Q: Equivalence criterion (Thevenin context): Two circuits are “equivalent” if and only if they produce the same terminal voltage for a given load, regardless of current. True or false?

Introduction / Context: When constructing a Thevenin (or Norton) equivalent, we want a simpler circuit that is indistinguishable from the original by any load connected at the terminals. This requires matching both voltage and current behaviors, not voltage alone. Given Data / Assumptions: Comparison is at the same two external terminals. Loads can vary; the equivalence should hold for any load in the scope of interest. Linear circuit assumptions commonly apply for Thevenin/Norton results. Concept / Approach: Two circuits are equivalent if their terminal V–I characteristics are identical. That means for any load, the voltage across and current through the load match in both circuits. Matching only the voltage for one particular load is insufficient; another load could reveal differences if the internal impedance does not match. Step-by-Step Solution: Compute open-circuit voltage (Voc) of both circuits; they must be equal.Compute short-circuit current (Isc) of both circuits; they must be equal.From Voc and Isc, deduce equal Thevenin impedance Zth = Voc / Isc.If these match, both voltage and current for any load will match. Verification / Alternative check: Choose two different loads and compare results from each circuit. Identical V and I in both cases confirms identical V–I relation, hence equivalence; differing results expose non-equivalence despite a single matched voltage case. Why Other Options Are Wrong: “True” lowers the standard of equivalence to a single-load voltage match, which is not sufficient and can be misleading. Common Pitfalls: Equating identical open-circuit voltage with full equivalence. Without matching internal impedance, the circuits will not behave the same under load. Final Answer: False

Q: Structure of a Thevenin AC equivalent: A Thevenin AC equivalent always consists of an AC voltage source in series with an equivalent impedance (not a specific capacitance). True or false?

Introduction / Context: Thevenin equivalents are widely used to simplify linear AC circuits. In the frequency domain, resistors, inductors, and capacitors are represented by complex impedances, and the Thevenin structure remains a voltage source in series with a single equivalent impedance. Given Data / Assumptions: Linear, time-invariant circuit at a fixed frequency. Impedances: R, jωL, and 1/(jωC) combine to form a single equivalent Zth. Dependent sources are allowed; Zth may be found via test-source methods. Concept / Approach: The Thevenin model is always a voltage source Vth in series with Zth. Zth can be resistive, inductive, capacitive, or any complex combination; but it is represented as a single impedance, not as a particular discrete capacitor or inductor unless that happens to be the reduced form. Step-by-Step Solution: Find Vth by open-circuiting the load and computing terminal voltage.Find Zth by deactivating independent sources and computing input impedance or by Voc/Isc.Construct the equivalent: Vth in series with Zth (a single complex impedance).Recognize that Zth may include effects of R, L, and C. Verification / Alternative check: Convert to Norton: In = Vth / Zth and Zn = Zth. Both models must predict identical load behavior, confirming that Thevenin uses a series impedance, not a designated “capacitance.” Why Other Options Are Wrong: “False” would suggest that Thevenin requires a specific capacitor element. In reality, any combination of reactive and resistive effects is lumped into the single equivalent impedance Zth. Common Pitfalls: Misinterpreting Zth as a single component. It is a complex number representing the net effect; its physical realization could involve multiple components. Final Answer: True

Q: Scope of superposition: The superposition theorem is applicable only to AC circuits, not DC circuits. True or false?

Introduction / Context: Superposition is a consequence of linearity, not of waveform type. Whether sources are DC, AC, or a mix, linear circuits allow responses from each independent source to be summed to obtain the total response. Given Data / Assumptions: The circuit is linear (resistors, linear dependent sources, and linear models of L and C). Sources may be constant (DC) or time-varying (AC) at one or more frequencies. Power is not superposed directly because it is nonlinear in voltage/current. Concept / Approach: To apply superposition, deactivate all but one independent source at a time: DC voltage sources become shorts, DC current sources become opens (same rules as for AC). Compute each partial response (voltages/currents) and add them algebraically. This works equally well in DC-only circuits, AC-only circuits, or mixed-signal cases. Step-by-Step Solution: Identify all independent sources (DC and AC).Deactivate the others and solve for the response due to the active one.Repeat for every source and sum the individual responses.If needed, handle different AC frequencies using phasors per frequency and then superpose in time domain. Verification / Alternative check: Compare with nodal analysis solutions including all sources simultaneously. The summed superposition result will match the full solution, demonstrating applicability beyond AC-only cases. Why Other Options Are Wrong: “True” confuses superposition with phasor techniques; phasors are for steady-state sinusoidal AC, but superposition itself is a linear principle valid for DC and AC. Common Pitfalls: Trying to superpose powers or deactivating dependent sources (they remain active). Also, ensure the circuit models remain linear over the operating range. Final Answer: False

Q: Purpose of Norton’s theorem: Like Thevenin’s theorem, Norton’s theorem reduces a complex linear network to a simpler, equivalent form that is easier to analyze at a pair of terminals. True or false?

