a quick check, 36 * 13 Ω ≈ 468 Ω, close to the actual 470 Ω, confirming the arithmetic. Why Other Options Are Wrong: 78 Ω or 7.8 Ω or 1.3 Ω: These correspond to using the wrong ratio or misplacing the decimal. 28 Ω: Not consistent with the required 1/36 factor. Common Pitfalls: Using Ns/Np instead of Np/Ns in the square. Forgetting to square the turns ratio when reflecting resistance. Final Answer: 13 Ω

Question 3

Ideal transformer voltage step-up: With a turns ratio of 20 (secondary-to-primary) and a primary AC voltage of 12 V, what is the secondary voltage produced?

Accepted Answer

Introduction / Context: This question tests the fundamental transformer relationship between turns ratio and voltage ratio in an ideal transformer. Understanding this relationship is essential for sizing transformers and predicting output voltages in power and electronics applications. Given Data / Assumptions: Primary (input) voltage, Vp = 12 V AC. Turns ratio r = Ns/Np = 20 (secondary turns to primary turns). Ideal transformer (no losses), so voltage scales directly with turns. Concept / Approach: For an ideal transformer, the voltage ratio equals the turns ratio. Hence Vs/Vp = Ns/Np. If Ns/Np > 1, the transformer steps the voltage up. If Ns/Np < 1, it steps the voltage down. Step-by-Step Solution: Vs/Vp = Ns/NpVs = Vp * (Ns/Np)Vs = 12 * 20Vs = 240 V Verification / Alternative check: Because the secondary has twenty times as many turns as the primary, the secondary voltage must be twenty times the primary voltage. 12 V * 20 = 240 V confirms the result. Why Other Options Are Wrong: 12 V: No step-up; contradicts the stated turns ratio. 120 V: Only a 10:1 step, not 20:1. 2,400 V: Off by a factor of 10; that would be a 200:1 ratio. Common Pitfalls: Mixing up the ratio as Np/Ns instead of Ns/Np, or assuming transformers change DC (they require AC to function properly). Final Answer: 240 V

Question 4

Coefficient of coupling (k): If 4% of the primary flux does not link the secondary, what is the coupling coefficient of the transformer?

Accepted Answer

Introduction / Context: The coefficient of coupling, k, measures how effectively magnetic flux produced by a primary winding links to the secondary winding. It ranges from 0 (no coupling) to 1 (perfect coupling). Practical transformers seek high k for efficient energy transfer. Given Data / Assumptions: Flux not linked to secondary = 4% of total primary flux. Thus, flux that does link = 96% of total. Assume standard definition: k equals the fraction of total primary flux linking the secondary in magnitude. Concept / Approach: By definition, k ≈ linked flux / total primary flux for tightly coupled windings. If 4% is lost (leakage flux), 96% couples into the secondary. Step-by-Step Solution: Leakage flux = 4% ⇒ Coupled fraction = 100% - 4% = 96%k = 0.96 Verification / Alternative check: A well-designed transformer typically has k close to 1 but not exactly 1 due to leakage inductance; 0.96 is realistic and consistent with the statement of 4% uncoupled flux. Why Other Options Are Wrong: 0.4 and 9.6: Outside the valid 0–1 range for k (9.6 is impossible; 0.4 would imply 60% leakage). 4: Dimensionally and numerically incorrect; k must be ≤ 1. Common Pitfalls: Confusing percent with fractional values (e.g., taking k = 96 instead of 0.96) or forgetting that k cannot exceed 1. Final Answer: 0.96

Question 5

Transformer fundamentals — identifying input and output windings: In a conventional transformer used for power or signal isolation, which statement correctly identifies the typical roles of the windings?

