Question 1

Series R–L–C notch output: In a series-tuned notch configuration where the output is taken across the series L–C branch, what is the ideal output voltage at exact resonance (assume a stiff voltage source and negligible component loss)?

Accepted Answer

Introduction / Context: A common way to realize a band-stop (notch) response is to use a series R–L–C path and take the output across the reactive branch (L–C). At the notch (resonant) frequency, the inductive and capacitive reactances cancel in magnitude, dramatically changing the voltage division. This question asks for the ideal output at that frequency. Given Data / Assumptions: Series R–L–C driven by an ideal voltage source. Output measured across the L–C series sub-branch. Ideal components (no resistance in L or leakage in C) to illustrate the principle. Concept / Approach: In a series R–L–C, the total impedance is Z = R + j(XL − XC). At resonance, XL = XC so the net reactive term is zero and Z ≈ R. Importantly, the inductor and capacitor voltages are equal and opposite at resonance, so the net voltage across the L–C series pair is ideally zero, even though the individual element voltages can be large and opposite in sign. Therefore, a detector connected across the L–C pair reads a deep minimum (the notch). Step-by-Step Solution: Set resonance: XL = XC. Write v_L = i * XL and v_C = i * (−XC). At resonance: v_L + v_C = i * (XL − XC) = 0. Hence the measured output across the L–C pair is ideally 0 V. Verification / Alternative check: Phasor diagram shows v_L and v_C 180 degrees apart with equal magnitude at f0, summing to zero. Practical parts have finite Q, so a small but nonzero residual appears; the ideal target remains 0 V. Why Other Options Are Wrong: 557 mV, 6.08 V, 6.5 V: these imply a nonzero residual inconsistent with the ideal cancellation at resonance in a pure series L–C branch. Common Pitfalls: Forgetting that while the L and C each can have large voltages at f0, their sum across the series pair is zero. Confusing this notch with the band-pass response seen when measuring across the resistor. Final Answer: 0 V

Question 2

Definition check for series resonance: In a series R–L–C circuit, which condition defines resonance?

Accepted Answer

Introduction / Context: Series resonance is a key operating point where a series R–L–C presents minimum impedance and maximum current. Recognizing the proper equality that marks this condition is essential for setting center frequency and predicting bandwidth and Q. Given Data / Assumptions: Series connection of R, L, and C driven by a sinusoidal source. Inductive reactance XL = 2 * pi * f * L; capacitive reactance XC = 1 / (2 * pi * f * C). No exotic elements; standard notation (XT sometimes denotes total reactance). Concept / Approach: In a series path, the total reactance is XT = XL − XC. Resonance occurs when the reactive terms cancel, i.e., XL = XC, making XT = 0. The loop impedance collapses to approximately R, maximizing current and producing a band-pass profile when the output is taken across R. Alternatives like XC = XC (tautology), XL = XB (undefined symbol), or XT = R (mixing reactance with resistance) do not define resonance. Step-by-Step Solution: Write total reactance: XT = XL − XC. Set XT = 0 for resonance ⇒ XL = XC. Solve f0 = 1 / (2 * pi * sqrt(L * C)) if needed for design. Use Q = (1/R) * sqrt(L/C) to relate selectivity and bandwidth. Verification / Alternative check: At resonance, current is maximal and the phase angle between source voltage and current is zero (power factor unity), confirming XT = 0 and XL = XC. Why Other Options Are Wrong: XC = XC: identity, gives no frequency condition. XL = XB: symbol XB not defined; meaningless here. XT = R: mixing reactive and resistive quantities; resonance requires XT = 0, not equality with R. Common Pitfalls: Forgetting that series and parallel resonance share the XL = XC magnitude equality, but differ in whether impedance is minimized or maximized. Final Answer: XL = XC

Question 3

Tank-in-series filter behavior: When a parallel L–C tank is placed in series with an output load resistor and driven by a voltage source, what kind of filter response results at the load?

