§ 3.1Network Elements
Classification axes (each pair is exam material):
- Linear / nonlinear: linear elements obey superposition and homogeneity (R, L, C with constant values); diode/transistor are nonlinear.
- Active / passive: passive elements cannot deliver net energy \( \big(\int_{-\infty}^{t} v\,i\,dt \ge 0\big) \) — R dissipates, L and C only store; active elements (sources, transistors) can supply energy.
- Bilateral / unilateral: bilateral elements behave identically for either current direction (R, L, C); diode is unilateral.
- Lumped / distributed: lumped if physical size ≪ λ; otherwise distributed (transmission lines, Unit-VII).
- Time-invariant / time-varying.
| Resistor R | Inductor L | Capacitor C | |
|---|---|---|---|
| v–i law | \( v = Ri \) | \( v = L\,di/dt \) | \( i = C\,dv/dt \) |
| Energy | dissipates \(i^2R\) | \( W = \tfrac12 Li^2 \) | \( W = \tfrac12 Cv^2 \) |
| Cannot change abruptly | — | current \(i_L\) | voltage \(v_C\) |
| DC steady state | — | short circuit | open circuit |
| At t = 0⁺ (de-energized) | — | open circuit | short circuit |
| Series / parallel | add / reciprocal | add / reciprocal | reciprocal / add |
Sources
- Independent: ideal voltage source (R_int = 0; never short it) and ideal current source (R_int = ∞; never open it).
- Dependent (controlled): VCVS (gain µ), VCCS (g_m), CCVS (r_m), CCCS (β) — essential for transistor models; they are not deactivated in superposition/Thevenin source-killing.
- Source transformation: \( V_s \) in series with R ⇌ \( I_s = V_s/R \) in parallel with R.
UGC NET focus
"Which quantity cannot change instantaneously?" (i_L, v_C); inductor as SC / capacitor as OC in DC steady state and the t = 0⁺ duals; identifying the four dependent-source types from transistor models; passivity definition via energy integral.
§ 3.2Network Graphs & Topology
Replace every element by a line (branch) between nodes → the graph captures interconnection only. For a connected graph with n nodes and b branches:
Fundamental counts (memorize)
$$ \text{Tree branches (twigs)} = n - 1, \qquad \text{Links (chords)} = b - n + 1 $$
- Tree: connected subgraph containing all n nodes with no loop; the remaining branches are the co-tree (links). Number of possible trees = det(A·Aᵀ) where A is the reduced incidence matrix.
- Fundamental loops (tie-sets): each link + the unique tree path closing it → b − n + 1 independent KVL equations (= number of independent mesh/loop currents).
- Fundamental cut-sets: each twig + the links it separates → n − 1 independent KCL equations (= number of independent node voltages).
- Incidence matrix A (n−1 × b): a_ij = +1 (branch j leaves node i), −1 (enters), 0 (not incident). KCL: \( A\,i_b = 0 \). Tie-set matrix B: KVL \( B\,v_b = 0 \); branch currents from loop currents \( i_b = B^{T}i_{loop} \). Cut-set matrix Q: \( Q\,i_b = 0 \), \( v_b = Q^{T}v_{tree} \). Orthogonality: \( B A^{T} = 0 \).
- Planar graph: drawable without branch crossings — mesh analysis applies only to planar networks.
- Duality: R↔G, L↔C, V↔I, series↔parallel, mesh↔node, SC↔OC — dual networks have interchanged equations.
UGC NET focus
Counting questions (given n, b → twigs, links, KCL/KVL equation counts); tree contains no loop; properties of A (column sums zero for full matrix); duality pair matching.
§ 3.3Nodal & Mesh Analysis
Nodal analysis (KCL-based)
- Choose a reference (ground); assign n − 1 node voltages; write KCL at each in conductance form: \( \mathbf{G}\,\mathbf{V} = \mathbf{I} \). For purely independent current sources, G is symmetric: G_kk = Σ conductances at node k; G_kj = −(conductance between k and j).
- Supernode: a voltage source (with no series R) between two non-reference nodes — enclose both nodes, write one KCL for the pair plus the constraint \( V_a - V_b = V_{source} \).
- Preferred when nodes ≪ meshes, with current sources, and for non-planar circuits; this is the method inside SPICE (modified nodal analysis).
Mesh analysis (KVL-based, planar only)
- Assign clockwise mesh currents to the b − n + 1 meshes; \( \mathbf{R}\,\mathbf{I} = \mathbf{V} \): R_kk = Σ resistances around mesh k; R_kj = −(shared resistance).
- Supermesh: a current source shared by two meshes — write KVL around the combined periphery plus the constraint \( I_a - I_b = I_{source} \).
UGC NET focus
Set up (not necessarily solve) the matrix by inspection — diagonal = self, off-diagonal = negative mutual; recognize when a supernode/supermesh is required; equation-count questions tie back to §3.2.
§ 3.4Network Theorems
Superposition
In a linear network with several independent sources, any response = algebraic sum of responses to each source acting alone, others deactivated (voltage source → short, current source → open; dependent sources stay active). Not applicable to power (P ∝ x², nonlinear in sources).
Thevenin's theorem
Any linear two-terminal network ≡ \( V_{Th} \) in series with \( Z_{Th} \): \(V_{Th}\) = open-circuit voltage; \( Z_{Th} \) = impedance looking in with independent sources killed. With dependent sources: \( Z_{Th} = V_{Th}/I_{SC} \) or apply a test source. Fails for nonlinear/unilateral elements and (for the impedance step) networks with controlled sources only handled by the test-source method.
Norton's theorem
Dual form: \( I_N = I_{SC} \) in parallel with \( Z_N = Z_{Th} \); conversion \( V_{Th} = I_N Z_{Th} \).
Maximum Power Transfer (MPT)
MPT conditions & maximum power (RMS quantities)
$$ \text{DC / resistive: } R_L = R_{Th} \Rightarrow P_{max} = \frac{V_{Th}^{2}}{4R_{Th}} \qquad \text{AC: } Z_L = Z_{Th}^{*} \Rightarrow P_{max} = \frac{|V_{Th}|^{2}}{4R_{Th}} $$
- Efficiency at MPT = 50% (half dissipates in R_Th) — acceptable in communication (signal level matters), not in power systems.