Accepted Answer

Introduction / Context: Transformers are passive electromagnetic devices that transfer energy between circuits through magnetic coupling. Understanding which winding is driven by a source and which delivers energy to a load is essential for interpreting datasheets, wiring equipment correctly, and predicting voltage/current transformation and impedance matching behavior. Given Data / Assumptions: Two distinct windings on a magnetic core (primary and secondary). Operation in linear region with sinusoidal or periodic excitation. No direct electrical connection between windings (for an isolation transformer). Concept / Approach: The primary winding is connected to the input source. The alternating current in the primary creates time-varying magnetic flux in the core. This flux links the secondary turns, inducing a voltage that drives the load connected to the secondary. The turns ratio determines the voltage ratio, while power balance (neglecting losses) dictates that increasing voltage conversion reduces current and vice versa. Therefore, primary is the input side; secondary is the output side. Step-by-Step Solution: Connect source to primary → establish alternating flux in the magnetic core.Flux links secondary turns → induces secondary voltage proportional to turns ratio.Load connected to secondary → power transferred magnetically from primary to secondary.Thus, primary = input, secondary = output. Verification / Alternative check: Check nameplate/datasheet notation: primary terminals are labeled for rated input (e.g., 230 V), secondary terminals are labeled for rated outputs (e.g., 12 V). Reversing roles is possible in some transformers if insulation and ratings allow, but the conventional designation remains primary as input and secondary as output. Why Other Options Are Wrong: Both input or both output: violates energy flow definition; one side must be driven. Primary as output and secondary as input: reversed from standard usage. Single winding serving both simultaneously: that describes an autotransformer, not a two-winding isolation transformer. Common Pitfalls: Confusing two-winding transformers with autotransformers where windings are shared. Ignoring insulation ratings when using windings in reverse. Final Answer: a primary winding used as an input and a secondary winding used as an output

Question 6

Transformer secondary current effect: If the number of turns on the secondary of an ideal transformer is increased (with the same primary drive and the same load impedance referred to its own side), what happens to the secondary current?

Accepted Answer

Introduction / Context: Transformers trade voltage for current according to their turns ratio. Designers often add turns to the secondary to raise output voltage, but this change also affects current capability and impedance seen by the load. Understanding this inverse relationship helps avoid overloading windings and ensures proper regulation. Given Data / Assumptions: Ideal transformer (negligible losses) for conceptual clarity. Primary drive and load impedance (on the secondary side) are unchanged. Frequency and core conditions remain the same. Concept / Approach: Voltage ratio V_s/V_p = N_s/N_p, while current ratio I_s/I_p = N_p/N_s. Increasing secondary turns N_s raises V_s but reduces I_s for a given power transfer. For the same load impedance, a higher V_s attempts to increase load current, but the transformer's current capability per ampere-turn decreases; overall, at fixed input, the secondary current rating per volt drops as turns increase. In the ideal relation, for a given primary current, I_s decreases when N_s increases because I_s scales inversely with N_s. Step-by-Step Solution: Voltage ratio: V_s ∝ N_s.Current ratio: I_s ∝ 1 / N_s (for a given I_p).Therefore, increasing N_s leads to a lower I_s capability.Conclusion: Secondary current decreases when secondary turns are increased. Verification / Alternative check: Power balance (ideal): P_p ≈ P_s → V_p * I_p ≈ V_s * I_s. If V_s rises with more turns and P stays similar (for the same primary drive limits), I_s must decrease proportionally, confirming the inverse relation. Why Other Options Are Wrong: Increase: contradicts the inverse current–turns relation. No effect: ignores transformer scaling laws. Increase the primary current: not necessarily true as a direct consequence of adding turns; depends on load and regulation, not the immediate current ratio law. Reverse polarity: changing turns count does not flip polarity; winding orientation does. Common Pitfalls: Assuming higher voltage automatically means more usable current; copper and core limits constrain current. Confusing load current change due to altered voltage with transformer current ratio capability. Final Answer: decrease the secondary current

Question 7

Mutual inductance from coupling coefficient: Two coils have L1 = 75 mH and L2 = 105 mH with coefficient of coupling k = 0.45. What is the mutual inductance M between the coils?

