Output Voltage: V₀ = D × Vₛ
Where: D = Duty cycle = T_on/T
Inductor Current Ripple: ΔI_L = (Vₛ - V₀) × D / (f × L)
Minimum Inductance: L_min = (1-D) × R / (2f)
Output Capacitor: C = (1-D) / (8 × L × f² × (ΔV₀/V₀))
Continuous Conduction Mode (CCM): I_L(avg) > ΔI_L/2
Output Voltage: V₀ = Vₛ / (1-D)
Where: D = Duty cycle
Voltage Gain: V₀/Vₛ = 1/(1-D)
Inductor Current: I_L(avg) = I₀ / (1-D)
Current Ripple: ΔI_L = Vₛ × D / (f × L)
Minimum Inductance: L_min = D(1-D)² × R / (2f)
Duty Cycle for given V₀: D = 1 - (Vₛ/V₀)
Output Voltage: V₀ = -D × Vₛ / (1-D)
Key Points:
- Output polarity reversed
- Can step-up or step-down
- V₀/Vₛ = D/(1-D)
When D < 0.5: Step-down
When D > 0.5: Step-up
Inductor Current: I_L(avg) = I₀ / (1-D)
Half-Wave Rectifier:
V_avg = V_m/π = 0.318V_m
V_rms = V_m/2 = 0.5V_m
Form Factor = π/2 = 1.57
Ripple Factor = 1.21
Full-Wave Rectifier (Center-Tap):
V_avg = 2V_m/π = 0.636V_m
V_rms = V_m/√2 = 0.707V_m
Form Factor = 1.11
Ripple Factor = 0.482
Bridge Rectifier:
Same as full-wave
PIV = V_m (not 2V_m like center-tap)
Controlled Rectifiers:
Single-Phase Half-Wave: V_avg = (V_m/2π)(1 + cosα)
Single-Phase Full-Wave: V_avg = (V_m/π)(1 + cosα)
Three-Phase Half-Wave: V_avg = (3V_mL/2π)cosα
Three-Phase Full-Wave: V_avg = (3V_mL/π)cosα
Class A: First quadrant (V₀ > 0, I₀ > 0) Class B: Second quadrant (V₀ > 0, I₀ < 0) Class C: Two-quadrant Class D: Two-quadrant Class E: Four-quadrant
Jones Chopper: For DC motor speed control Morgan Chopper: For low power applications
Single-Phase Half-Bridge: V₀(rms) = V_s/2
Single-Phase Full-Bridge: V₀(rms) = V_s
180° Conduction Mode:
V₀(avg) = 0
V₀(rms) = V_s√(2/3) = 0.816V_s
120° Conduction Mode:
V₀(avg) = 0
V₀(rms) = V_s/√3 = 0.577V_s
Three-Phase Inverter: Line voltage = √3 × Phase voltage
Firing Angle (α): Delay angle for triggering Conduction Angle: π - α (single-phase)
Latching Current (I_L): Minimum anode current to maintain conduction after gate pulse removal Holding Current (I_H): Minimum anode current to maintain conduction (I_H < I_L)
Turn-ON Methods:
- Gate triggering
- Forward voltage triggering
- dv/dt triggering
- Light triggering
- Temperature triggering
Turn-OFF Methods (Commutation):
- Natural/Line commutation
- Forced commutation
- Class A, B, C, D, E, F
EMF Equation: E = 4.44 × f × N × Φ_m Where:
- E = RMS voltage (V)
- f = Frequency (Hz)
- N = Number of turns
- Φ_m = Maximum flux (Wb)
Turns Ratio: E₁/E₂ = N₁/N₂ = V₁/V₂ = I₂/I₁
Efficiency:
η = Output Power / Input Power
η = Output / (Output + Losses)
Maximum efficiency occurs when: Copper Loss = Iron Loss
Voltage Regulation:
%Reg = [(E₂ - V₂)/V₂] × 100
Approximate: %Reg ≈ %R cosφ ± %X sinφ
(+ for lagging, - for leading)
OC Test (Open Circuit):
Measures: Core loss (Iron loss), Magnetizing current, No-load current
Conducted at: Rated voltage
SC Test (Short Circuit):
Measures: Copper loss, Equivalent impedance
Conducted at: Reduced voltage (5-10% rated)
Equivalent Circuit Parameters:
R₀₁ = V₁²/P₀ (from OC test)