- Restricted cases: if only |Z_L| adjustable (fixed angle): \( |Z_L| = |Z_{Th}| \); if R_L only (X_L fixed): \( R_L = \sqrt{R_{Th}^2 + (X_{Th}+X_L)^2} \).
Other theorems (one-liners that appear as MCQs)
- Reciprocity: in a single-source linear bilateral network, interchanging an ideal source and the response point leaves the response unchanged (Z_12 = Z_21 networks; no dependent sources).
- Millman: parallel voltage-source branches combine to \( V = \dfrac{\sum V_iG_i}{\sum G_i} \).
- Substitution: any branch may be replaced by a source equal to its existing v or i without disturbing the network.
- Tellegen: \( \sum_b v_b i_b = 0 \) — conservation of power, requires only KCL+KVL (any elements, even different networks with the same graph).
- Compensation: ΔZ in a branch carrying I ≡ adding source V = I·ΔZ opposing the current.
UGC NET focus
MPT numericals (find R_L and P_max — watch RMS vs peak: peak form gives V_m²/8R_Th); 50% efficiency statement; superposition not valid for power; Thevenin with dependent sources via I_SC; Tellegen needs no element information.
§ 3.5AC Circuit Analysis & Resonance
Phasors and impedance
For sinusoidal steady state at ω, replace \( v(t) = V_m\cos(\omega t + \phi) \) by phasor \( V = V_m\angle\phi \); differentiation → ×jω. Impedances: \( Z_R = R,\ Z_L = j\omega L,\ Z_C = 1/j\omega C = -j/\omega C \). Current lags by 90° in L, leads by 90° in C (CIVIL mnemonic).
AC power (RMS phasors)
$$ S = VI^{*} = P + jQ, \quad P = V_{rms}I_{rms}\cos\theta\ (\text{W}), \quad Q = V_{rms}I_{rms}\sin\theta\ (\text{VAR}), \quad |S| = V_{rms}I_{rms}\ (\text{VA}) $$
Power factor cosθ (lagging = inductive); PF correction adds parallel C supplying Q. \( V_{rms} = V_m/\sqrt2 \) for sinusoids; form factor 1.11, peak factor 1.414.
Series RLC resonance
Series resonance
$$ \omega_0 = \frac{1}{\sqrt{LC}}, \qquad Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0CR} = \frac{1}{R}\sqrt{\frac{L}{C}}, \qquad BW = \frac{\omega_0}{Q} = \frac{R}{L}, \qquad f_{1,2} \approx f_0 \mp \frac{BW}{2} $$
- At ω₀: Z = R (minimum), current maximum, circuit purely resistive, unity PF; voltage magnification: \( V_L = V_C = QV_s \) (can exceed supply — "acceptor circuit").
- Below ω₀ capacitive, above inductive; half-power frequencies where |Z| = √2·R; ω₀ = √(ω₁ω₂) (geometric mean).
Parallel RLC (anti)resonance
Parallel resonance
$$ \omega_0 = \frac{1}{\sqrt{LC}}\ (\text{ideal}), \qquad Z_{max} = R_{dyn} = \frac{L}{RC}\ (\text{coil-}R\ \text{tank}), \qquad \omega_0 = \frac{1}{\sqrt{LC}}\sqrt{1 - \frac{R^{2}C}{L}} $$
Impedance maximum, line current minimum ("rejector circuit"), current magnification Q inside the tank. Q_parallel = R/(ω₀L) for ideal parallel R.
UGC NET focus
Q and bandwidth numericals; V_C = QV at series resonance; dynamic resistance L/CR; series = minimum Z / parallel = maximum Z contrast; power-factor/complex-power computations; geometric-mean relation of half-power frequencies.
§ 3.6Transient Analysis
First-order circuits (RL, RC)
Universal single-time-constant solution
$$ x(t) = x(\infty) + \big[x(0^{+}) - x(\infty)\big]e^{-t/\tau}, \qquad \tau_{RC} = RC, \quad \tau_{RL} = \frac{L}{R} $$
- R seen by the storage element (Thevenin R) sets τ. One τ → 63.2% of the change; 5τ ≈ complete; initial slope reaches the final value in exactly τ.
- Continuity: \( v_C(0^+) = v_C(0^-) \), \( i_L(0^+) = i_L(0^-) \); everything else may jump.
- Charging capacitor: \( v_C = V(1 - e^{-t/\tau}) \), \( i = \tfrac{V}{R}e^{-t/\tau} \); energy delivered by source = CV², stored = ½CV² → 50% charging efficiency regardless of R.
Second-order circuits (series RLC)
Characteristic equation \( s^2 + 2\alpha s + \omega_0^2 = 0 \), with damping factor \( \alpha = R/2L \) (series) or \( 1/2RC \) (parallel) and \( \omega_0 = 1/\sqrt{LC} \). Damping ratio \( \zeta = \alpha/\omega_0 \):
| Condition | ζ | Roots | Response |
|---|---|---|---|
| \( R > 2\sqrt{L/C} \) | >1 | real, distinct | overdamped (two exponentials) |
| \( R = 2\sqrt{L/C} \) | =1 | real, repeated | critically damped (fastest non-oscillatory) |
| \( R < 2\sqrt{L/C} \) | <1 | complex \( -\alpha \pm j\omega_d \) | underdamped, rings at \( \omega_d = \omega_0\sqrt{1-\zeta^2} \) |
| R = 0 | 0 | ±jω₀ | undamped oscillation |
Complete response = natural (transient, source-free) + forced (steady-state, source form). Q-link: \( \zeta = 1/2Q \).
UGC NET focus
Evaluate x(0⁺), x(∞), τ by inspection and write the answer directly from the universal formula; critical resistance \(2\sqrt{L/C}\); 63.2%/5τ facts; the 50% capacitor-charging energy result; ω_d vs ω₀ distinction.