Accepted Answer

Introduction / Context: Mutual inductance quantifies how effectively a changing current in one coil induces voltage in another via shared magnetic flux. In coupled-coil problems, the coupling coefficient k (0 ≤ k ≤ 1) summarizes how much of the flux links both coils. Computing M from L1, L2, and k is a common task in transformer design and filter analysis. Given Data / Assumptions: L1 = 75 mH. L2 = 105 mH. Coefficient of coupling k = 0.45. Linear magnetic operation (no saturation) and sinusoidal excitation. Concept / Approach: The relationship between mutual inductance and the self-inductances is M = k * sqrt(L1 * L2). This follows from defining k as the fraction of flux linkage shared between the coils. Units must be consistent (use H, not mH) during calculations, then convert the result to mH for the final answer. Step-by-Step Solution: Convert to henry: L1 = 75 mH = 0.075 H; L2 = 105 mH = 0.105 H.Compute product: L1 * L2 = 0.075 * 0.105 = 0.007875 H^2.Take square root: sqrt(0.007875) ≈ 0.088877 H.Multiply by k: M = 0.45 * 0.088877 ≈ 0.03999 H.Convert to mH: 0.03999 H ≈ 39.9 mH. Verification / Alternative check: Since k < 1, M must be less than sqrt(L1 * L2) ≈ 88.9 mH; 39.9 mH satisfies this bound. Also, M must be greater than 0; the computed value is reasonable given moderate coupling. Why Other Options Are Wrong: 3.54 mH / 7.88 mH: Too small; would imply an unrealistically tiny k for the given L1 and L2. 189.3 mH / 450 mH: Exceed sqrt(L1 * L2) or even L values, impossible for k ≤ 1. Common Pitfalls: Mixing mH and H units, leading to errors by factors of 1000. Using M = k * (L1 + L2) incorrectly; the correct relation uses the geometric mean. Final Answer: 39.9 mH

Question 8

Impedance matching with an ideal transformer: What turns ratio (primary:secondary) is required to match a 1 kΩ source resistance to a 160 Ω load?

Accepted Answer

Introduction / Context: Transformers can match impedances between a source and a load to maximize power transfer or set a desired loading. The impedance seen at the primary is the secondary load scaled by the square of the turns ratio. Selecting the correct turns ratio prevents reflections, improves efficiency, and protects the source from excessive loading. Given Data / Assumptions: Source resistance R_s = 1 kΩ. Load resistance R_L = 160 Ω. Ideal transformer (lossless) for calculation. Concept / Approach: The impedance referred to the primary is Z_p = (N_p/N_s)^2 * Z_s, where Z_s is the secondary impedance (the load). For a perfect match, Z_p should equal the source resistance R_s. Solve for the ratio N_p/N_s using the equality (N_p/N_s)^2 * R_L = R_s. Step-by-Step Solution: Set up match: (N_p/N_s)^2 * 160 = 1000.Compute ratio squared: (N_p/N_s)^2 = 1000 / 160 = 6.25.Take square root: N_p/N_s = sqrt(6.25) = 2.5.Therefore, required turns ratio (primary:secondary) = 2.5:1. Verification / Alternative check: Check reflection: With 2.5:1, reflected load at primary is (2.5)^2 * 160 = 6.25 * 160 = 1000 Ω, exactly matching the source. Any other ratio would mis-match and change the power transfer conditions. Why Other Options Are Wrong: 0.4:1 or 1:2.5: These imply stepping down primary turns, reflecting the load to 25.6 Ω or 64 Ω equivalent, not 1 kΩ. 6.25:1 or 16:1: These ratios overshoot, reflecting loads to 6.25^2 * 160 or 16^2 * 160, far from 1 kΩ. Common Pitfalls: Using voltage ratio instead of the squared relationship for impedance. Inverting the ratio (secondary:primary) and obtaining the reciprocal result by mistake. Final Answer: 2.5:1

Question 9

Principle of mutual induction — key dependency: Mutual induction between two coils primarily depends on which factor in operation?