Z₀₁ = V_sc/I_sc (from SC test)
R₀₁ = P_sc/I_sc²
X₀₁ = √(Z₀₁² - R₀₁²)
All-Day Efficiency: η_all-day = (Output in kWh / Input in kWh) × 100
Back EMF:
Motor: E_b = V - I_aR_a
Generator: E_b = V + I_aR_a
EMF Equation: E_b = (PΦNZ)/(60A) Where:
- P = Number of poles
- Φ = Flux per pole (Wb)
- N = Speed (rpm)
- Z = Total conductors
- A = Parallel paths
Torque Equation:
T = (PΦZI_a)/(2πA)
T = E_b × I_a / ω (where ω = 2πN/60)
Speed Equation: N ∝ (V - I_aR_a)/Φ For shunt: N ∝ V (approximately, if R_a negligible)
Types:
- Separately Excited: Field winding separate supply
- Shunt: Field parallel to armature, constant speed
- Series: Field series with armature, high starting torque
- Compound: Both shunt and series field
Speed Control:
- Armature voltage control
- Field flux control
- Armature resistance control
Starters:
- Three-point starter
- Four-point starter Purpose: Limit starting current
Losses:
- Copper loss (I²R)
- Iron loss (Hysteresis + Eddy current)
- Mechanical loss (Friction + Windage)
- Stray load loss
Synchronous Speed: N_s = 120f/P Where f = frequency, P = poles
Slip:
s = (N_s - N_r)/N_s
% Slip = s × 100
At start: s = 1
At synchronous: s = 0
Full load: s = 0.02 to 0.05
Rotor Speed: N_r = N_s(1 - s)
Slip Frequency: f_r = s × f
Torque Equation: T = (3V²R₂/s) / [ω_s((R₂/s)² + X₂²)] Where ω_s = 2πN_s/60
Starting Torque (s=1): T_st = (3V²R₂) / [ω_s(R₂² + X₂²)]
Maximum Torque: T_max = (3V²) / (2ω_sX₂) Occurs at: s_m = R₂/X₂
Condition for Maximum Starting Torque: R₂ = X₂ (external resistance added)
Torque-Slip Relation: T/T_max = 2 / [(s/s_m) + (s_m/s)]
Power Flow:
Air Gap Power (P_g) = 3I₂²(R₂/s)
Rotor Copper Loss = s × P_g
Mechanical Power = (1-s) × P_g
Torque = P_g / ω_s
Circle Diagram: Shows relation between current, power factor, slip, torque
Types:
- Squirrel Cage: Simple, rugged, low starting torque
- Wound Rotor: External resistance, high starting torque
Star-Delta Starter:
Starting current = 1/3 of DOL
Starting torque = 1/3 of DOL
Speed: N_s = 120f/P (Always synchronous, independent of load)
EMF Equation:
E_f = V + I_aR_a + jI_aX_s (cylindrical rotor)
E_f = V + I_aR_a + jI_aX_d (salient pole, d-axis)
Power Angle Equation:
P = (EV/X_s) sinδ (cylindrical)
P = (EV/X_d) sinδ + (V²/2)(1/X_q - 1/X_d) sin2δ (salient)
Where:
- E = Excitation voltage
- V = Terminal voltage
- δ = Power angle (torque angle)
- X_s = Synchronous reactance
Voltage Regulation: %Reg = [(|E| - |V|)/|V|] × 100
Parallel Operation Conditions:
- Same voltage magnitude
- Same frequency
- Same phase sequence
- Same phase angle
Synchronizing Power: P_syn = (EV/X_s) cosδ (per mechanical radian)
Hunting: Oscillation of rotor around synchronous speed Damper windings (amortisseur) prevent hunting
V-Curves: Plot of I_a vs I_f at constant P Shows relation between armature current and field current
Inverted V-Curves: Plot of power factor vs I_f
Synchronous Condenser: Synchronous motor running at no-load, used for power factor correction Operates at: Leading power factor (over-excited)