§ 3.7Laplace Transform
Definition (unilateral)
$$ F(s) = \int_{0^{-}}^{\infty} f(t)\,e^{-st}\,dt, \qquad s = \sigma + j\omega $$
Converts linear constant-coefficient ODEs into algebra and automatically includes initial conditions — the natural tool for transient circuit analysis.
| f(t), t ≥ 0 | F(s) | f(t) | F(s) |
|---|---|---|---|
| δ(t) | 1 | e^{−at} | 1/(s+a) |
| u(t) | 1/s | t e^{−at} | 1/(s+a)² |
| t | 1/s² | sin ωt | ω/(s²+ω²) |
| tⁿ | n!/s^{n+1} | cos ωt | s/(s²+ω²) |
| e^{−at}sin ωt | ω/[(s+a)²+ω²] | e^{−at}cos ωt | (s+a)/[(s+a)²+ω²] |
| Property | Statement |
|---|---|
| Linearity | af₁ + bf₂ → aF₁ + bF₂ |
| Time shift | f(t−a)u(t−a) → e^{−as}F(s) |
| Frequency shift | e^{−at}f(t) → F(s+a) |
| Scaling | f(at) → (1/a)F(s/a) |
| Differentiation | f′(t) → sF(s) − f(0⁻); f″ → s²F − sf(0⁻) − f′(0⁻) |
| Integration | ∫₀ᵗf dτ → F(s)/s |
| t-multiplication | t f(t) → −dF/ds |
| Convolution | f₁ * f₂ → F₁(s)·F₂(s) |
Initial & final value theorems
$$ f(0^{+}) = \lim_{s\to\infty} sF(s), \qquad f(\infty) = \lim_{s\to 0} sF(s)\ \ (\text{valid only if all poles of } sF(s) \text{ in LHP}) $$
s-domain circuit models
- R → R; L → sL in series with source \( Li(0^-) \) (or parallel current source i(0⁻)/s); C → 1/sC in series with \( v(0^-)/s \) (or parallel Cv(0⁻)).
- Solve by nodal/mesh in s, then invert by partial fractions; repeated and complex poles handled by standard expansions.
UGC NET focus
Final-value-theorem validity check (e.g., fails for sin ωt — poles on jω axis); pair lookups; the L and C initial-condition equivalent sources; convolution↔multiplication duality.
§ 3.8Fourier Series & Fourier Transform
Fourier series (periodic signals, period T, ω₀ = 2π/T)
Trigonometric & exponential forms
$$ f(t) = a_0 + \sum_{n=1}^{\infty}(a_n\cos n\omega_0 t + b_n\sin n\omega_0 t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}, \qquad c_n = \frac{1}{T}\int_T f(t)e^{-jn\omega_0 t}dt $$
- Dirichlet conditions guarantee convergence; at discontinuities the series converges to the midpoint; truncation causes ~9% Gibbs overshoot.
- Symmetry shortcuts: even → only a_n (cosines); odd → only b_n; half-wave symmetry f(t±T/2) = −f(t) → only odd harmonics. Square wave: \( \frac{4A}{\pi}\big(\sin\omega_0t + \tfrac13\sin3\omega_0t + \cdots\big) \).
- Parseval: average power \( = \sum |c_n|^2 = a_0^2 + \tfrac12\sum(a_n^2+b_n^2) \).
Fourier transform (aperiodic, energy signals)
Definition & inverse
$$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt, \qquad x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega $$
| x(t) | X(ω) | Property | Statement |
|---|---|---|---|
| δ(t) | 1 | Duality | X(t) ↔ 2πx(−ω) |
| 1 | 2πδ(ω) | Time shift | x(t−t₀) ↔ e^{−jωt₀}X(ω) |
| e^{−at}u(t) | 1/(a+jω) | Modulation | x(t)e^{jω₀t} ↔ X(ω−ω₀) |
| rect(t/τ) | τ·sinc(ωτ/2) | Scaling | x(at) ↔ (1/|a|)X(ω/a) |
| e^{−a|t|} | 2a/(a²+ω²) | Convolution | x*h ↔ XH; xh ↔ (1/2π)X*H |
| cos ω₀t | π[δ(ω−ω₀)+δ(ω+ω₀)] | Differentiation | x′ ↔ jωX |
| Gaussian e^{−t²/2σ²} | Gaussian (self-FT shape) | Parseval | ∫|x|²dt = (1/2π)∫|X|²dω |
- Rect ↔ sinc pair embodies the time–bandwidth inverse relationship; narrow pulse → wide spectrum.
- Real x(t) → conjugate-symmetric X(−ω) = X*(ω): |X| even, phase odd.
- FT = Laplace on the jω axis (when ROC includes it); FS = sampled FT of one period: \( c_n = \tfrac1T X_{1\,period}(n\omega_0) \).
UGC NET focus
Symmetry → which coefficients vanish; square-wave harmonic amplitudes (1/n, odd only); Gibbs 9%; rect↔sinc and first-null bandwidth 2π/τ; Parseval power computations.
§ 3.9Z-Transform
Definition (bilateral)
$$ X(z) = \sum_{n=-\infty}^{\infty} x[n]\,z^{-n} $$
The discrete-time counterpart of Laplace. The ROC (region of convergence) is part of the answer — the same X(z) with different ROCs corresponds to different sequences.
| x[n] | X(z) | ROC |
|---|---|---|
| δ[n] | 1 | all z |
| u[n] | z/(z−1) | |z| > 1 |
| aⁿu[n] | z/(z−a) | |z| > |a| (right-sided) |
| −aⁿu[−n−1] | z/(z−a) | |z| < |a| (left-sided) |
| n aⁿu[n] | az/(z−a)² | |z| > |a| |
| aⁿcos(ω₀n)u[n] | (z²−az cosω₀)/(z²−2az cosω₀+a²) | |z| > |a| |
- ROC rules: right-sided → outside outermost pole; left-sided → inside innermost; two-sided → annulus; finite-length → all z (except possibly 0/∞). ROC never contains poles.
- Properties: time shift \( x[n-k] \leftrightarrow z^{-k}X(z) \) (the "delay = z⁻¹" identity behind filter structures); scaling aⁿx[n] ↔ X(z/a); convolution ↔ product; time reversal x[−n] ↔ X(1/z); differentiation n·x[n] ↔ −z dX/dz.
- Initial value: x[0] = lim_{z→∞} X(z) (causal); final value: lim_{z→1}(z−1)X(z) if poles of (z−1)X(z) inside unit circle.