Accepted Answer

Introduction / Context: Placing a parallel resonant (tank) circuit in series with a load is a classic way to create a notch or band-stop response. At its resonant frequency, the tank's impedance becomes large, impeding current to the load and thus attenuating the output. Away from resonance, the tank's impedance is lower and the signal passes more freely. Given Data / Assumptions: Ideal parallel L–C tank with finite but high Q. Tank is in series between source and load resistor. We read the output across the load resistor. Concept / Approach: For a parallel L–C at resonance, the inductor and capacitor currents circulate within the tank, making its input impedance very high. Because the tank is in series with the load, this high impedance throttles current, yielding a deep attenuation (notch) at f0. At frequencies well below or above f0, the tank's impedance decreases, so more current flows and the load voltage recovers, which is characteristic of a band-stop filter centered at f0. Step-by-Step Solution: Tune L and C so XL = XC at f0 ⇒ tank input impedance is large. Place the tank in series with the load: series path current minimizes at f0. Load voltage v_out = i * R_load therefore dips at f0 (notch). Away from f0 the tank impedance falls ⇒ v_out increases. Verification / Alternative check: Frequency sweeps of such a network show a distinct dip at f0 in |V_out/V_in|. Moving the tank to a shunt branch produces complementary band-pass behavior across the load. Why Other Options Are Wrong: Low-pass/high-pass: these have monotonic pass/stop behavior, not a notch centered at f0. Band-pass: would require a topology that maximizes output near f0, not minimizes it (e.g., series RLC with output across R). Common Pitfalls: Confusing series vs shunt placement of the tank; overlooking how loading alters Q and notch depth. Final Answer: band-stop

Question 4

RLC filter terminology: which property of an electrical circuit best describes its ability to respond strongly to selected (desired) frequencies while rejecting or greatly attenuating all other frequencies?

Accepted Answer

Introduction / Context: In RLC circuits and filters, engineers often need a concise term for how well a network responds to certain frequencies and rejects others. This capability is crucial in radio receivers, instrumentation, and audio crossovers. The question asks for the correct property name that captures this frequency discrimination ability. Given Data / Assumptions: Circuit context: linear time-invariant RLC or active filter networks. Desired behavior: pass (or amplify) a target band and suppress out-of-band content. No specific numeric values are required; this is a definition/terminology check. Concept / Approach: “Selectivity” describes how selectively a circuit accepts desired frequencies while rejecting undesired ones. It is a qualitative attribute, often quantified by shape factor, out-of-band attenuation, and skirt steepness around the passband or notch. While bandwidth and quality factor are related metrics (e.g., Q = f0 / BW for narrowband resonators), they do not by themselves state the overall ability to reject out-of-band signals; sensitivity refers to signal magnitude response, not frequency discrimination. Step-by-Step Solution: 1) Identify the capability being described: strong response at desired frequencies, weak response elsewhere. 2) Map terms: “selectivity” is the accepted descriptor of frequency discrimination in filters and tuned circuits. 3) Recognize supporting metrics: bandwidth and quality factor relate to the sharpness near resonance but are not the general descriptor. 4) Conclude that “Selectivity” is the correct term. Verification / Alternative check: Receiver specifications list “selectivity” as the measure of adjacent-channel rejection, often using dB attenuation at offsets from the center frequency. This aligns precisely with the verbal definition in the stem. Why Other Options Are Wrong: Bandwidth: measures the width of the passband but not overall rejection behavior. Sensitivity: indicates required input level for a given output; not frequency discrimination. Quality factor: relates to resonance sharpness; a high Q suggests narrow bandwidth but is not the general property name. Dynamic range: span between minimum detectable and maximum allowable signal; not about frequency selectivity. Common Pitfalls: Equating narrow bandwidth or high Q with complete selectivity; ignoring skirt steepness and stopband attenuation which are also part of real-world selectivity performance. Final Answer: Selectivity

Question 5

Series vs. parallel RLC far from resonance: at frequencies well above and well below the resonant frequency, the series RLC network behaves like ______ while the parallel RLC network behaves like ______.