Introduction / Context: Norton’s theorem is the current-source dual of Thevenin’s theorem. Both are powerful reduction techniques that replace a complicated network with an equivalent seen at two terminals, streamlining analysis and design such as matching and load sensitivity checks. Given Data / Assumptions: Linear circuit behavior is assumed at the operating point/frequency. Any combination of R, L, C, and controlled sources is allowed if treated correctly. We are interested in terminal behavior as “seen” by a load. Concept / Approach: Norton’s equivalent consists of a current source In in parallel with an equivalent impedance Zn. Thevenin’s consists of a voltage source Vth in series with Zth. They are related by source transformation: Vth = In * Zth and Zth = Zn. Either form leads to the same load voltage and current for any load, so both simplify the original network to a compact, manageable model. Step-by-Step Solution: Find Isc (short-circuit current) at the terminals; this is In.Find Zn by deactivating independent sources or using Voc/Isc.Construct the Norton model: In in parallel with Zn.Optionally convert between Norton and Thevenin to suit analysis preferences. Verification / Alternative check: Attach a test load RL and compute V and I using both models; results will match, confirming that both theorems serve the same simplification purpose with dual source representations. Why Other Options Are Wrong: “False” would deny Norton’s role as a general reduction method equivalent in power to Thevenin’s theorem. Common Pitfalls: Forgetting that dependent sources must remain active when finding Zn; use a test source if necessary. Also, ensure the selected model (Norton or Thevenin) matches the analysis convenience (e.g., current-driven loads favor Norton). Final Answer: True

Question 1

Definition check: In network theorems, the Norton equivalent current corresponds to which quantity at the output terminals?

Accepted Answer

Introduction / Context: The Norton and Thevenin equivalents simplify complex linear circuits. Correctly identifying the Norton current is essential for quick hand analysis and for converting between Norton and Thevenin forms. Given Data / Assumptions: Linear bilateral network assumptions apply. We are concerned with the equivalent seen at two output terminals. Independent sources are active; dependent sources are treated per standard procedures. Concept / Approach: The Norton form is a current source IN in parallel with an equivalent resistance RN. The defining current IN is by definition the short-circuit current at the output terminals (i_sc). RN equals the Thevenin resistance RTH, found by deactivating independent sources or via test sources when dependent sources exist. Step-by-Step Solution: To find IN, short the output terminals.Compute the resulting current from the network into the short: IN = i_sc.Keep RN = RTH to complete the Norton pair. Verification / Alternative check: Conversion check: The Thevenin equivalent voltage VTH equals IN * RN. Measuring either i_sc or VTH with RN provides mutual confirmation of the equivalence. Why Other Options Are Wrong: 'The current through the load' depends on load value and is not an invariant equivalent parameter. 'Open-current from the source' is undefined; open-circuit relates to Thevenin voltage, not Norton current. 'None of the above' is incorrect because a standard definition exists. Common Pitfalls: Confusing i_sc (Norton current) with I_load at arbitrary loads; forgetting that RTH = RN during conversions. Final Answer: the short circuit current

Question 2

In AC circuit analysis using Thevenin’s theorem, the reduced Thevenin equivalent seen by a load consists of which two components? Select the most accurate pair for sinusoidal steady-state (phasor) conditions.