- Stability/causality: causal LTI stable ⟺ all poles strictly inside the unit circle ⟺ ROC includes |z| = 1. DTFT exists when ROC contains the unit circle: \( X(e^{j\Omega}) = X(z)|_{z=e^{j\Omega}} \).
- s↔z mapping: \( z = e^{sT} \) — LHP → inside unit circle, jω-axis → unit circle.
- Inversion: partial fractions (in z⁻¹ or X(z)/z form), power-series, residues.
UGC NET focus
Same X(z), different ROC → different x[n]; stability = poles inside unit circle; z = e^{sT} mapping facts; delay property; final-value validity.
§ 3.10Time & Frequency Domain Response
- Impulse response h(t) fully characterizes an LTI system: \( y = x * h \); step response s(t) = ∫h; frequency response \( H(j\omega) = |H|e^{j\angle H} \) — output sinusoid is scaled by |H(jω₀)| and shifted by ∠H(jω₀).
- First-order \( H(s) = \tfrac{1}{1+s\tau} \): step response \(1-e^{-t/\tau}\); 10–90% rise time \( t_r = 2.2\tau \); −3 dB bandwidth \( \omega_c = 1/\tau \) →
Rise-time–bandwidth product $$ t_r \times BW(\text{Hz}) \approx 0.35 $$
- Second-order \( H(s) = \dfrac{\omega_n^{2}}{s^{2} + 2\zeta\omega_n s + \omega_n^{2}} \): underdamped step response overshoots —
Second-order step-response metrics $$ M_p = e^{-\pi\zeta/\sqrt{1-\zeta^{2}}}, \quad t_p = \frac{\pi}{\omega_d}, \quad t_s \approx \frac{4}{\zeta\omega_n}\ (2\%), \quad \omega_d = \omega_n\sqrt{1-\zeta^{2}} $$Frequency domain: resonant peak \( M_r = 1/(2\zeta\sqrt{1-\zeta^2}) \) at \( \omega_r = \omega_n\sqrt{1-2\zeta^2} \) (exists for ζ < 0.707).
- Distortionless transmission: |H| constant and phase linear in ω (constant group delay \( \tau_g = -d\phi/d\omega \)); deviation → amplitude/phase (delay) distortion.
- Trade-off intuition: wider bandwidth ↔ faster response; higher Q/lower ζ ↔ peaking and ringing.
UGC NET focus
0.35/BW rise time; overshoot from ζ (ζ = 0.707 → ~4.3%, no resonance peak boundary); settling 4/ζωₙ; linear phase ⇒ pure delay; output of LTI to sinusoid stays a sinusoid at the same frequency.
§ 3.11Transfer Functions, Poles & Zeros
Transfer function (zero initial conditions)
$$ H(s) = \frac{Y(s)}{X(s)} = K\,\frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}, \qquad n \ge m \ \text{(proper)} $$
- Poles (denominator roots) = natural frequencies — they fix the form of the transient: real LHP pole → decaying exponential; complex pair −α±jω_d → damped sinusoid; jω-axis pair → sustained oscillation; RHP → growth (instability).
- Zeros shape the magnitude/phase but not stability; a zero at the excitation frequency blocks transmission (notch).
- Stability: BIBO-stable ⟺ all poles strictly in the LHP ⟺ ∫|h(t)|dt < ∞. Marginal: simple jω poles. (Routh–Hurwitz testing: Unit-IX.)
- Minimum-phase systems: all zeros also in LHP — magnitude determines phase uniquely; RHP zeros add excess phase (non-minimum-phase: initial undershoot).
- For RLC driving-point immittances: poles and zeros are simple, lie in the closed LHP, and (for LC networks) alternate on the jω axis — Foster/Cauer realizability facts.
- h(t) = ℒ⁻¹{H(s)}; DC gain = H(0); high-frequency behaviour from n − m excess poles (−20(n−m) dB/dec eventual slope).
UGC NET focus
Match pole locations ↔ response sketches; stability classification from a given H(s); LC immittance pole–zero alternation; "poles of the transfer function = natural frequencies of the network."
§ 3.12Bode Plots
Asymptotic log–log plots of |H| (dB = 20 log|H|) and phase vs log ω. Decompose H(jω) into standard factors; magnitudes add in dB, phases add directly.
| Factor | Magnitude asymptote | Phase |
|---|---|---|
| Gain K | 20 log K (flat) | 0° (180° if K<0) |
| s (zero at origin) | +20 dB/dec through 0 dB at ω = 1 | +90° constant |
| 1/s (pole at origin) | −20 dB/dec | −90° constant |
| (1 + s/ω_c) zero | 0 until ω_c, then +20 dB/dec | 0→+90° (+45° at ω_c), over ω_c/10 … 10ω_c |
| 1/(1 + s/ω_c) pole | 0 until ω_c, then −20 dB/dec | 0→−90° (−45° at ω_c) |
| Quadratic pole pair (ζ, ωₙ) | −40 dB/dec past ωₙ; peak \(1/2\zeta\sqrt{1-\zeta^2}\) if ζ<0.707 | 0→−180° (−90° at ωₙ) |
- Corner (break) frequency: asymptotes intersect; true curve deviates −3 dB at ω_c for a simple pole (−1 dB at ω_c/2, exact curve symmetric in log ω).
- Slope bookkeeping: each pole adds −20 dB/dec, each zero +20 dB/dec; final slope −20(n−m) dB/dec; −20 dB/dec ≡ −6 dB/octave.
- Reading specs: gain margin = −|H| (dB) at the phase-crossover (∠ = −180°); phase margin = 180° + ∠H at the gain-crossover (|H| = 1). Positive margins → closed-loop stability (control context, Unit-IX).
- Reverse problems: reconstruct H(s) from a given asymptotic plot — count slope changes (±20: simple, ±40: double or quadratic) and read corner frequencies; identify type (number of origin poles) from the low-frequency slope.
UGC NET focus
−3 dB at the corner; slope arithmetic (e.g., two poles + one zero → net −20 dB/dec at HF); −45°/decade phase ramp approximation; reconstructing H(s) from a sketched plot is a recurring full numericals.
