📘 Suchenow Academy | TGPSC AEE Electrical
Subject 1 → Chapter 1.7: Fourier, Laplace & Z Transforms
| Expected questions | 3–5 every exam — Laplace partial fractions and properties are highest yield |
| Upgraded features | Live Fourier spectrum visualiser · Searchable transform pair table · Pole-zero plot explorer · Partial fraction solver · Convolution animator |
| Target exams | TGPSC AEE, TSGENCO AE, GATE EE, ESE |
| Syllabus line | Fourier series, Fourier transform, Laplace transform, Z-transform |
1. Why Transforms? — The Big Picture First
Differential equations in time domain are hard. Transforms convert them to algebraic equations in frequency/complex domain — solve easily, then convert back. This is the single most powerful idea in circuit analysis.
📡
Fourier Transform
Periodic & non-periodic signals → frequency spectrum. "What frequencies are in this signal?"
F(ω) = ∫f(t)e⁻ʲωᵗdt
⚡
Laplace Transform
Circuit analysis with initial conditions. "Solve this ODE instantly using algebra."
F(s) = ∫f(t)e⁻ˢᵗdt
🔢
Z-Transform
Discrete-time sequences — sampled signals, digital filters, difference equations.
X(z) = Σx[n]z⁻ⁿ
🌾 తెలుగులో: Transform అంటే translation — Telugu ని English కి translate చేసినట్లు. Time domain (కష్టమైన ODE) ని frequency domain (సులభమైన algebra) కి translate చేస్తుంది. Solve చేసి తిరిగి translate చేస్తే answer వస్తుంది!
2. Fourier Series — Periodic Signals
Any periodic signal f(t) with period T can be written as:
f(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)] where ω₀=2π/T
aₙ = (2/T)∫f(t)cos(nω₀t)dt bₙ = (2/T)∫f(t)sin(nω₀t)dt
Exponential form: f(t) = Σ Cₙ e^(jnω₀t) where Cₙ = (1/T)∫f(t)e^(-jnω₀t)dt
f(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)] where ω₀=2π/T
aₙ = (2/T)∫f(t)cos(nω₀t)dt bₙ = (2/T)∫f(t)sin(nω₀t)dt
Exponential form: f(t) = Σ Cₙ e^(jnω₀t) where Cₙ = (1/T)∫f(t)e^(-jnω₀t)dt
🌾 తెలుగులో: Fourier series అంటే ఏ signal నైనా sine మరియు cosine waves యొక్క sum గా రాయవచ్చు. Square wave = fundamental + 3rd harmonic + 5th harmonic + ... (odd harmonics only!)
🎮 Lab 1 — Fourier Spectrum Visualiser (Live)
Select a waveform — watch it built from harmonics in real time. Drag harmonics slider to add more:
FOURIER SERIES PROPERTIES:
Even function f(−t)=f(t): bₙ=0 (only cosine terms)
Odd function f(−t)=−f(t): aₙ=0 (only sine terms)
Half-wave symmetry: only ODD harmonics present
Full-wave rectified sine: only EVEN harmonics + DC
COMMON PAIRS (Fourier):
Square wave: Cₙ = 2A/nπ (odd n), 0 (even n)
Triangle wave: Cₙ = 4A/n²π² (odd n), alternating sign
Sawtooth: Cₙ = A/nπ (all n, alternating)
Impulse δ(t): F(ω) = 1 (flat spectrum — all frequencies!)
Gate function: F(ω) = A·τ·sinc(ωτ/2)
3. Laplace Transform ⭐ (Highest Exam Yield)
Definition: F(s) = L{f(t)} = ∫₀^∞ f(t)e^(-st) dt (s = σ + jω, complex frequency)
Inverse: f(t) = L⁻¹{F(s)} = (1/2πj) ∫ F(s)eˢᵗ ds (Bromwich integral — never computed directly, use tables!)
Inverse: f(t) = L⁻¹{F(s)} = (1/2πj) ∫ F(s)eˢᵗ ds (Bromwich integral — never computed directly, use tables!)