Accepted Answer

Introduction / Context: Mutual induction is the foundation of transformer action, inductive sensors, and many coupling phenomena in electronics. It concerns how a changing current in one coil produces a changing magnetic flux that links a second coil, thereby inducing a voltage according to Faraday’s law and Lenz’s law. Given Data / Assumptions: Two coils placed such that some flux from one links the other. Time-varying current in at least one coil (AC or switched DC). Finite coupling coefficient k > 0 determined by geometry and core material. Concept / Approach: Induced voltage e_2 in coil 2 is proportional to the time rate of change of the flux linked with it. Since flux linkage from coil 1 is driven by its current, the key operational dependency is on di/dt of the primary (exciting) coil. Faster changes (higher frequency or sharp switching) yield larger induced voltages; constant DC (no change) produces no sustained induced voltage after transients. Step-by-Step Solution: Start with Faraday relation: induced emf ∝ dΦ/dt.Flux Φ from coil 1 is proportional to i_1 (in linear operation). Therefore, dΦ/dt ∝ di_1/dt.Thus, mutual induction depends directly on current changes (di/dt) in the exciting coil.Higher frequency or rapid switching → larger induced voltage for the same current amplitude. Verification / Alternative check: Experimentally, connect a scope to the secondary while stepping the primary current: only during transitions (turn-on/turn-off) does the secondary show a pulse. With sinusoidal excitation, the induced voltage follows frequency and amplitude, confirming dependence on di/dt. Why Other Options Are Wrong: Winding ratios affect voltage scaling but do not create induction without changing current. Output polarities result from winding orientation and dot convention, not the presence of induction itself. DC voltage levels alone produce no sustained induction if current is constant. Core weight is not the determining factor; material and geometry affect coupling, but induction requires change in current. Common Pitfalls: Expecting continuous induced voltage from steady DC; only transients induce until currents stabilize. Confusing mutual inductance value (a parameter) with the operational condition (di/dt) required for induced emf. Final Answer: current changes

Question 10

Signal interfacing in RF and audio systems: A special transformer designed to convert an unbalanced (single-ended) signal to a balanced (differential) signal is called a:

Accepted Answer

Introduction / Context: In RF, audio, and instrumentation systems, we frequently need to interface single-ended sources (coaxial cables, unbalanced outputs) with differential or balanced inputs (twisted pair, balanced lines). The device engineered for this purpose is a special transformer known as a balun, a contraction of ”balanced to unbalanced.” Given Data / Assumptions: The application requires converting between unbalanced and balanced transmission. We desire good impedance matching and minimal common-mode radiation. Operation may cover narrowband (transformer-wound) or broadband (transmission-line) designs. Concept / Approach: A balun performs two roles: mode conversion (balanced ↔ unbalanced) and, when designed appropriately, impedance transformation. By presenting equal and opposite voltages at its balanced ports, it suppresses common-mode currents and reduces radiation and susceptibility to interference. Autotransformers and center-tapped transformers have other uses but are not generic mode converters from single-ended to differential unless specifically implemented as a balun topology. Step-by-Step Solution: Identify the need: convert unbalanced source to balanced load (or vice versa).Select the device that inherently provides balanced outputs with equal magnitude and 180° phase difference.Conclusion: the correct choice is the balun. Verification / Alternative check: Practical RF antennas (e.g., dipoles) connect to coax through a balun to prevent feedline radiation. Audio DI boxes internally use transformer baluns to feed balanced microphone inputs from unbalanced instruments, providing ground isolation and noise rejection. Why Other Options Are Wrong: Autotransformer: Shares windings between primary and secondary; not inherently a balanced/unbalanced converter. Center-tapped transformer: Useful for creating dual-polarity outputs, but not specifically a balun without proper winding/connection. Step-across transformer: Nonstandard term; does not describe mode conversion. Common Pitfalls: Assuming any transformer creates a balanced output; without symmetric winding and proper connection, common-mode rejection will be poor. Final Answer: balun

Question 11

Maximum power transfer theorem in circuit design: Under what condition does a source deliver maximum power to a load (assume resistive source/load)?