Accepted Answer

Introduction / Context: Thevenin’s theorem is a cornerstone of network reduction in electrical engineering. For any linear, bilateral network seen from two terminals, we can compute an equivalent that behaves identically at those terminals. In alternating current (AC) phasor analysis, this equivalent is expressed with complex quantities to capture both resistance and reactance, allowing us to predict current, voltage, and power for any attached load quickly. Given Data / Assumptions: We are working under sinusoidal steady-state, so impedances (not just resistances) represent elements. The original network is linear and bilateral. We want the correct composition of the Thevenin equivalent as seen by a load. Concept / Approach: The Thevenin equivalent across two terminals is formed by: (1) the open-circuit (no-load) voltage at those terminals, called V_TH, and (2) an equivalent series impedance Z_TH that accounts for the network’s internal opposition to current (including resistive and reactive parts). Z_TH is found by deactivating independent sources (voltage sources shorted, current sources opened) and calculating the equivalent impedance seen into the network. Step-by-Step Solution: Identify Thevenin voltage: V_TH = V_OC (phasor open-circuit terminal voltage).Identify Thevenin impedance: Z_TH = equivalent series impedance seen at the terminals with independent sources deactivated.Therefore, the Thevenin equivalent is a single ideal voltage source V_TH in series with Z_TH. Verification / Alternative check: Replace the reduced model with an arbitrary load Z_L. Compute I = V_TH / (Z_TH + Z_L). If you instead analyze the original network directly, you obtain the same terminal current and load voltage. This identity validates that the pair {V_TH, Z_TH in series} completely characterizes terminal behavior. Why Other Options Are Wrong: 'the equivalent voltage source and the equivalent series resistance': In AC phasor form, opposition is complex; using only resistance ignores reactance, so this is incomplete. 'the equivalent voltage source and the equivalent parallel impedance': Thevenin uses a series element; a parallel form corresponds to Norton after source conversion. 'the equivalent voltage source and the equivalent parallel resistance': Again, parallel form is a Norton construct and resistance alone is insufficient in AC. Common Pitfalls: Confusing Thevenin (series source + series impedance) with Norton (current source + parallel admittance/impedance) forms. Forgetting to deactivate sources correctly when computing Z_TH (short ideal voltage sources, open ideal current sources). Final Answer: the equivalent voltage source and the equivalent series impedance

Question 3

According to the superposition theorem for linear circuits in phasor (AC) analysis, how should multiple independent sources be handled when determining currents or voltages at a given element?

Accepted Answer

Introduction / Context: The superposition theorem enables stepwise analysis of linear circuits containing multiple independent sources. It is especially useful in AC phasor analysis where each source may have distinct magnitude and phase. By evaluating one source at a time while properly deactivating the others, we can sum the resulting phasor contributions to find the total response. Given Data / Assumptions: The circuit is linear and time-invariant. We are performing AC phasor analysis (complex impedances). Multiple independent sources (voltage and/or current) exist. Concept / Approach: Superposition states that the total response equals the sum of individual responses due to each independent source acting alone. When considering one source, all other independent voltage sources are replaced by their internal impedance (ideal voltage source → short circuit; finite source resistance/impedance remains). Independent current sources are opened (ideal current source → open circuit; any internal impedance remains). Dependent sources, if present, are not deactivated because they depend on circuit variables. Step-by-Step Solution: Select one independent source; deactivate the others by replacing them with their internal impedances.Solve the circuit (using KVL/KCL, Ohm’s law, or impedance combinations) to find the partial response.Repeat for each source and sum all phasor results algebraically to get the total voltage/current. Verification / Alternative check: As a check, you can analyze the full circuit with all sources active at once (e.g., by nodal analysis). The resulting voltages and currents will match the phasor sum obtained via superposition if the circuit is linear. Why Other Options Are Wrong: 'all sources are considered independently': Too vague; the method requires specific deactivation rules using internal impedance. 'all sources are considered simultaneously': That is a valid direct solve, but not the definition of superposition. 'internal resistance': In AC phasor analysis, elements contribute complex impedance; limiting to resistance omits reactance and is incorrect. Common Pitfalls: Deactivating dependent sources (should remain active). Forgetting that ideal voltage source → short, ideal current source → open. Final Answer: the sources are considered one at a time with all others replaced by their internal impedance

Question 4

Definition check: In Thevenin’s theorem, the Thevenin equivalent voltage seen at two terminals equals which measurable quantity of the original network?

Accepted Answer

Introduction / Context: Thevenin’s theorem allows any linear two-terminal network to be represented by a single ideal voltage source in series with an equivalent impedance. Correctly identifying the Thevenin voltage, V_TH, is essential to build this model and to predict how different loads will behave when connected across the terminals. Given Data / Assumptions: We are analyzing a linear, bilateral circuit in steady state. No load is connected while determining V_TH. Measurements are made at the two terminals of interest. Concept / Approach: The Thevenin equivalent voltage, V_TH, is by definition the terminal voltage with the load removed, i.e., the open-circuit voltage V_OC. With no load, there is no current through the external terminals; internal source and network interactions fix the terminal potential, which becomes V_TH. Once V_TH is known and Z_TH computed, any load response follows from simple voltage division in the Thevenin model. Step-by-Step Solution: Remove the load to create an open circuit at the terminals.Compute or measure the terminal voltage under open-circuit conditions: V_OC.Assign V_TH = V_OC (phasor in AC analysis). Verification / Alternative check: Connect a test load Z_L and predict the load voltage using the Thevenin model: V_L = V_TH * Z_L / (Z_TH + Z_L). If you instead compute V_L directly on the original network, results will match, confirming that V_TH was correctly identified as V_OC. Why Other Options Are Wrong: 'equal to the source voltage': Internal drops may make the terminal voltage differ from the source value; only in trivial cases are they equal. 'the same as the load voltage': Load voltage depends on Z_L and is not defined for the open-circuit condition. 'none of the above': Incorrect because V_TH explicitly equals the open-circuit terminal voltage. Common Pitfalls: Confusing V_TH with the internal source value instead of the measurable terminal voltage without load. Attempting to find V_TH with the load still connected. Final Answer: the open circuit voltage

Question 5

According to network theorems in electrical engineering, Norton's theorem replaces any linear two-terminal network by which form of equivalent source-and-impedance model?