🎮 Lab 2 — Searchable Transform Pair Library
Type a signal name or function (e.g. "step", "sin", "e^at", "ramp") — pairs appear instantly:
f(t) (time domain)
F(s) (Laplace)
ROC / notes
KEY LAPLACE PROPERTIES:
Linearity: L{af(t)+bg(t)} = aF(s)+bG(s)
Time shift: L{f(t−a)u(t−a)} = e⁻ᵃˢF(s)
Freq shift: L{e^(at)f(t)} = F(s−a)
Scaling: L{f(at)} = (1/a)F(s/a)
Differentiation:L{f'(t)} = sF(s)−f(0⁻) ← initial condition!
Integration: L{∫f(t)dt} = F(s)/s + f⁻¹(0)/s
Convolution: L{f*g} = F(s)·G(s) ← multiplication in s-domain!
Initial value: f(0⁺) = lim[s→∞] sF(s)
Final value: f(∞) = lim[s→0] sF(s) ← only if poles in left half plane
⚠️ Final value theorem trap: f(∞)=lim[s→0]sF(s) ONLY if all poles of sF(s) are in the LEFT half s-plane. If any pole on jω-axis or RHP → final value theorem INVALID. Very common exam trap.
🎮 Lab 3 — Partial Fraction Decomposition (Live)
Most exam Laplace problems need partial fractions. Enter numerator and denominator coefficients — see decomposition step by step:
Numerator (highest power first)
Denominator (highest power first)
Enter coefficients and click Decompose. Example: N=2s+8, D=s²+3s+2=(s+1)(s+2)
🎮 Lab 4 — Pole-Zero Plot & System Stability (Live)
Drag the pole/zero slider — see how pole location determines stability. Left half plane = stable, Right = unstable, jω-axis = marginally stable:
POLE-ZERO & STABILITY:
Pole in LEFT half s-plane (σ<0): STABLE — response decays
Pole on jω-AXIS (σ=0): MARGINALLY STABLE — sustained oscillation
Pole in RIGHT half s-plane (σ>0): UNSTABLE — response grows
Response types:
Real pole σ: f(t) ∝ e^(σt) — exponential decay/growth
Complex pair σ±jω: f(t) ∝ e^(σt)cos(ωt) — damped/growing oscillation
Repeated pole: f(t) ∝ t·e^(σt) — polynomial × exponential
4. Circuit Analysis Using Laplace
s-domain element models:
Resistor R: Z(s) = R (same)
Inductor L: Z(s) = sL, initial current I₀ → voltage source L·I₀ in series
Capacitor C: Z(s) = 1/(sC), initial voltage V₀ → voltage source V₀/s in series
Procedure: Replace elements → write KVL/KCL in s-domain → solve for I(s) or V(s) → inverse Laplace → i(t) or v(t)
Resistor R: Z(s) = R (same)
Inductor L: Z(s) = sL, initial current I₀ → voltage source L·I₀ in series
Capacitor C: Z(s) = 1/(sC), initial voltage V₀ → voltage source V₀/s in series
Procedure: Replace elements → write KVL/KCL in s-domain → solve for I(s) or V(s) → inverse Laplace → i(t) or v(t)
🎮 Lab 5 — RC Circuit Laplace Solver (Live)
Series RC with step input Vs and initial capacitor voltage V₀. Drag values — see V_C(s), partial fractions and v_C(t) instantly:
5. Z-Transform — Discrete Domain
Definition: X(z) = Z{x[n]} = Σₙ x[n]z⁻ⁿ
Key pairs:
δ[n] → 1 u[n] → z/(z−1) aⁿu[n] → z/(z−a) n·u[n] → z/(z−1)²
cos(ω₀n)u[n] → z(z−cosω₀)/(z²−2zcosω₀+1)
Inverse Z: Use partial fractions on X(z)/z, then multiply by z. Or use power series.
Key pairs:
δ[n] → 1 u[n] → z/(z−1) aⁿu[n] → z/(z−a) n·u[n] → z/(z−1)²
cos(ω₀n)u[n] → z(z−cosω₀)/(z²−2zcosω₀+1)
Inverse Z: Use partial fractions on X(z)/z, then multiply by z. Or use power series.
🌾 తెలుగులో: Z-transform అంటే discrete signals కి Laplace transform వంటిది. Continuous time కి Laplace, Discrete time కి Z-transform. DSP, digital filters, sampled systems కి వాడతారు.
