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| Question | Answer |
|---|---|
| What is impedance Z in a circuit? | Z is the total opposition to AC current: Z=R+jX, where R=resistance, X=reactance. Units: Ohms. |
| What is the impedance of a resistor R? | Z_R = R (purely real, no frequency dependence) |
| What is the impedance of an inductor L at frequency omega? | Z_L = jωL (purely imaginary, increases with frequency) |
| What is the impedance of a capacitor C at frequency omega? | Z_C = 1/(jωC) (purely imaginary, decreases with frequency) |
| What does an ideal transformer with turns ratio n:1 do to impedance? | It reflects secondary impedance to primary as Z_primary = n² × Z_secondary |
| In a 2:1 transformer, if secondary has impedance Z_s, what is the primary impedance? | Z_primary = (2)² × Z_s = 4 × Z_s |
| How do you find total impedance of series elements? | Add all impedances: Z_total = Z1 + Z2 + Z3 + ... |
| What is resonant frequency ω₀ of a series RLC circuit? | ω₀ = 1/√(LC). At resonance, inductive and capacitive reactances cancel. |
| What is the damping factor ζ (zeta) in a series RLC circuit? | ζ = R/(2) × √(C/L). It describes how oscillations decay over time. |
| What does ζ < 1 mean? | Underdamped: circuit oscillates with decaying amplitude |
| What does ζ = 1 mean? | Critically damped: fastest return to equilibrium without oscillation |
| What does ζ > 1 mean? | Overdamped: slow return to equilibrium, no oscillation |
| What is a pole of a transfer function X(s)? | A value of s where X(s) → ∞ (denominator = 0) |
| What is a zero of a transfer function X(s)? | A value of s where X(s) = 0 (numerator = 0) |
| For X(s)=(s²+3s+2)/(s+5), what are the zeros? | Factor numerator: (s+1)(s+2)=0, so zeros at s=-1 and s=-2 |
| For X(s)=(s²+3s+2)/(s+5), what is the pole? | Denominator s+5=0, so pole at s=-5 |
| Why does X(s)=(s²+3s+2)/(s+5) have equal poles and zeros? | Numerator is degree 2, denominator degree 1, so add a zero at infinity to balance: 2 zeros, 1 finite pole + 1 at ∞ |
| What is the inverse Laplace transform used for? | Converting s-domain expressions back to time-domain signals x(t) |
| What is partial fraction decomposition? | Breaking a complex fraction into simpler fractions to apply known Laplace pairs |
| For X(s)=(s²+3s+2)/(s+5), rewrite as partial fractions. | Do polynomial long division first since degree of num ≥ degree of den: X(s)=s-2+12/(s+5) |
| What is the inverse Laplace of 1/(s+a)? | e^(-at)·u(t) |
| What is the inverse Laplace of s/(s²+ω²)? | cos(ωt)·u(t) |
| What is the inverse Laplace of ω/(s²+ω²)? | sin(ωt)·u(t) |
| What is the Laplace transform of cos(ωt)·u(t)? | s/(s²+ω²) |
| What is the Laplace transform of sin(ωt)·u(t)? | ω/(s²+ω²) |
| What is the Laplace transform of cos(ωt+θ)·u(t)? | Use Euler: cos(ωt+θ)=cos(θ)cos(ωt)-sin(θ)sin(ωt), then transform each term. |
| For x(t)=cos(3t+45°)·u(t), what is X(s)? | cos45°·[s/(s²+9)] - sin45°·[3/(s²+9)] = (s/√2 - 3/√2)/(s²+9) = (s-3)/(√2·(s²+9)) |
| For X(s)=(s-3)/(√2·(s²+9)), find poles and zeros. | Zero: s=3. Poles: s²+9=0 → s=±j3 (imaginary poles on jω axis) |
| What is u(t)? | Unit step function: u(t)=0 for t<0, u(t)=1 for t≥0 |
| What is initial condition of a capacitor at t=0⁻? | Voltage across capacitor just before switch changes: V_C(0⁻). Capacitor voltage cannot change instantaneously. |
| What is initial condition of an inductor at t=0⁻? | Current through inductor just before switch changes: I_L(0⁻). Inductor current cannot change instantaneously. |
| Why can't capacitor voltage change instantaneously? | i=C·dv/dt; infinite current would be required for instantaneous voltage change |