§ 3.13Passive Filters
First-order RC/RL prototypes
- Low-pass RC (output across C): \( H = \frac{1}{1+j\omega RC} \), \( f_c = \frac{1}{2\pi RC} \), −20 dB/dec, phase 0→−90°. Acts as an integrator well above f_c.
- High-pass RC (output across R): \( H = \frac{j\omega RC}{1+j\omega RC} \), same f_c, +20 dB/dec below; differentiator well below f_c.
- Band-pass: series RLC output across R — center ω₀ = 1/√(LC), BW = R/L, Q = ω₀L/R. Band-stop (notch): output across the LC pair; twin-T RC notch is the classic RC-only realization.
Image-parameter (classical) filters
- Constant-k prototype: series Z₁ and shunt Z₂ with \( Z_1Z_2 = k^2 = R_0^2 \) (frequency-independent product).
Constant-k low-pass (T or π section) $$ f_c = \frac{1}{\pi\sqrt{LC}}, \qquad R_0 = \sqrt{\frac{L}{C}}; \qquad \text{HP: } f_c = \frac{1}{4\pi\sqrt{LC}} $$Pass-band when Z₁/4Z₂ lies between −1 and 0. Drawbacks: slow cutoff, image impedance varies over the pass band (mismatch).
- m-derived sections: modify with parameter m (0 < m < 1) to place a pole of attenuation \( f_\infty \) just beyond cutoff → sharp transition: \( m = \sqrt{1 - (f_c/f_\infty)^2} \) (LP). Attenuation falls again past f_∞, so practice cascades constant-k (high far-out attenuation) + m-derived (sharp edge) + m = 0.6 half-sections for near-constant terminating impedance.
- Modern polynomial responses (named, frequently asked): Butterworth — maximally flat, \( |H|^2 = 1/(1+(\omega/\omega_c)^{2n}) \); Chebyshev — equiripple pass band, steeper; elliptic — ripple both bands, steepest; Bessel — maximally flat delay (best pulse fidelity). Active versions: Unit-IV.
UGC NET focus
f_c formulas: RC 1/2πRC, constant-k LP 1/π√LC (note the missing 2!), HP 1/4π√LC; R₀ = √(L/C); m-derived purpose and m-formula; Butterworth/Chebyshev/Bessel property matching; LP↔integrator, HP↔differentiator regimes.
§ 3.14Two-Port Network Parameters
Port convention: I₁, I₂ both flow into the network; V₁ input, V₂ output.
| Set | Equations | Found with | Natural use |
|---|---|---|---|
| Z (open-circuit) | V₁ = z₁₁I₁ + z₁₂I₂ V₂ = z₂₁I₁ + z₂₂I₂ | open ports (I = 0) | series connection (Z's add) |
| Y (short-circuit) | I₁ = y₁₁V₁ + y₁₂V₂ I₂ = y₂₁V₁ + y₂₂V₂ | short ports (V = 0) | parallel connection (Y's add); Y = Z⁻¹ |
| ABCD (transmission) | V₁ = AV₂ − BI₂ I₁ = CV₂ − DI₂ | output open/short | cascade: matrices multiply; lines (Unit-VII) |
| h (hybrid) | V₁ = h₁₁I₁ + h₁₂V₂ I₂ = h₂₁I₁ + h₂₂V₂ | input short / output open mix | transistor models: h_ie, h_re, h_fe, h_oe |
Reciprocity & symmetry tests (memorize the whole row)
$$ \textbf{Reciprocal: } z_{12}=z_{21};\ \ y_{12}=y_{21};\ \ AD-BC=1;\ \ h_{12}=-h_{21} $$
$$ \textbf{Symmetric: } z_{11}=z_{22};\ \ y_{11}=y_{22};\ \ A=D;\ \ \Delta h = h_{11}h_{22}-h_{12}h_{21} = 1 $$
- Networks of R, L, C, M only are reciprocal; dependent sources break reciprocity (transistors: h₂₁ = h_fe ≫ −h₁₂).
- Standard results worth caching: series impedance Z alone → ABCD = [1, Z; 0, 1]; shunt admittance Y → [1, 0; Y, 1]; ideal transformer n:1 → [n, 0; 0, 1/n].
- T-network ↔ Z parameters (z₁₁ = Z_a + Z_c etc.); π-network ↔ Y parameters; Δ↔Y (star–delta) conversion as the underlying identity.
- Interconnections: series → add Z; parallel → add Y; series–parallel → add h; cascade → multiply ABCD (validity requires port-current condition — Brune test).
- Conversions exist among all sets, e.g. \( z_{11} = \Delta h/h_{22} \), \( y_{21} = -h_{21}/h_{11} \), \( A = z_{11}/z_{21} \), \( B = \Delta z/z_{21} \).
- Image impedance / characteristic impedance of symmetric sections: \( Z_0 = \sqrt{Z_{OC}Z_{SC}} \) — bridges to filter and line theory.
UGC NET focus
Reciprocity/symmetry condition table (AD − BC = 1 is the most-asked single fact); which connection adds which parameters; compute Z or ABCD of simple T/π/L sections; h-parameter ↔ transistor quantity mapping.
§ 3.15State Variable Method
nth-order network → n coupled first-order ODEs. State vector x = the memory of the system; the natural choice in circuits: capacitor voltages and inductor currents (number of states = number of independent storage elements, minus excess C-loops/L-cutsets).
State-space form
$$ \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}, \qquad \mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u} $$
- Transfer function from state space: \( H(s) = \mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B} + \mathbf{D} \); poles = eigenvalues of A (det(sI − A) = characteristic polynomial).
- Time solution: \( \mathbf{x}(t) = e^{\mathbf{A}t}\mathbf{x}(0) + \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{B}\mathbf{u}(\tau)d\tau \); the state-transition matrix \( \Phi(t) = e^{\mathbf{A}t} = \mathcal{L}^{-1}\{(s\mathbf{I}-\mathbf{A})^{-1}\} \) with Φ(0) = I, Φ⁻¹(t) = Φ(−t), Φ(t₁+t₂) = Φ(t₁)Φ(t₂).