Accepted Answer

Introduction / Context: The maximum power transfer theorem is a cornerstone in analog design, audio, and communications. It tells us how to choose or ”match” a load to a real (non-ideal) source so that the load receives the greatest possible power. This principle also underpins impedance matching in RF systems and small-signal sensor interfacing. Given Data / Assumptions: A Thevenin source characterized by V_s and internal resistance R_s. A purely resistive load R_L (no reactive components). Goal: maximize power dissipated in R_L. Concept / Approach: Power in the load is P_L = V_L^2 / R_L. With a Thevenin source, V_L = V_s * R_L / (R_s + R_L). Substituting gives P_L = (V_s^2 * R_L) / (R_s + R_L)^2. Taking the derivative of P_L with respect to R_L and setting it to zero yields the optimal condition R_L = R_s. Physically, this splits the source voltage equally and maximizes the product of current and load voltage at the load. Step-by-Step Solution: Write P_L = (V_s^2 * R_L) / (R_s + R_L)^2.Differentiate with respect to R_L and set dP_L/dR_L = 0.Solve to obtain R_L = R_s as the extremum; second-derivative test confirms a maximum. Verification / Alternative check: Try numerical values: let V_s = 10 V, R_s = 10 Ω. With R_L = 10 Ω, I = 10 / (10 + 10) = 0.5 A, so P_L = I^2 * R_L = 0.25 * 10 = 2.5 W. Choosing R_L much larger or smaller reduces the load power, confirming the theorem. Why Other Options Are Wrong: Greater or less than: Both move away from the matched condition and decrease delivered power. Negligible source resistance: This maximizes voltage delivered, not necessarily power for a fixed R_L landscape; the theorem concerns a given R_s. Common Pitfalls: Confusing maximum power delivery with maximum efficiency; at R_L = R_s, only 50% efficiency is achieved because the other half is dissipated in R_s. Final Answer: When the source resistance equals the load resistance

Question 12

Ideal transformer power conservation: An ideal transformer has a 2:1 voltage ratio (primary:secondary) and delivers 100 W at the primary. What is the secondary power output?

Accepted Answer

Introduction / Context: Understanding how power behaves in an ideal transformer is critical for power supply design and isolation applications. Ideal transformers are lossless: while voltage and current scale with the turns ratio, real power in equals real power out (ignoring negligible magnetizing power in the ideal model). Given Data / Assumptions: Ideal transformer (no copper, core, or stray losses). Voltage ratio N_p:N_s = 2:1 (so V_s = V_p/2). Primary power P_p = 100 W. Concept / Approach: For the ideal transformer, P_p = P_s. Voltage and current scale inversely to keep power constant: if the secondary voltage halves (2:1 step-down), the secondary current doubles for the same power transfer. Therefore, regardless of the voltage ratio, the secondary power equals the primary power in the ideal case. Step-by-Step Solution: Given P_p = 100 W, ideal: P_s = P_p.Hence P_s = 100 W.Turns ratio only affects V and I, not P in the ideal model. Verification / Alternative check: Let V_p = 200 V, I_p = 0.5 A → P_p = 100 W. With 2:1 step-down, V_s = 100 V. For power conservation, I_s must be 1 A → P_s = 100 W, confirming equality. Why Other Options Are Wrong: 50 W / 75 W / 200 W: These imply losses or gains not present in the ideal model. Common Pitfalls: Assuming voltage ratio changes power directly; in ideal transformers, current adjusts so power stays constant (ignoring phase considerations with reactive loads). Final Answer: 100 W

Question 13

Transformer efficiency calculation: A 120 V rms primary draws 300 mA rms. The secondary supplies 18 V across a 10 Ω load. Compute the efficiency η = P_out / P_in.