Accepted Answer

Introduction / Context: Norton's theorem is a cornerstone of circuit simplification in electrical engineering. It allows you to replace any linear, bilateral two-terminal network (containing independent/dependent sources and resistances/impedances) with an equivalent model that is easier to analyze—especially for varying loads and parallel interconnections. Given Data / Assumptions: The original network is linear and time-invariant. All frequency-domain impedances are allowed (R, L, C) as a single equivalent impedance. The theorem is applied at a specific pair of output terminals. Concept / Approach: Norton's theorem states that a two-port seen from its output terminals is equivalent to a current source IN in parallel with an impedance RN (also written ZN). IN is the short-circuit current at the terminals, and RN equals the Thevenin resistance RTH (found by deactivating independent sources, or by test-source methods when dependent sources are present). Step-by-Step Solution: Identify the two terminals where the load connects; remove the load.Find IN: short the terminals and compute the resulting current (short-circuit current).Find RN: deactivate independent sources (voltage sources → short, current sources → open) and compute the equivalent resistance seen into the network; or apply a test source if dependent sources exist.Form the Norton equivalent as IN in parallel with RN. Verification / Alternative check: Convert between Thevenin and Norton to verify: VTH = IN * RN and RTH = RN. Analyzed with any load RL, the Norton and Thevenin forms yield identical terminal currents and voltages. Why Other Options Are Wrong: 'Voltage source in series with impedance' is Thevenin, not Norton. 'Current source in series with impedance' is not a standard canonical form. 'Voltage source in parallel with impedance' also does not match Norton's canonical form. Common Pitfalls: Confusing IN with load current (IN is the short-circuit current), and forgetting that RN equals RTH. Also, failing to treat dependent sources correctly when computing RN. Final Answer: an equivalent current source in parallel with an equivalent impedance

Question 6

Instantaneous direction matters: If two time-varying currents flow in the same direction through a single branch at an instant, what is the instantaneous net branch current?

Accepted Answer

Introduction / Context: Superposition and current addition are foundational concepts in circuit theory. In practical signal paths, multiple sources (or coupled subcircuits) can drive current through the same branch. Understanding how instantaneous currents combine ensures correct sizing of components and accurate waveform analysis. Given Data / Assumptions: Two currents, i1(t) and i2(t), exist in the same branch at the same instant. Their directions at that instant are the same (defined by the same reference direction). Linear network behavior is assumed for superposition reasoning, but instantaneous algebra applies generally. Concept / Approach: Currents are signed quantities. When two currents share the same direction at an instant, their instantaneous values add arithmetically. The net branch current equals i_net(t) = i1(t) + i2(t). If one were opposite in direction, it would subtract based on the chosen sign convention. Step-by-Step Solution: Choose a reference direction for the branch current (e.g., left to right).At a given time t0, measure i1(t0) and i2(t0). Both are positive in the reference direction.Compute instantaneous net: i_net(t0) = i1(t0) + i2(t0).Resulting net waveform is the pointwise sum over time: i_net(t) = i1(t) + i2(t). Verification / Alternative check: Kirchhoff's Current Law (KCL) at any node confirms current conservation. In a series branch, currents must be equal through elements; when multiple source contributions are present, they superimpose linearly with signs set by direction. Why Other Options Are Wrong: 'Difference' applies only when currents are in opposite directions under the chosen sign. 'Zero' would require equal magnitudes but opposite directions. 'Cannot be determined' is incorrect because direction and magnitudes at the instant uniquely define the sum. Common Pitfalls: Mixing phasor addition with instantaneous addition; forgetting to apply consistent sign conventions; assuming RMS values can be simply added without considering phase. Final Answer: is the sum of the two currents

Question 7

Maximum power transfer from a capacitive source: What must the load impedance be relative to the source impedance to achieve maximum power transfer?