| Why can't inductor current change instantaneously? | v=L·di/dt; infinite voltage would be required for instantaneous current change |
| How do you find V₀ (initial capacitor voltage) in the circuit at t=0⁻? | Analyze the circuit before switching (DC steady-state): capacitor is open circuit, inductor is short circuit. |
| In DC steady state, how does a capacitor behave? | Open circuit (no current flows through it) |
| In DC steady state, how does an inductor behave? | Short circuit (wire with no voltage drop) |
| What is the general form of underdamped capacitor voltage response? | Vc(t) = e^(-αt)[A·cos(ωd·t) + B·sin(ωd·t)] + Vf, where α=damping coeff, ωd=damped frequency |
| What is α (damping coefficient) in a series RLC? | α = R/(2L) |
| What is ωd (damped natural frequency)? | ωd = √(ω₀² - α²), valid when underdamped (ω₀ > α) |
| What is ω₀ (natural frequency) in series RLC? | ω₀ = 1/√(LC) |
| What is a difference amplifier? | Op-amp circuit that amplifies the difference between two input voltages |
| What is the virtual short principle in an ideal op-amp? | The voltage difference between + and - input terminals is zero (V+ = V-) |
| What is the virtual open principle in an ideal op-amp? | No current flows into the op-amp input terminals |
| How does a capacitor behave at very high frequencies (ω→∞)? | Impedance Z_C=1/(jωC)→0, so capacitor acts like a short circuit |
| How does an inductor behave at very high frequencies (ω→∞)? | Impedance Z_L=jωL→∞, so inductor acts like an open circuit |
| What is a lowpass filter (LPF)? | Circuit that passes low-frequency signals and attenuates high-frequency signals |
| What is the s-domain transfer function H(s)? | H(s) = V_out(s)/V_in(s), the ratio of output to input in Laplace domain |
| What is the standard form of a 2nd order LPF transfer function? | H(s) = ω₀²/(s² + 2ζω₀s + ω₀²) |
| How do you determine damping from the transfer function denominator? | Compare s²+2ζω₀s+ω₀² with your denominator to extract ζ and ω₀ |
| What are natural frequencies/characteristic roots? | Roots of the denominator polynomial (poles) of H(s); they determine the natural response. |
| How do you classify damping from the poles of H(s)? | Complex conjugate poles → underdamped; real distinct poles → overdamped; repeated real poles → critically damped |
| What is a delta function δ(t)? | Impulse of infinite amplitude, zero duration, unit area; models instantaneous events |
| What is the Laplace transform of δ(t)? | L{δ(t)} = 1 |
| What is the impulse response of a circuit? | Output when input is δ(t); tells you how the circuit responds to any input via convolution |
| In a series RLC with δ(t) input, what current flows? | i(t) is the impulse response; find I(s)=1/(Ls+R+1/Cs) and take inverse Laplace |
| Energy stored in an inductor formula? | W_L = ½·L·i² |
| Energy stored in a capacitor formula? | W_C = ½·C·v² |
| How do you find energy deposited by δ(t) at t=0⁺? | Find i(0⁺) and v_C(0⁺) after the impulse, then compute W_L=½Li² and W_C=½Cv² |
| For series RLC with δ(t): what is i(0⁺)? | The impulse delivers charge q=1 to the circuit; since δ(t) is voltage source, i(0⁺)=1/L (from v=L·di/dt) |
| What is KVL (Kirchhoff's Voltage Law)? | Sum of all voltages around any closed loop = 0 |
| What is KCL (Kirchhoff's Current Law)? | Sum of all currents entering a node = 0 |
| What is the s-domain equivalent of a capacitor C with initial voltage V₀? | Impedance 1/(sC) in series with voltage source V₀/s |