Accepted Answer

Introduction / Context: Transformer efficiency quantifies how effectively input power is converted to useful output power. Real transformers exhibit copper losses (I^2R), core losses (hysteresis and eddy currents), and stray losses. A quick efficiency check validates design sizing and reveals abnormal loading or faults. Given Data / Assumptions: Primary voltage V_p = 120 V rms. Primary current I_p = 0.300 A rms. Secondary voltage V_s = 18 V rms across R_L = 10 Ω. Assume sinusoidal steady state and that meter readings are true rms. Concept / Approach: Compute input power P_in = V_p * I_p (power factor assumed near unity for simplicity; many small transformers under resistive loading are close). Compute output power P_out = V_s^2 / R_L. Efficiency η = P_out / P_in expressed as a percentage. Step-by-Step Solution: P_in = 120 * 0.300 = 36.0 W.P_out = V_s^2 / R_L = 18^2 / 10 = 324 / 10 = 32.4 W.η = P_out / P_in = 32.4 / 36.0 = 0.9 = 90%. Verification / Alternative check: Secondary current I_s = V_s / R_L = 18 / 10 = 1.8 A. If the turns ratio were measured, P_out should still be near 32.4 W; any large discrepancy from 36 W input indicates realistic losses yielding efficiency below 100%. Why Other Options Are Wrong: 88% / 92% / 95%: Close but inconsistent with the direct calculation; only 90% matches P_out and P_in. Common Pitfalls: Multiplying V and I on the secondary without considering the load, or forgetting to square V when using P = V^2 / R; also, mixing rms and peak values leads to errors. Final Answer: 90%

Question 14

What does a transformer's turns ratio determine in practical circuits (assume linear operation)?

Accepted Answer

Introduction / Context: The turns ratio N_p:N_s is the key parameter of a transformer. It dictates how voltage, current, and impedance appear from one side to the other. Recognizing these relationships allows designers to step voltages up or down, match impedances, and manage current levels safely and efficiently. Given Data / Assumptions: Linear transformer behavior around the intended operating point. Idealized relationships are used to explain the principle (real transformers include losses). Sinusoidal steady-state operation. Concept / Approach: Fundamental equations: V_s / V_p = N_s / N_p. Currents are inversely proportional: I_s / I_p = N_p / N_s. Impedance reflects by the square of the turns ratio: Z_reflected = (N_p / N_s)^2 * Z_load (as seen from the primary). Therefore, turns ratio simultaneously sets voltage ratio, current ratio, and impedance transformation. Step-by-Step Solution: Voltage: V_s = V_p * (N_s / N_p).Current: I_s = I_p * (N_p / N_s).Impedance: Z_p = (N_p / N_s)^2 * Z_s. Verification / Alternative check: Example: N_p:N_s = 2:1 → V halves, I doubles, and a 16 Ω load looks like 64 Ω from the primary. Measurements on a bench transformer corroborate these proportionalities. Why Other Options Are Wrong: Each individual statement (voltage, current, impedance) is true, but considering the whole picture, the correct comprehensive choice is ”all of the above.” Common Pitfalls: Forgetting the square-law for impedance reflection or mixing up the direction of the ratio (Ns/Np vs. Np/Ns). Final Answer: all of the above

Question 15

Transformer Construction Basics  Evaluate the statement: “A transformer consists of two or more cores that are electrically coupled on a common core.”

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Introduction / Context: Transformers enable voltage level changes and isolation using electromagnetic induction. This question distinguishes the roles of windings and core, and the difference between electrical versus magnetic coupling. Given Data / Assumptions: Conventional iron-core or ferrite-core transformer. Sinusoidal AC excitation for standard operation. No special multi-core coupled structures implied. Concept / Approach: A standard transformer has two or more windings (coils) magnetically coupled via a common magnetic core. The coupling mechanism is magnetic (via changing flux), not electrical. Saying “electrically coupled cores” confuses the terminology: it is windings that are placed on one core, and they are magnetically coupled. Step-by-Step Solution: Primary winding on a magnetic core is energized with AC, producing alternating flux.This flux links to other windings on the same core, inducing voltages proportional to turns.Coupling is magnetic: V_secondary / V_primary = N_secondary / N_primary (ideally).There is no requirement for “two or more cores,” and the coupling is not electrical between cores. Verification / Alternative check: Open any basic transformer diagram: a single core with multiple windings. Equivalent circuits use magnetizing inductance and mutual inductance, emphasizing magnetic, not electrical, coupling. Why Other Options Are Wrong: All “True …” choices misstate the fundamental structure. Three-phase cores may use three legs but still rely on magnetic coupling of windings placed on the same laminated core, not “electrically coupled cores.” Common Pitfalls: Confusing the phrase “electromagnetic induction” with “electrical coupling,” and mixing up the role of the core (provides a flux path) versus windings (provide turns ratio and carry current). Final Answer: False

Question 16

Autotransformer Definition Check  Evaluate the statement: “An autotransformer is one in which the primary and secondary are automatically coupled.”