Accepted Answer

Introduction / Context: The maximum power transfer (MPT) condition is crucial in communications, RF design, and audio interfaces. When reactive elements are present, merely matching magnitudes is insufficient; the phase (reactive sign) must be addressed to ensure full power delivery from source to load. Given Data / Assumptions: Source represented by Thevenin or Norton equivalent with complex impedance Zs = Rs + jXs. Load impedance ZL is selectable. Sinusoidal steady-state operation in the frequency domain. Concept / Approach: For AC circuits with reactance, the MPT condition is ZL = Zs* (complex conjugate). This means RL = Rs (equal resistive parts) and XL = −Xs (reactive parts cancel). For a capacitive source (Xs < 0), the load must be inductive with equal magnitude reactance (XL > 0) to neutralize the reactive component at the interface. Step-by-Step Solution: Let Zs = Rs − j|Xs| (capacitive).Choose ZL = Rs + j|Xs| (inductive) so ZL = Zs*.The combined transfer at the operating frequency sees purely resistive interaction, maximizing power delivered to RL.Result: Maximum average power occurs when ZL = Zs*. Verification / Alternative check: Differentiate the real power delivered to the load with respect to RL and XL; the optimum occurs at RL = Rs and XL = −Xs. RF matching networks implement this with L- or pi-networks that create the conjugate at the design frequency. Why Other Options Are Wrong: 'Capacitive reactance equal to circuit resistance' mixes units and ignores the reactive sign. 'Be as capacitive as it is inductive' is meaningless without phase relation. 'None of the above' is false because a well-defined condition exists: conjugate matching. Common Pitfalls: Confusing DC (where XL = 0) with AC matching; attempting magnitude-only matching; forgetting that MPT can increase losses in some systems where efficiency, not power, is prioritized. Final Answer: have an impedance that is the complex conjugate of the source impedance

Question 8

Identify Thevenin vs. Norton: Thevenin’s theorem reduces any linear AC circuit to an equivalent current source in parallel with an equivalent impedance. True or false?

Accepted Answer

Introduction / Context: Thevenin and Norton theorems provide dual, interchangeable equivalents for linear circuits. Knowing which is which prevents modeling mistakes when simplifying networks for load analysis or maximum power transfer problems. Given Data / Assumptions: The circuit is linear and bilateral over the range of interest. Dependent sources are allowed if properly handled when finding equivalents. Frequency-domain AC analysis uses complex impedances. Concept / Approach: Thevenin equivalent: a single voltage source in series with an equivalent impedance (Vth in series with Zth). Norton equivalent: a single current source in parallel with an equivalent impedance (In in parallel with Zn). The two are related by source transformation: Vth = In * Zth and Zth = Zn. Step-by-Step Solution: Find Vth: open-circuit the load and compute the terminal voltage.Find Zth: zero independent sources (voltage sources → short, current sources → open) and compute equivalent impedance seen at the terminals (or use Voc/Isc).Form Thevenin model: Vth in series with Zth. This is not a current source. Verification / Alternative check: Compute the short-circuit current Isc. Then In = Isc for the Norton model and Vth = In * Zth; both models predict identical load currents and voltages for any load, confirming duality. Why Other Options Are Wrong: Choosing “True” confuses Norton with Thevenin. A current source in parallel corresponds to Norton, not Thevenin. Common Pitfalls: Mixing up the required test condition: open-circuit for Vth, short-circuit for In. Also, incorrectly deactivating dependent sources when finding Zth—use a test source method if dependents exist. Final Answer: False

Question 9

Applicability of superposition: The superposition theorem is primarily useful for single-source circuits. True or false?

Accepted Answer

Introduction / Context: The superposition theorem is a linear-circuit tool used to analyze the combined effect of multiple independent sources by summing individual contributions. Understanding when it helps is essential for efficient problem solving. Given Data / Assumptions: We consider linear circuits (R, L, C, linear dependent sources). Superposition applies to voltages and currents but not directly to powers (since power is nonlinear in V and I). Independent sources may be voltage or current sources at different locations or frequencies. Concept / Approach: Superposition shines in multi-source circuits: deactivate all but one independent source at a time (voltage sources → short; current sources → open), compute the response, then sum all contributions. In a single-source circuit, superposition offers no advantage; there is only one contribution to compute, so the theorem is trivially satisfied but not particularly useful. Step-by-Step Solution: Identify all independent sources.Analyze the circuit with only one active source at a time.Sum all individual responses to obtain the total response.Avoid applying superposition to powers; compute power from the final V and I. Verification / Alternative check: Compare with nodal or mesh analysis. For many sources, superposition can simplify algebra by reusing partial solutions, confirming its utility specifically in multi-source scenarios. Why Other Options Are Wrong: “True” incorrectly suggests superposition is best for single-source circuits, where it is unnecessary. Common Pitfalls: Incorrectly “turning off” dependent sources. Dependent sources remain active because their values depend on circuit variables. Also, attempting to superimpose power directly leads to errors. Final Answer: False

Question 10

Definition of an equivalent circuit: An equivalent circuit is any replacement network that delivers the same terminal voltage and current to a specified load as the original. True or false?