Z-TRANSFORM KEY PROPERTIES:
Linearity: Z{ax[n]+by[n]} = aX(z)+bY(z)
Time delay: Z{x[n−k]} = z⁻ᵏX(z)
Time advance: Z{x[n+k]} = zᵏX(z) − zᵏx[0] − ... − z·x[k−1]
Scaling: Z{aⁿx[n]} = X(z/a)
Convolution: Z{x[n]*h[n]} = X(z)·H(z)
Initial value: x[0] = lim[z→∞] X(z)
Final value: x[∞] = lim[z→1] (z−1)X(z) (if stable)
Laplace vs Z: s=jω → z=e^(jω). s-plane jω-axis → z-plane unit circle
Left half s-plane → inside unit circle z-plane (stable region)
6. "3 Questions That ALWAYS Appear" ⭐
⭐ Guaranteed every TGPSC/TSGENCO/GATE exam
Q-TYPE 1: Find inverse Laplace via partial fractions (always)
F(s) = (2s+8)/[(s+1)(s+2)]. Find f(t).
Method: F(s) = A/(s+1) + B/(s+2)
A=(2s+8)|_{s=−1}=6, B=(2s+8)|_{s=−2}=4
f(t) = 6e⁻ᵗ + 4e⁻²ᵗ (for t≥0)
Q-TYPE 2: Initial/Final value theorem (always)
F(s) = 10/[s(s+2)]. Find f(0⁺) and f(∞).
f(0⁺) = lim[s→∞] sF(s) = lim[s→∞] 10/(s+2) = 0
f(∞) = lim[s→0] sF(s) = lim[s→0] 10/(s+2) = 5
Check: poles of sF(s) = 10/(s+2) → pole at s=−2 (LHP) ✓ FVT valid
Q-TYPE 3: Laplace of circuit (always)
RC series: R=1Ω, C=1F, Vs=u(t), V₀=0. Find V_C(s) and v_C(t).
KVL: Vs/s = I(s)[R + 1/(sC)] → I(s) = (1/s)/(R+1/sC) = C/(RCs+1)
V_C(s) = I(s)/(sC) = 1/[s(s+1)] = 1/s − 1/(s+1)
v_C(t) = (1−e⁻ᵗ)u(t) ✓ matches standard RC charging formula
7. Solved Problems — Try First! 🎯
P1. Find L{t²e^(−3t)}.
L{tⁿ}=n!/s^(n+1). Frequency shift: L{e^(at)f(t)}=F(s−a). So L{t²}=2/s³ → L{t²e^(−3t)}=2/(s+3)³. Answer: 2/(s+3)³.
P2. Find L{sin(2t)·u(t)}.
L{sin(ωt)}=ω/(s²+ω²). With ω=2: 2/(s²+4).
P3. Find L⁻¹{(s+3)/[(s+1)(s+2)]}.
PF: (s+3)/[(s+1)(s+2)] = A/(s+1)+B/(s+2). A=(s+3)|_{s=−1}=2. B=(s+3)|_{s=−2}=1. f(t)=2e⁻ᵗ+e⁻²ᵗ.
P4. F(s)=5/[s(s+5)]. Find f(0⁺) and f(∞). Is FVT valid?
f(0⁺)=lim[s→∞]sF(s)=lim 5/(s+5)=0. f(∞)=lim[s→0]sF(s)=5/5=1. sF(s)=5/(s+5), pole at s=−5 (LHP) → FVT valid ✓. (Represents step response of 1st-order system.)
P5. F(s)=10/[s(s²+4)]. Is FVT valid? Find f(∞) if valid.
sF(s)=10/(s²+4). Poles at s=±j2 (on jω-axis). FVT NOT valid — poles not in strict LHP. f(∞) does not exist (sustained oscillation). FVT would give 10/4=2.5, which is WRONG.
P6. Find Laplace of a full-wave rectified sine: f(t)=|sin(ωt)|.
F(s)=ω/(s²+ω²) · coth(πs/2ω) — using periodic function Laplace formula. Contains only even harmonics (0, 2ω, 4ω...) as noted by Fourier series of full-wave rectified sine.
P7. Find Z{aⁿu[n]}.
Z{aⁿu[n]}=Σaⁿz⁻ⁿ=Σ(a/z)ⁿ=1/(1−a/z)=z/(z−a), |z|>|a|. This is the Z-transform analogue of Laplace's 1/(s+a).
P8. Stability: Z-transform system function H(z)=(z+0.5)/[(z−0.8)(z+1.2)]. Stable?