Accepted Answer

Introduction / Context: Autotransformers are frequently used for voltage adjustment with higher efficiency and smaller size than two-winding transformers. The word “auto” often leads to a misunderstanding that they are “automatic;” in fact, it means “self.” Given Data / Assumptions: Conventional single-winding autotransformer topology. Shared winding section between input and output. Sinusoidal AC operation. Concept / Approach: An autotransformer has a single continuous winding with one or more taps. Primary and secondary are portions of the same winding, so they are electrically connected (not isolated). The term does not mean “automatically coupled;” it means “self-transformer,” i.e., one winding serving both functions. Step-by-Step Solution: Identify that the primary and secondary share turns on the same coil.Voltage ratio equals the ratio of turns between the tap points.Because they share a conductor, there is electrical continuity and no galvanic isolation.Therefore, the phrase “automatically coupled” is incorrect usage; the correct idea is “self-coupled via a common winding.” Verification / Alternative check: Manufacturer datasheets state “single winding” and highlight lack of isolation as a caution, confirming the definition. Why Other Options Are Wrong: Any “True …” option relies on the wrong etymology. Frequency or turns ratio does not change the fact that “auto” means self, not automatic. Common Pitfalls: Assuming autotransformers provide isolation like two-winding transformers; they do not. Isolation requires separate windings. Final Answer: False

Question 17

Reflected Load Concept (Transformers)  Evaluate the statement: “A reflective load is a load as it appears to the source in the primary of a transformer.”

Accepted Answer

Introduction / Context: The idea of a “reflected load” (often phrased as “load referred to the primary”) lets engineers analyze a transformer-coupled system from one side. It is essential in audio output stages, power converters, and impedance matching networks. Given Data / Assumptions: Two-winding transformer with turns ratio Np:Ns. Secondary load ZL attached. Operation in sinusoidal steady state for simplicity. Concept / Approach: The transformer “reflects” the secondary load to the primary by the square of the turns ratio. The effective impedance seen at the primary terminals is Z_reflected = (Np/Ns)^2 * ZL (ideal case). This is exactly what is meant by how the load “appears” to the source at the primary side. Step-by-Step Solution: Let a = Np/Ns.Ideal transformer relations: Vp/Vs = a and Ip/Is = 1/a.Impedance transfer: Zp_seen = Vp/Ip = (a * Vs) / (Is / a) = a^2 * (Vs/Is) = a^2 * ZL.Hence the load on the secondary is “reflected” to the primary multiplied by a^2. Verification / Alternative check: Measure input current for a known ZL and compare with the predicted primary current using Z_reflected; bench tests with various turns ratios confirm the squared relationship. Why Other Options Are Wrong: “False” denies a standard transformer principle. Limiting it to “ideal only” is unnecessary for the concept; real transformers include copper and core losses, but the referred-impedance method still applies with added series/parallel loss elements. Common Pitfalls: Forgetting that power is approximately conserved (minus losses) and that increasing primary-side impedance via turns ratio trades current for voltage and vice versa. Final Answer: True

Question 18

Does a Transformer Respond Only to DC?  Evaluate the statement: “A transformer responds only to dc.”