- Advantages over transfer functions: handles MIMO, time-varying and nonlinear extensions, initial conditions naturally, internal (not just I/O) description; basis of controllability/observability (Kalman): controllable if rank[B AB … Aⁿ⁻¹B] = n; observable if rank[C; CA; …; CAⁿ⁻¹]ᵀ = n.
- Canonical forms (controllable/observable/diagonal) and the companion-matrix construction from a differential equation are standard derivation asks.
UGC NET focus
Pick state variables (v_C, i_L); eigenvalues of A = poles; Φ(t) properties; write A, B for a simple RC/RLC circuit; rank tests for controllability/observability.
§ 3.16Signal Representation & Continuous-Time Signals
Standard signals
- Unit impulse δ(t): ∫δ = 1, sifting \( \int x(t)\delta(t-t_0)dt = x(t_0) \), δ(at) = δ(t)/|a|; unit step u(t) = ∫δ; ramp r(t) = t·u(t); exponential, sinusoid, complex exponential \(e^{j\omega t}\) (eigenfunction of LTI systems); sinc, rect, sgn, Gaussian.
Classification (definitions are direct MCQs)
- Energy vs power: \( E = \int|x|^2dt \), \( P = \lim_{T\to\infty}\tfrac{1}{2T}\int_{-T}^{T}|x|^2dt \). Energy signal: 0 < E < ∞ (P = 0) — pulses; power signal: 0 < P < ∞ (E = ∞) — periodic/random. A signal can be neither (e.g., e^{t}u(t)), never both.
- Periodic: x(t+T) = x(t); sum of sinusoids periodic iff frequency ratios rational.
- Even/odd decomposition: \( x_e = \tfrac12[x(t)+x(-t)] \), \( x_o = \tfrac12[x(t)-x(-t)] \).
- Deterministic vs random; causal/anticausal.
- Operations: shift, scale, reversal — order matters: x(at − b) = shift by b then compress by a (or compress then shift by b/a).
System properties (test definitions)
- Linearity (superposition), time-invariance, causality (h(t) = 0 for t < 0), stability (BIBO: ∫|h| < ∞), memory, invertibility. LTI systems: y = x * h, with convolution properties (commutative, associative; width = sum of widths; convolution with δ(t − t₀) = shift).
UGC NET focus
Energy/power classification of given signals; periodicity of sums (rational-ratio test); sifting-property evaluations; convolution length/limits; testing a given y = f(x) for linearity/TI/causality.
§ 3.17Sampling Theorem
Nyquist–Shannon sampling theorem
A signal band-limited to f_m can be completely reconstructed from its samples if the sampling rate \( f_s \ge 2f_m \). Nyquist rate = 2f_m (minimum f_s); Nyquist interval = 1/2f_m (maximum spacing); f_s/2 is the folding frequency.
- Ideal (impulse) sampling: \( x_s(t) = x(t)\sum\delta(t-nT_s) \) → spectrum becomes periodic replicas: \( X_s(f) = \tfrac{1}{T_s}\sum_k X(f - kf_s) \).
- Aliasing: if f_s < 2f_m the replicas overlap; a component at f > f_s/2 masquerades as \( |f - kf_s| \) — e.g., 7 kHz sampled at 10 kHz appears at 3 kHz. Prevention: analog anti-aliasing low-pass pre-filter before the sampler (cannot be fixed after).
- Reconstruction: ideal LP filter of bandwidth f_s/2 ↔ sinc interpolation \( x(t) = \sum x(nT_s)\,\mathrm{sinc}\!\big(\tfrac{t-nT_s}{T_s}\big) \).
- Practical sampling: natural (chopper) and flat-top (sample-and-hold) — flat-top multiplies the spectrum by a sinc → aperture effect, equalized by an x/sin x post-filter.
- Bandpass sampling: signals occupying (f_L, f_H) can be sampled as low as ~2·BW (not 2f_H) under integer-band placement conditions.
- Applications: PCM/digital audio (speech 8 kHz; CD 44.1 kHz for 20 kHz audio), digital scopes, SDR; oversampling relaxes the anti-alias filter and spreads quantization noise.
UGC NET focus
Nyquist-rate computations for composite signals (rate set by the highest frequency; for products of sinc/band-limited signals, bandwidths add for multiplication); locate the alias frequency; why anti-alias filtering precedes sampling; CD 44.1 kHz example.
§ 3.18Discrete-Time Signals & the DFT
DT signal essentials
- x[n] defined on integers; δ[n] (Kronecker, no scaling subtlety), u[n], aⁿ. DT sinusoid periodicity: cos(Ω₀n) periodic iff Ω₀/2π is rational; period N = smallest integer with Ω₀N = 2πk. DT frequencies are unique only modulo 2π; Ω = π is the highest rate of oscillation.
- DTFT: \( X(e^{j\Omega}) = \sum x[n]e^{-j\Omega n} \) — continuous and 2π-periodic in Ω. Relation to analog frequency after sampling: Ω = ωT_s = 2πf/f_s.
Discrete Fourier Transform
DFT & IDFT (length N, W_N = e^{−j2π/N})
$$ X[k] = \sum_{n=0}^{N-1} x[n]\,W_N^{kn}, \qquad x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]\,W_N^{-kn} $$
- The DFT = N samples of the DTFT of one period: \( X[k] = X(e^{j\Omega})|_{\Omega = 2\pi k/N} \); bin spacing (frequency resolution) = f_s/N.
- Everything is circular (mod N): shifts, symmetry, and especially convolution — multiplying DFTs gives circular convolution. Linear convolution of lengths L and M via DFTs requires zero-padding to N ≥ L + M − 1 (overlap-add / overlap-save for streaming).
- Properties: linearity; circular shift x[(n−m)_N] ↔ W_N^{km}X[k]; Parseval \( \sum|x|^2 = \tfrac1N\sum|X|^2 \); real x[n] → conjugate symmetry X[N−k] = X*[k]; X[0] = Σx[n] (DC bin).
- W_N (twiddle) properties: periodicity \(W_N^{k+N}=W_N^k\), symmetry \(W_N^{k+N/2} = -W_N^k\) — the two identities that make the FFT possible.
- Leakage: a tone not centred on a bin smears across the spectrum (rect-window sinc sidelobes); windows (Hann, Hamming, Blackman) trade main-lobe width vs sidelobe level; zero-padding interpolates the display but adds no true resolution.