Accepted Answer

Introduction / Context: Equivalent circuits (Thevenin, Norton, small-signal models) are used to simplify analysis without changing what the connected load experiences. This concept underpins modular design and black-box testing in electronics. Given Data / Assumptions: The “equivalence” is evaluated at the same two external terminals. For any chosen load connected to those terminals, the replacement must yield the same voltage and current as the original. Linearity is often assumed for Thevenin/Norton, but equivalence can be defined more broadly if conditions are specified. Concept / Approach: If two networks present identical terminal V–I characteristics for the loads of interest, they are equivalent. Thevenin represents the network by Vth in series with Zth; Norton uses In in parallel with Zn. Both produce the same V and I for any load when their parameters satisfy Vth = In * Zth and Zth = Zn. Step-by-Step Solution: Choose the load or class of loads for which equivalence is required.Compare the terminal V–I relationship of the original and candidate equivalent.If V_L and I_L match for all loads in scope, the circuits are equivalent.Document the equivalent parameters (e.g., Vth, Zth). Verification / Alternative check: Compute responses to at least two different loads (e.g., open-circuit and short-circuit). If both match and the system is linear, all intermediate loads will match as well, confirming equivalence. Why Other Options Are Wrong: “False” would deny the standard definition used in circuit theory and measurement. Common Pitfalls: Forgetting to specify the terminals; equivalence is terminal-specific. Also, for nonlinear circuits, equivalence may hold only over a defined operating region. Final Answer: True

Question 11

Maximum power transfer with reactive sources: For maximum power transfer from a capacitive source (negative reactance part), the load impedance must be the complex conjugate of the source impedance. True or false?

Accepted Answer

Introduction / Context: In AC circuits with reactive elements, matching for maximum power transfer extends beyond equal resistances. Conjugate matching ensures that reactive parts cancel, optimizing power delivered to the load—crucial in RF, audio power stages, and impedance-matched sensors. Given Data / Assumptions: The Thevenin equivalent seen by the load is Zs = Rs + jXs. A “capacitive source” means Xs < 0 (capacitive reactance). We can choose a load impedance ZL = RL + jXL. Concept / Approach: Maximum power transfer in AC occurs when ZL = Zs* (complex conjugate), i.e., RL = Rs and XL = −Xs. If Xs is negative (capacitive), then XL must be positive (inductive) and equal in magnitude, canceling the net reactance at the interface. This yields a purely resistive combined impedance, maximizing real power flow into the load. Step-by-Step Solution: Let Zs = Rs − j|Xs| (capacitive).Choose ZL = RL + j|Xs| with RL = Rs.Series combination at the interface: Zs ||? No—looking into the source-load boundary, the reactive parts cancel in the power transfer condition because ZL = Zs* for conjugate matching.Result: effective reactance cancels, and power delivered to RL is maximized for a given source. Verification / Alternative check: Differentiate load power with respect to RL and XL using phasor current I = Vth / (Zs + ZL). The optimum occurs at RL = Rs and XL = −Xs, confirming conjugate matching. Why Other Options Are Wrong: “False” ignores the reactive cancellation requirement, which is standard in AC maximum power transfer derivations. Common Pitfalls: Maximizing voltage or current alone does not maximize power; both magnitude and phase must be considered. Also, conjugate matching is about power, not necessarily efficiency, which may require different criteria. Final Answer: True

Question 12

Equivalence criterion (Thevenin context): Two circuits are “equivalent” if and only if they produce the same terminal voltage for a given load, regardless of current. True or false?

Accepted Answer

Introduction / Context: When constructing a Thevenin (or Norton) equivalent, we want a simpler circuit that is indistinguishable from the original by any load connected at the terminals. This requires matching both voltage and current behaviors, not voltage alone. Given Data / Assumptions: Comparison is at the same two external terminals. Loads can vary; the equivalence should hold for any load in the scope of interest. Linear circuit assumptions commonly apply for Thevenin/Norton results. Concept / Approach: Two circuits are equivalent if their terminal V–I characteristics are identical. That means for any load, the voltage across and current through the load match in both circuits. Matching only the voltage for one particular load is insufficient; another load could reveal differences if the internal impedance does not match. Step-by-Step Solution: Compute open-circuit voltage (Voc) of both circuits; they must be equal.Compute short-circuit current (Isc) of both circuits; they must be equal.From Voc and Isc, deduce equal Thevenin impedance Zth = Voc / Isc.If these match, both voltage and current for any load will match. Verification / Alternative check: Choose two different loads and compare results from each circuit. Identical V and I in both cases confirms identical V–I relation, hence equivalence; differing results expose non-equivalence despite a single matched voltage case. Why Other Options Are Wrong: “True” lowers the standard of equivalence to a single-load voltage match, which is not sufficient and can be misleading. Common Pitfalls: Equating identical open-circuit voltage with full equivalence. Without matching internal impedance, the circuits will not behave the same under load. Final Answer: False

Question 13

Structure of a Thevenin AC equivalent: A Thevenin AC equivalent always consists of an AC voltage source in series with an equivalent impedance (not a specific capacitance). True or false?