| What is the s-domain equivalent of an inductor L with initial current I₀? | Impedance sL in series with voltage source L·I₀ |
| What is sinusoidal steady-state analysis? | Analysis using phasors/jω substitution; replace s with jω in transfer functions |
| What does substituting s=jω do? | Converts s-domain transfer function to frequency response H(jω) |
| What is the magnitude of H(jω)? | |H(jω)| = magnitude of output phasor / magnitude of input phasor (gain) |
| What is the phase of H(jω)? | ∠H(jω) = phase of output minus phase of input |
| In the difference amplifier with capacitor feedback, what happens at ω→0 (DC)? | Capacitor is open circuit → no feedback → op-amp saturates (very high gain) |
| In the difference amplifier with capacitor feedback, what happens at ω→∞? | Capacitor is short circuit → acts like resistive feedback → finite gain |
| How do you apply superposition in op-amp circuits? | Find output due to each input separately (set others to zero), then add results |
| What is the node voltage method? | Assign voltages at each node, apply KCL at each node, solve system of equations |
| What is an RC ladder network? | Cascaded R and C stages; used in filter design (like the 2nd-order LPF in Q5) |
| For the 2-stage RC LPF (Q5), how do you find H(s)? | Write KCL at the middle node and output node in s-domain, solve for V_o/V_s |
| What is the characteristic equation of a 2nd order circuit? | s² + 2αs + ω₀² = 0, where roots give the natural frequencies |
| How do you factor s²+3s+2? | Find two numbers multiplying to 2 and adding to 3: (s+1)(s+2) |
| What is polynomial long division? | Dividing numerator by denominator when degree of numerator ≥ degree of denominator |
| For X(s)=(s²+3s+2)/(s+5), perform long division. | s²+3s+2 ÷ (s+5) = s - 2 remainder 12, so X(s) = s - 2 + 12/(s+5) |
| What is the inverse Laplace of s? | δ'(t) (derivative of delta function) — but in practice, proper fractions are easier to handle |
| What is the final value theorem? | lim(t→∞) x(t) = lim(s→0) s·X(s), valid if poles of sX(s) are in left half plane |
| What is the initial value theorem? | x(0⁺) = lim(s→∞) s·X(s) |
| In the Q3 circuit before switching (t=0⁻), which elements are DC sources? | 2800V source on left side; 1A current source on right side |
| At t=0⁻ in Q3, capacitor is open: how to find V₀? | Use voltage divider / KVL with DC sources; inductor is short, capacitor is open |
| At t=0⁻ in Q3, inductor is short: how to find I₀? | The current through the short-circuit inductor found by circuit analysis of left loop |
| After t=0 in Q3, the switch moves to b: what changes? | Left side (2800V, 100Ω) is disconnected; right side (1mH, 1nF, 1kΩ, 1A) forms new circuit |
| What is the forced response of a circuit? | The particular solution due to the input source; what the circuit does at steady state |
| What is the natural response of a circuit? | The homogeneous solution; how the circuit behaves due to initial conditions (decaying transient) |
| What is the complete response? | Complete response = natural response + forced response |
| How do you find α and ω₀ for Q3 RLC (R=1kΩ, L=1mH, C=1nF)? | α=R/2L=1000/(2×0.001)=500,000; ω₀=1/√(LC)=1/√(10⁻³×10⁻⁹)=1,000,000 rad/s |
| Is Q3 underdamped or overdamped? | α=500k, ω₀=1M: since α < ω₀, the circuit is underdamped |
| What is the Laplace transform of e^(-at)? | 1/(s+a), valid for t≥0 |
| What is the Laplace transform of t·u(t)? | 1/s² |
| What is the Laplace transform of e^(-at)·cos(ωt)·u(t)? | (s+a)/((s+a)²+ω²) |
| What is the Laplace transform of e^(-at)·sin(ωt)·u(t)? | ω/((s+a)²+ω²) |
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