Accepted Answer

Introduction / Context: Understanding the limiting behavior of RLC circuits far from resonance allows quick mental checks when sketching Bode plots or predicting filter action. This question contrasts the series and parallel forms when the operating frequency is much lower or much higher than the resonant frequency f0. Given Data / Assumptions: Linear, ideal R, L, and C elements. Frequency well below and well above resonance (|ω − ω0| large). Qualitative “looks like” refers to net impedance magnitude relative to load: open (very large impedance) vs short (very low impedance). Concept / Approach: For a series RLC, the net impedance is Z = R + j(ωL − 1/(ωC)). Far below resonance (ω → 0), 1/(ωC) is very large in magnitude, so |Z| is large; far above resonance (ω → ∞), ωL is very large; either way the series path is “large impedance,” akin to an open. For a parallel RLC, the net admittance is the sum of branch admittances; away from resonance either the capacitor (at high ω) or the inductor (at low ω) provides a very low-impedance path, so the network tends toward a “short.” Step-by-Step Solution: 1) Series RLC: consider limits ω → 0 and ω → ∞; the reactive magnitude becomes very large in either limit. 2) Conclude series RLC appears open-like (very high |Z|) far from resonance. 3) Parallel RLC: consider branch impedances; at low ω the inductor is near short, at high ω the capacitor is near short. 4) Conclude parallel RLC appears short-like (very low |Z|) far from resonance. Verification / Alternative check: Impedance loci or Bode plots corroborate: series resonance gives a deep impedance minimum at ω0 and rises sharply away; parallel resonance gives a large impedance maximum at ω0 and falls off away, approaching a short at extremes. Why Other Options Are Wrong: Like a short, like an open: reverses the true limiting behavior. Inductive, capacitive / Capacitive, inductive: describe phase, not the open/short magnitude analogy across both extremes. Resistive, reactive: too vague and incorrect for the limits described. Common Pitfalls: Mixing series vs parallel intuition; focusing on phase sign rather than overall magnitude; forgetting that “open/short” here is a qualitative magnitude analogy, not an exact boundary condition. Final Answer: Like an open, like a short

Question 6

Filter behavior using a tank: when a parallel resonant (tank) circuit is inserted in series with an output load resistor, what overall filter type does the network implement around the tank’s resonant frequency?

Accepted Answer

Introduction / Context: Resonant tanks (LC networks) are the building blocks of many passive filters. A “parallel tank” has a very high impedance at its resonant frequency and a much lower impedance away from resonance. The way it is placed with respect to the signal path determines whether it passes or rejects a band of frequencies. Given Data / Assumptions: The tank is a parallel LC tuned to a resonant frequency f0. The tank is placed in series with the output load resistor. Components are ideal; we reason qualitatively around f0. Concept / Approach: A parallel LC exhibits maximum impedance at f0. If this high-impedance element is in series with the load, the signal path “blocks” at f0, producing a deep attenuation (a notch). Frequencies far from f0 see a comparatively low impedance through the series path and are less attenuated. This is the defining behavior of a band-stop (notch) filter with the stopband centered on f0. Step-by-Step Solution: 1) Recognize the series placement of a high-Z element at f0. 2) At f0: Z_tank → large; series path attenuation is high; output drops. 3) Away from f0: Z_tank → small; series path allows signal to reach the load. 4) Therefore the network rejects a narrow band around f0 and passes frequencies on either side. Verification / Alternative check: Plotting the transfer function magnitude shows a minimum (notch) at f0. Swapping the topology (placing the tank in shunt across the load) would produce a bandpass behavior, confirming the role of placement. Why Other Options Are Wrong: Low-pass / High-pass: these pass one side of the spectrum and reject the other, not a narrow band around f0. Bandpass: would require low impedance at f0 in the series path (a series-resonant element), not a high impedance. All-pass: preserves magnitude while shifting phase; not applicable here. Common Pitfalls: Confusing parallel vs series resonance; assuming any tank in the path yields bandpass; forgetting that parallel tanks have high impedance at resonance while series tanks have low impedance at resonance. Final Answer: Band-stop

RLC Circuits and Resonance Questions