UGC NET focus
Compute small DFTs (N = 2, 4) directly; resolution f_s/N numericals; circular vs linear convolution and the L+M−1 padding rule; conjugate-symmetry of real-signal DFTs; periodicity condition for DT sinusoids.
§ 3.19Fast Fourier Transform (FFT)
The FFT (Cooley–Tukey, 1965) is an algorithm computing the DFT exactly — same output, fewer operations — by recursively splitting the N-point DFT using twiddle-factor symmetry/periodicity.
Complexity (the headline result)
$$ \text{Direct DFT: } N^{2}\ \text{complex multiplications} \quad\longrightarrow\quad \text{Radix-2 FFT: } \frac{N}{2}\log_2 N \ \text{mult},\ \ N\log_2 N\ \text{add} $$
N = 1024: ~10⁶ → ~5×10³ multiplications (factor ≈ 200 saving).
- Radix-2 DIT (decimation-in-time): split x[n] into even/odd-indexed halves: \( X[k] = E[k] + W_N^k O[k] \), \( X[k+N/2] = E[k] - W_N^k O[k] \) — the butterfly. Requires N = 2^m; log₂N stages × N/2 butterflies.
- DIT: input in bit-reversed order, output natural; DIF (decimation-in-frequency): input natural, output bit-reversed. Both in-place (memory-efficient).
- Variants: radix-4, split-radix (fewer mults), mixed-radix/prime-factor for composite N; Goertzel algorithm when only a few bins are needed (DTMF detection); real-input FFTs halve work.
- Applications: fast spectral analysis, fast convolution/correlation (filtering long sequences), OFDM modems (IFFT/FFT at transmitter/receiver), radar/sonar Doppler processing, image processing.
UGC NET focus
Counting questions: stages = log₂N, butterflies/stage = N/2, total mult = (N/2)log₂N (e.g., N = 8 → 12); bit-reversal ordering (for N = 8: 0,4,2,6,1,5,3,7); FFT computes the DFT exactly — it is not an approximation.
§ 3.20DSP Basics & Digital Filters: IIR and FIR
DSP system anatomy
x(t) → anti-alias LPF → S/H + ADC (quantization: SQNR ≈ 6.02b + 1.76 dB for b bits) → processor (MAC-centric DSP/FPGA) → DAC → reconstruction (smoothing) LPF → y(t). Merits over analog: programmability, exact repeatability, no drift/aging, easy storage, adaptive and linear-phase filters possible; costs: bandwidth limited by f_s, quantization/round-off noise, power.
LTI difference equation and transfer function:
General digital filter
$$ y[n] = \sum_{k=0}^{M} b_k\,x[n-k] \; - \; \sum_{k=1}^{N} a_k\,y[n-k] \quad\Longleftrightarrow\quad H(z) = \frac{\sum b_kz^{-k}}{1 + \sum a_kz^{-k}} $$
FIR filters
- All a_k = 0 → output from a finite window of inputs (non-recursive, all-zero, "moving average"); h[n] has length M+1; always stable.
- Exactly linear phase if coefficients are symmetric or antisymmetric: h[n] = ±h[M−n] (four types); group delay constant = M/2 samples → distortionless filtering (audio, biomedical, data).
- Design: windowing of the ideal sinc impulse response (rect → −13 dB sidelobes; Hamming −43 dB; Blackman −58 dB; Kaiser adjustable), frequency sampling, optimal equiripple Parks–McClellan (Remez).
- Cost: high order (often 10× IIR) for sharp cutoffs → more MACs/delay.
IIR filters
- Feedback (a_k ≠ 0) → infinite impulse response; poles anywhere inside the unit circle; stable iff all poles inside |z| = 1.
- Designed from analog prototypes (Butterworth/Chebyshev/elliptic) via:
- Impulse invariance: h[n] = T·h_a(nT); preserves time response; aliases — LP/BP only.
- Bilinear transform: \( s = \tfrac{2}{T}\tfrac{1-z^{-1}}{1+z^{-1}} \) — one-to-one jω-axis → unit circle, no aliasing, but frequency warping \( \omega_a = \tfrac{2}{T}\tan(\Omega/2) \) (pre-warp the critical frequencies).
- Efficient (low order, low delay) but nonlinear phase; coefficient quantization can destabilize → implement as cascaded second-order sections (biquads).
| FIR | IIR | |
|---|---|---|
| Feedback | None (non-recursive) | Recursive |
| Stability | Guaranteed | Conditional (poles in unit circle) |
| Linear phase | Exact (symmetric h) | Not achievable (approx. only) |
| Order for sharp cutoff | High | Low (efficient) |
| Analog heritage | None needed | Butterworth/Chebyshev/elliptic |
| Quantization sensitivity | Low | Higher (limit cycles possible) |
| Typical use | Linear-phase, adaptive, multirate | Sharp selective filtering at low cost |
UGC NET focus
6 dB/bit SQNR; FIR linear-phase symmetry condition and delay M/2; bilinear warping formula and pre-warping; impulse-invariance aliasing flaw; stability tests on given H(z); window sidelobe ordering.