Accepted Answer

Introduction / Context: Thevenin equivalents are widely used to simplify linear AC circuits. In the frequency domain, resistors, inductors, and capacitors are represented by complex impedances, and the Thevenin structure remains a voltage source in series with a single equivalent impedance. Given Data / Assumptions: Linear, time-invariant circuit at a fixed frequency. Impedances: R, jωL, and 1/(jωC) combine to form a single equivalent Zth. Dependent sources are allowed; Zth may be found via test-source methods. Concept / Approach: The Thevenin model is always a voltage source Vth in series with Zth. Zth can be resistive, inductive, capacitive, or any complex combination; but it is represented as a single impedance, not as a particular discrete capacitor or inductor unless that happens to be the reduced form. Step-by-Step Solution: Find Vth by open-circuiting the load and computing terminal voltage.Find Zth by deactivating independent sources and computing input impedance or by Voc/Isc.Construct the equivalent: Vth in series with Zth (a single complex impedance).Recognize that Zth may include effects of R, L, and C. Verification / Alternative check: Convert to Norton: In = Vth / Zth and Zn = Zth. Both models must predict identical load behavior, confirming that Thevenin uses a series impedance, not a designated “capacitance.” Why Other Options Are Wrong: “False” would suggest that Thevenin requires a specific capacitor element. In reality, any combination of reactive and resistive effects is lumped into the single equivalent impedance Zth. Common Pitfalls: Misinterpreting Zth as a single component. It is a complex number representing the net effect; its physical realization could involve multiple components. Final Answer: True

Question 14

Scope of superposition: The superposition theorem is applicable only to AC circuits, not DC circuits. True or false?

Accepted Answer

Introduction / Context: Superposition is a consequence of linearity, not of waveform type. Whether sources are DC, AC, or a mix, linear circuits allow responses from each independent source to be summed to obtain the total response. Given Data / Assumptions: The circuit is linear (resistors, linear dependent sources, and linear models of L and C). Sources may be constant (DC) or time-varying (AC) at one or more frequencies. Power is not superposed directly because it is nonlinear in voltage/current. Concept / Approach: To apply superposition, deactivate all but one independent source at a time: DC voltage sources become shorts, DC current sources become opens (same rules as for AC). Compute each partial response (voltages/currents) and add them algebraically. This works equally well in DC-only circuits, AC-only circuits, or mixed-signal cases. Step-by-Step Solution: Identify all independent sources (DC and AC).Deactivate the others and solve for the response due to the active one.Repeat for every source and sum the individual responses.If needed, handle different AC frequencies using phasors per frequency and then superpose in time domain. Verification / Alternative check: Compare with nodal analysis solutions including all sources simultaneously. The summed superposition result will match the full solution, demonstrating applicability beyond AC-only cases. Why Other Options Are Wrong: “True” confuses superposition with phasor techniques; phasors are for steady-state sinusoidal AC, but superposition itself is a linear principle valid for DC and AC. Common Pitfalls: Trying to superpose powers or deactivating dependent sources (they remain active). Also, ensure the circuit models remain linear over the operating range. Final Answer: False

Question 15

Purpose of Norton’s theorem: Like Thevenin’s theorem, Norton’s theorem reduces a complex linear network to a simpler, equivalent form that is easier to analyze at a pair of terminals. True or false?