Poles at z=0.8 and z=−1.2. Stability condition: ALL poles inside unit circle |z|<1. |0.8|=0.8<1 ✓ but |−1.2|=1.2>1 ✗. System is UNSTABLE.
P9. Fourier series of square wave (A=1, T=2π). Find C₃.
Cₙ=2A/nπ for odd n, 0 for even n. C₃=2×1/(3π)=2/3π=0.212. Square wave has only odd harmonics (half-wave symmetry). C₁=2/π (fundamental is largest).
P10. L{f'(t)} when f(0⁻)=3 and L{f(t)}=F(s). Find L{f'(t)}.
Differentiation property: L{f'(t)}=sF(s)−f(0⁻)=sF(s)−3. The initial condition f(0⁻)=3 appears as a subtracted constant — this is how circuit initial conditions enter the Laplace domain.
P11. Convolution: f(t)=e⁻ᵗu(t), g(t)=e⁻²ᵗu(t). Find (f*g)(t).
L{f}=1/(s+1), L{g}=1/(s+2). Convolution → multiplication: F(s)G(s)=1/[(s+1)(s+2)]=1/(s+1)−1/(s+2). Inverse: (f*g)(t)=(e⁻ᵗ−e⁻²ᵗ)u(t). Convolution in time = multiplication in s-domain ✓.
P12 (Full TGPSC pattern). Series RL circuit: R=2Ω, L=1H, v(t)=10u(t)V, i(0⁻)=2A. Find i(t) using Laplace.
s-domain: V(s)=10/s. Inductor: Z_L=sL=s, initial current → L·i(0⁻)=2 (series source).
KVL: V(s)=I(s)(R+sL)−L·i(0⁻) → 10/s=I(s)(2+s)−2
I(s)=[10/s+2]/(s+2)=(10+2s)/[s(s+2)]
PF: I(s)=A/s+B/(s+2). A=(10+2s)|_{s=0}/1·(wait—) A=10/2=5. B=(10+2s)|_{s=−2}/(−2)=(10−4)/(−2)=−3.
Wait: I(s)=(2s+10)/[s(s+2)]. A=lim[s→0]s·I(s)=10/2=5. B=lim[s→−2](s+2)I(s)=(−4+10)/(−2)=−3.
i(t)=(5−3e⁻²ᵗ)u(t) A. Check: i(0⁺)=5−3=2A ✓ (matches initial condition). i(∞)=5A=V/R=10/2 ✓.
KVL: V(s)=I(s)(R+sL)−L·i(0⁻) → 10/s=I(s)(2+s)−2
I(s)=[10/s+2]/(s+2)=(10+2s)/[s(s+2)]
PF: I(s)=A/s+B/(s+2). A=(10+2s)|_{s=0}/1·(wait—) A=10/2=5. B=(10+2s)|_{s=−2}/(−2)=(10−4)/(−2)=−3.
Wait: I(s)=(2s+10)/[s(s+2)]. A=lim[s→0]s·I(s)=10/2=5. B=lim[s→−2](s+2)I(s)=(−4+10)/(−2)=−3.
i(t)=(5−3e⁻²ᵗ)u(t) A. Check: i(0⁺)=5−3=2A ✓ (matches initial condition). i(∞)=5A=V/R=10/2 ✓.
8. PYQ Bank
- [TSGENCO 2015] L{e^(at)sinωt} = ω/[(s−a)²+ω²].
- [TSSPDCL 2018] L{u(t)} = 1/s. L{δ(t)} = 1.
- [TSTRANSCO 2018] Final value theorem requires all poles of sF(s) in left half plane.
- [APPSC 2016] Convolution in time domain = multiplication in s-domain.
- [GATE-style] Z-transform stability: all poles inside unit circle |z|<1.
- [ESE pattern] L{tⁿ} = n!/s^(n+1).
- [TGPSC 2022] Half-wave symmetry → only odd harmonics present.
- [TGPSC 2022] L{f'(t)} = sF(s)−f(0⁻).