§ 3.21Unit-3 Formula Sheet
| Topic | Formula | Notes |
|---|---|---|
| Graph counts | twigs = n − 1; links = b − n + 1 | KCL / KVL equation counts |
| Max power transfer | \( Z_L = Z_{Th}^* ;\ P_{max} = V_{Th}^2/4R_{Th} \) (RMS) | η = 50% |
| Millman | \( V = \sum V_iG_i / \sum G_i \) | parallel branches |
| Series resonance | \( \omega_0 = 1/\sqrt{LC};\ Q = \omega_0L/R;\ BW = \omega_0/Q \) | V_L = V_C = QV_s |
| Parallel tank | \( R_{dyn} = L/RC \) | Z max, I min |
| First-order transient | \( x(t) = x_\infty + (x_{0^+} - x_\infty)e^{-t/\tau} \) | τ = RC or L/R; 63.2% at τ |
| RLC damping | \( \alpha = R/2L;\ \zeta = \alpha/\omega_0 = \tfrac{R}{2}\sqrt{C/L} \) | critical R = 2√(L/C) |
| Laplace IVT/FVT | \( f(0^+) = \lim_{s\to\infty}sF;\ f(\infty) = \lim_{s\to0}sF \) | FVT: poles of sF in LHP |
| FS coefficients | \( c_n = \tfrac1T\int_T f e^{-jn\omega_0t}dt \) | Parseval P = Σ|c_n|² |
| FT pair (gate) | rect(t/τ) ↔ τ sinc(ωτ/2) | time–BW inverse |
| Z-pair | aⁿu[n] ↔ z/(z−a), |z|>|a| | stability: poles in unit circle |
| Rise time | \( t_r \approx 0.35/BW \) (Hz); \( t_r = 2.2\tau \) | first-order |
| 2nd-order metrics | \( M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}};\ t_s = 4/\zeta\omega_n \) | ω_d = ω_n√(1−ζ²) |
| Bode slopes | pole −20 dB/dec; zero +20; quadratic −40 | −3 dB at corner |
| RC cutoff | \( f_c = 1/2\pi RC \) | LP/HP first order |
| Constant-k LP | \( f_c = 1/\pi\sqrt{LC};\ R_0 = \sqrt{L/C} \) | HP: 1/4π√LC |
| m-derived | \( m = \sqrt{1-(f_c/f_\infty)^2} \) | m = 0.6 for matching |
| Two-port tests | reciprocal: AD−BC = 1; symmetric: A = D | z₁₂ = z₂₁; h₁₂ = −h₂₁ |
| Cascade | [ABCD] = [ABCD]₁ × [ABCD]₂ | series→Z add; parallel→Y add |
| State space | \( H(s) = C(sI−A)^{-1}B + D \) | poles = eig(A); Φ = e^{At} |
| Sampling | \( f_s \ge 2f_m \); alias at |f − kf_s| | resolution after DFT: f_s/N |
| DFT | \( X[k] = \sum x[n]W_N^{kn} \) | circular conv.; pad to L+M−1 |
| FFT cost | (N/2)log₂N mult vs N² | stages = log₂N |
| SQNR | ≈ 6.02b + 1.76 dB | ~6 dB per bit |
| Bilinear transform | \( s = \tfrac{2}{T}\tfrac{1-z^{-1}}{1+z^{-1}};\ \omega_a = \tfrac{2}{T}\tan\tfrac{\Omega}{2} \) | pre-warp; no aliasing |
| FIR linear phase | h[n] = ±h[M−n]; delay M/2 | always stable |
§ 3.22Quick Revision Notes — Unit 3 in 25 Points
Rapid-fire recap (last-day revision)
- i_L and v_C cannot jump; DC steady state: L = short, C = open; at t = 0⁺ (relaxed): L = open, C = short.
- Graph: twigs n − 1 (KCL count), links b − n + 1 (KVL count); a tree spans all nodes, no loops.
- Supernode = voltage source between non-reference nodes; supermesh = shared current source; dependent sources are never killed.
- Superposition: responses add, power does not; valid only for linear networks.
- Thevenin/Norton: V_Th = V_OC, Z_Th = V_OC/I_SC with dependent sources; conversion V_Th = I_N Z_Th.
- MPT: Z_L = Z_Th* → P_max = V_Th²/4R_Th (RMS), efficiency 50%; magnitude-only case |Z_L| = |Z_Th|.
- Tellegen Σv_bi_b = 0 needs only KCL + KVL; reciprocity needs linear bilateral (no controlled sources).
- Series resonance: Z min, I max, unity PF, V_L = V_C = QV; parallel: Z max = L/CR, I min.
- Q = ω₀L/R = (1/R)√(L/C); BW = ω₀/Q; ω₀ = √(ω₁ω₂).
- One-τ rule: 63.2%; 5τ done; capacitor charging wastes exactly half the source energy.
- RLC: critical R = 2√(L/C); ζ = R/2 · √(C/L) (series); ringing at ω_d = ω₀√(1−ζ²); ζ = 1/2Q.
- Laplace: f′ → sF − f(0⁻); FVT only if poles of sF(s) in LHP (fails for oscillations).
- Fourier symmetry: even→cos, odd→sin, half-wave→odd harmonics only; Gibbs ≈ 9% overshoot.
- rect ↔ sinc; narrow in time = wide in frequency; real signal → |X| even, ∠X odd.
- Z-transform: ROC determines the sequence; causal+stable ⇔ poles inside unit circle ⇔ ROC ⊇ |z| = 1; z = e^{sT}.
- t_r·BW ≈ 0.35; overshoot e^{−πζ/√(1−ζ²)}; settling ≈ 4/ζωₙ; resonant peak only for ζ < 0.707.
- Poles = natural frequencies; LHP = stable; jω simple = marginal; LC immittances: poles/zeros alternate on jω axis.
- Bode: ±20 dB/dec per simple zero/pole, −40 for quadratic; −3 dB and −45° at the corner; −20 dB/dec = −6 dB/octave.
- Filter cutoffs: RC → 1/2πRC; constant-k LP → 1/π√LC; HP → 1/4π√LC; R₀ = √(L/C); m-derived adds f_∞ for a sharp edge, m = 0.6 ends for matching.
- Two-port: reciprocity z₁₂ = z₂₁ / AD − BC = 1 / h₁₂ = −h₂₁; symmetry z₁₁ = z₂₂ / A = D; cascade multiplies ABCD.
- State space: x = (v_C, i_L); poles = eigenvalues of A; Φ(t) = e^{At}, Φ(0) = I.
- Sampling: f_s ≥ 2f_m; alias of f appears at |f − kf_s|; anti-alias filter must precede the ADC; CD = 44.1 kHz.
- DFT bin spacing = f_s/N; DFT products give circular convolution — zero-pad to L + M − 1 for linear.
- FFT: exact DFT in (N/2)log₂N mults (N = 8 → 12), log₂N stages, bit-reversed I/O; basis of OFDM.
- FIR: always stable, exact linear phase if h[n] = ±h[M−n], delay M/2. IIR: efficient but conditional stability, nonlinear phase; bilinear transform warps — pre-warp critical frequencies. SQNR ≈ 6 dB/bit.