Accepted Answer

Introduction / Context: Norton’s theorem is the current-source dual of Thevenin’s theorem. Both are powerful reduction techniques that replace a complicated network with an equivalent seen at two terminals, streamlining analysis and design such as matching and load sensitivity checks. Given Data / Assumptions: Linear circuit behavior is assumed at the operating point/frequency. Any combination of R, L, C, and controlled sources is allowed if treated correctly. We are interested in terminal behavior as “seen” by a load. Concept / Approach: Norton’s equivalent consists of a current source In in parallel with an equivalent impedance Zn. Thevenin’s consists of a voltage source Vth in series with Zth. They are related by source transformation: Vth = In * Zth and Zth = Zn. Either form leads to the same load voltage and current for any load, so both simplify the original network to a compact, manageable model. Step-by-Step Solution: Find Isc (short-circuit current) at the terminals; this is In.Find Zn by deactivating independent sources or using Voc/Isc.Construct the Norton model: In in parallel with Zn.Optionally convert between Norton and Thevenin to suit analysis preferences. Verification / Alternative check: Attach a test load RL and compute V and I using both models; results will match, confirming that both theorems serve the same simplification purpose with dual source representations. Why Other Options Are Wrong: “False” would deny Norton’s role as a general reduction method equivalent in power to Thevenin’s theorem. Common Pitfalls: Forgetting that dependent sources must remain active when finding Zn; use a test source if necessary. Also, ensure the selected model (Norton or Thevenin) matches the analysis convenience (e.g., current-driven loads favor Norton). Final Answer: True

Question 16

Circuit theorems – What does Norton's theorem actually state? Statement: Norton's theorem provides a method for reducing any AC circuit to an equivalent that consists of an equivalent voltage source in series with an equivalent impedance. Decide if …

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Introduction / Context: Norton's and Thevenin's theorems are dual tools for simplifying linear circuits at a specified frequency. Students often confuse which theorem yields a voltage source in series versus a current source in parallel. This item clarifies that distinction and when each applies in AC analysis with phasors and complex impedances. Given Data / Assumptions: Linear, bilateral network seen from two terminals at a single frequency (phasor domain). Independent/dependent sources and complex impedances may be present. We want the form defined by Norton's theorem. Concept / Approach: Thevenin's theorem: replace the network by V_th in series with Z_th. Norton's theorem: replace the same network by I_n in parallel with Z_n, where Z_n = Z_th and I_n = V_th / Z_th. Therefore, the statement that Norton's theorem yields a voltage source in series is incorrect; that is precisely Thevenin's form. Step-by-Step Solution: Find Z_th (or Z_n) by deactivating independent sources and measuring input impedance.Find V_th across the open-circuited terminals.Compute I_n = V_th / Z_th.Form Norton equivalent: a current source I_n in parallel with impedance Z_n = Z_th. Verification / Alternative check: Convert between forms: series V_th–Z_th ↔ parallel I_n–Z_n using I_n = V_th / Z_th. Both predict identical terminal behavior, confirming the duality and the incorrectness of the statement. Why Other Options Are Wrong: “True” swaps the forms of the two theorems. The qualifiers about resistive networks or single frequency do not fix the misstatement; the topology (series vs. parallel) is still wrong for Norton. Common Pitfalls: Memorizing without understanding: Norton → current source in parallel; Thevenin → voltage source in series. Also forgetting that Z_th = Z_n. Final Answer: False.

Question 17

AC maximum power transfer – Complex-conjugate matching Statement: In an AC circuit, power delivered to the load is maximized at the frequency where the load impedance is the complex conjugate of the source (output) impedance. Is this correct?

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Introduction / Context: Maximum power transfer is a cornerstone in RF, audio, and power electronics. In AC systems with reactance, matching is not just about magnitudes; the phase (reactive part) must be addressed via conjugation. This question confirms the complex-conjugate matching condition for peak power delivery. Given Data / Assumptions: Linear AC circuit analyzed at a specific frequency using phasors. Source (output) impedance Zs = Rs + jXs, load impedance ZL = RL + jXL. Goal: maximize average power absorbed by the load. Concept / Approach: The average power to the load is maximized when ZL = Zs* (the complex conjugate of Zs). This cancels reactive parts (XL = −Xs) and matches resistive parts (RL = Rs), maximizing the magnitude of load voltage/current product that contributes to real power without reactive circulation. Step-by-Step Solution: Let Zs = Rs + jXs and ZL = RL + jXL.Power to the load P = |Vth|^2 * RL / |Zs + ZL|^2 (Thevenin form).To maximize P with respect to RL and XL, set XL = −Xs (reactive cancellation).Differentiate P with respect to RL; optimum occurs at RL = Rs.Thus ZL,opt = Rs − jXs = Zs* (complex conjugate). Verification / Alternative check: At the match: input seen by source is purely resistive; voltage standing wave ratio is minimized in transmission-line terms, confirming peak power transfer. Why Other Options Are Wrong: “False” and “must equal exactly (not conjugate)” ignore reactive cancellation. “True only for resistive sources” is unnecessary; the conjugate form explicitly covers reactive sources. Common Pitfalls: Matching magnitudes only; forgetting that reactive parts must cancel to avoid circulating reactive power and suboptimal real power absorption. Final Answer: True.

Circuit Theorems in AC Analysis Questions