9. Traps & Memory Hooks
⚠️ TOP TRAPS:
(1) FVT invalid if ANY pole of sF(s) on jω-axis or RHP
(2) L{sin(ωt)}=ω/(s²+ω²), NOT 1/(s²+ω²) — missing ω in numerator
(3) Z-stable = poles inside unit circle; Laplace-stable = poles in LHP
(4) Convolution in time → MULTIPLY in s/z domain (not add)
(5) Differentiation: L{f'(t)}=sF(s)−f(0⁻), not f(0⁺)
(6) Repeated poles: (n−1)! not n! in residue formula
(1) FVT invalid if ANY pole of sF(s) on jω-axis or RHP
(2) L{sin(ωt)}=ω/(s²+ω²), NOT 1/(s²+ω²) — missing ω in numerator
(3) Z-stable = poles inside unit circle; Laplace-stable = poles in LHP
(4) Convolution in time → MULTIPLY in s/z domain (not add)
(5) Differentiation: L{f'(t)}=sF(s)−f(0⁻), not f(0⁺)
(6) Repeated poles: (n−1)! not n! in residue formula
- "Laplace solves ODEs as algebra" — the whole point
- "Convolution time = Multiplication s-domain"
- "FVT needs LHP poles of sF(s)"
- "Z-stable = inside circle; Laplace-stable = left half"
- "Half-wave symmetry = odd harmonics only"
🌾 Final Telugu hook: "Time domain కష్టం → s/z domain కి వెళ్ళు → solve చేయి → తిరిగి రా. Convolution గుణకారం అవుతుంది, differentiation 's' multiply అవుతుంది. FVT కి LHP poles తప్పనిసరి!" — పరీక్ష లో ముందు రాయండి.
MASTER CHEAT SHEET:
LAPLACE PAIRS (must memorise):
δ(t)→1 | u(t)→1/s | t→1/s² | tⁿ→n!/s^(n+1)
e^(at)→1/(s−a) | sinωt→ω/(s²+ω²) | cosωt→s/(s²+ω²)
e^(at)sinωt→ω/[(s−a)²+ω²] | e^(at)cosωt→(s−a)/[(s−a)²+ω²]
te^(at)→1/(s−a)² | ramp t·u(t)→1/s²
PROPERTIES:
Time shift: f(t−a)u(t−a)→e^(−as)F(s)
Freq shift: e^(at)f(t)→F(s−a)
Diff: f'(t)→sF(s)−f(0⁻) | f''(t)→s²F(s)−sf(0⁻)−f'(0⁻)
Integ: ∫f→F(s)/s | Convolve→multiply
IVT: f(0⁺)=lim[s→∞]sF(s) | FVT: f(∞)=lim[s→0]sF(s) [LHP only!]
Z-PAIRS: δ[n]→1 | u[n]→z/(z−1) | aⁿu[n]→z/(z−a) | nu[n]→z/(z−1)²
Stable Z: |poles|<1 | Stable Laplace: Re(poles)<0
🎯 Chapter 1.7 Quiz — 10 Questions
⏱ Exam Timer Drill — 5 Questions · 6 Minutes
TGPSC speed: ~72 sec/question.
6:00
D1. L{e^(-2t)·sin(3t)} = ?
a) 3/(s²+9) b) 3/[(s+2)²+9] c) (s+2)/[(s+2)²+9] d) 3/(s+2)²Frequency shift: L{e^(at)f(t)}=F(s−a). L{sin3t}=3/(s²+9) → L{e^(−2t)sin3t}=3/[(s+2)²+9].
D2. F(s)=6/[s(s+3)]. Find f(∞) using FVT.
a) 0 b) 2 c) 6 d) FVT not validsF(s)=6/(s+3). Pole at s=−3 (LHP ✓ FVT valid). f(∞)=lim[s→0]6/(s+3)=6/3=2.
D3. Convolution in time domain corresponds to what in s-domain?
a) Addition b) Multiplication c) Division d) DifferentiationL{f*g}=F(s)·G(s). Convolution in time = Multiplication in s-domain. This makes it very useful for system analysis.
D4. Z-transform system is stable when all poles are:
a) In left half z-plane b) Inside unit circle |z|<1 c) On real axis d) Outside unit circleZ-stable: all poles inside unit circle |z|<1. Laplace-stable: all poles in LHP (Re(s)<0). The jω-axis maps to the unit circle.
D5. Square wave has only odd harmonics because of:
a) Even symmetry b) Odd symmetry c) Half-wave symmetry d) DC offsetHalf-wave symmetry f(t+T/2)=−f(t) → only odd harmonics (1st, 3rd, 5th...). Even harmonics cancel. Square, triangle waves both have half-wave symmetry.
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