Accepted Answer

Introduction / Context: Transformers rely on changing magnetic flux to induce voltages. Understanding whether they work with AC or DC is foundational for correct application and preventing core saturation and overheating. Given Data / Assumptions: Conventional iron or ferrite core transformer. Faraday’s law of electromagnetic induction applies. Steady DC is a constant (unchanging) signal. Concept / Approach: The induced voltage in a winding is proportional to the time rate of change of magnetic flux. DC does not change with time (di/dt = 0 after transients), so it does not sustain induction. AC, which changes with time, produces alternating flux that induces voltages in the coupled winding. Step-by-Step Solution: Apply Faraday’s law: induced voltage ∝ dΦ/dt.With AC, Φ varies with time; therefore, induced secondary voltage is sustained.With steady DC, Φ tends toward a constant; dΦ/dt ≈ 0, hence no induced secondary voltage after initial transient.DC excitation can overheat and saturate the core because current may rise without counter-emf from changing flux. Verification / Alternative check: Practical observation: connect a transformer primary to DC and you get a brief transient followed by no useful secondary voltage, often accompanied by excessive primary current and potential damage. Why Other Options Are Wrong: Any claim that transformers “respond only to DC” reverses the truth. Air gaps or autotransformer topology do not change the requirement for changing flux to induce voltage. Common Pitfalls: Confusing rectifiers (which convert AC to DC) with transformers, and assuming DC can be “stepped” with passive magnetic devices—it cannot without switching to create changing flux. Final Answer: False

Question 19

Function of a Transformer  Evaluate the statement: “A transformer changes AC to DC.”

Accepted Answer

Introduction / Context: Separating the roles of magnetic and semiconductor devices is crucial in power electronics. Transformers alter AC voltage and provide isolation, while conversion between AC and DC requires rectification and filtering. Given Data / Assumptions: Two-winding power transformer operating on AC. Optional downstream rectifier and filter not included unless specified. Steady-state operation at line frequency or switching frequency. Concept / Approach: A transformer alone cannot produce DC from AC. It changes AC voltage level (and provides isolation) via electromagnetic induction. AC-to-DC conversion requires a rectifier (diodes or controlled devices) and typically a smoothing network (capacitor, inductor) to reduce ripple. Step-by-Step Solution: Transformer action: V_secondary ∝ N_secondary/N_primary * V_primary (AC quantities).Rectification: convert alternating polarity to unidirectional current using diodes (half-wave, full-wave, bridge).Filtering/regulation: use capacitors/inductors and regulators to reduce ripple and stabilize DC output.Therefore, the transformer by itself does not perform AC-to-DC conversion. Verification / Alternative check: Bench experiment: a transformer secondary waveform remains AC; attach a diode bridge and a filter to obtain DC, confirming role separation. Why Other Options Are Wrong: Center taps, filters, or frequency do not change the fact that rectification is necessary; the transformer alone cannot create DC. Common Pitfalls: Calling the entire supply “the transformer,” when the rectifier and filter stages are actually responsible for DC output. Final Answer: False

Question 20

Does a Step-Up Transformer Increase Power?  Evaluate the statement: “A step-up transformer steps up power.”

Accepted Answer

Introduction / Context: Understanding how voltage, current, and power transform is essential for safe and efficient power system design. This question targets the common misconception that raising voltage automatically raises power inside an ideal transformer. Given Data / Assumptions: Ideal transformer unless losses are explicitly considered. Input and output measured at transformer terminals. External load determines real power transfer. Concept / Approach: A step-up transformer increases secondary voltage and proportionally decreases secondary current such that (neglecting losses) input power approximately equals output power. Real devices have small copper and core losses; therefore, output power is slightly less than input power. The transformer does not create power; it changes voltage and current levels. Step-by-Step Solution: Ideal relation: V2/V1 = N2/N1 and I2/I1 = N1/N2.Compute power: P_in ≈ V1 * I1 and P_out ≈ V2 * I2.Substitute: V2 * I2 ≈ (N2/N1) * V1 * (N1/N2) * I1 = V1 * I1 = P_in.Conclusion: the transformer does not “step up” power; it transfers power with minor losses. Verification / Alternative check: Efficiency tests (load tests) show P_out = η * P_in with η slightly less than 1. When voltage is stepped up, current steps down, maintaining similar power levels. Why Other Options Are Wrong: Claims that power increases ignore current reduction. Power factor does not enable power creation. Ideal operation certainly does not increase power; it conserves it (minus losses). Common Pitfalls: Using P = V^2/R blindly without recognizing that the effective R “seen” at the secondary is transformed by the turns ratio, which also changes current accordingly. Final Answer: False

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