ABCD Parameters:
V_s = AV_r + BI_r
I_s = CV_r + DI_r
For short line (length < 80 km): A = D = 1, B = Z, C = 0
For medium line (80-240 km): Nominal π or T methods
For long line (>240 km): Hyperbolic functions
Surge Impedance: Z_c = √(L/C) = √(Z/Y) Where Z = series impedance/km, Y = shunt admittance/km
Surge Impedance Loading (SIL): P_SIL = V²/Z_c = V²√(C/L)
Ferranti Effect: Receiving end voltage > Sending end voltage Occurs at: Light load or no-load, long lines
Wavelength: λ = 1/(f√(LC))
Propagation Constant: γ = α + jβ = √(ZY) Where α = attenuation constant, β = phase constant
Velocity of Propagation: v = 1/√(LC) ≈ 3×10⁸ m/s (for overhead lines)
ABCD Relations:
AD - BC = 1 (for passive, linear network)
A = D (for symmetrical network)
Voltage Regulation: %Reg = [(|V_rNL| - |V_rFL|)/|V_rFL|] × 100
Types of Faults:
- Symmetrical: 3-phase (LLL), 3-phase to ground (LLLG)
- Unsymmetrical: LG, LL, LLG
Occurrence Frequency:
LG: 70-80%
LL: 15-20%
LLG: Rare
LLL or LLLG: 5%
Fault Current: I_f = E/Z (for solid fault)
Sequence Networks:
- Positive Sequence: Normal operating sequence (abc)
- Negative Sequence: Reverse sequence (acb)
- Zero Sequence: All phases in phase
Sequence Impedances:
Z₁ = Positive sequence impedance
Z₂ = Negative sequence impedance
Z₀ = Zero sequence impedance
For synchronous machines: Z₁ ≠ Z₂
For transformers and lines: Z₁ = Z₂
Fault Calculations:
Single Line to Ground (LG):
I_a1 = I_a2 = I_a0 = E/(Z₁ + Z₂ + Z₀ + 3Z_f)
I_f = 3I_a1
Line to Line (LL):
I_a1 = -I_a2 = E/(Z₁ + Z₂ + Z_f)
I_f = I_b = -I_c = √3 × I_a1
Double Line to Ground (LLG): I_a1 = E/[Z₁ + Z₂||(Z₀ + 3Z_f)]
Three Phase (LLL): I_f = E/Z₁
Fault MVA: Fault MVA = Base MVA / Z_pu
Base Quantities: Base MVA × Base kV = Base kA × Base kV (not consistent, recalculate)
Actually: S_base = √3 × V_base × I_base
Z_base = V_base²/S_base
Conversion: Z_pu(new) = Z_pu(old) × (S_base(new)/S_base(old)) × (V_base(old)/V_base(new))²
Advantages:
- Simplified calculations
- Transformer turns ratio eliminated
- Equipment ratings comparable
- Numerical values manageable
Bus Types:
- Slack/Swing Bus: Specified V and δ, find P and Q
- PV Bus (Generator): Specified P and |V|, find Q and δ
- PQ Bus (Load): Specified P and Q, find |V| and δ
Methods:
Gauss-Seidel:
Iterative method
V_i^(k+1) = (1/Y_ii)[(P_i - jQ_i)/V_i^(k)* - Σ(Y_ij × V_j)]
Newton-Raphson:
Faster convergence
Uses Jacobian matrix
Decoupled Load Flow:
P depends mainly on δ
Q depends mainly on |V|
Fast Decoupled: Assumes: |V| ≈ 1 pu, δ small
DC Load Flow:
Linear approximation
Neglects losses and Q
Relay Types:
Overcurrent Relay:
Operates when: I > I_pickup
Types: Instantaneous, Definite time, Inverse time, IDMT
Distance Relay:
Operates when: Impedance < Reach
Zones: Zone 1 (80-90%), Zone 2 (120%), Zone 3 (backup)
Differential Relay:
Operates on difference between two currents
Used for: Transformers, generators, buses
Directional Relay: Checks direction of power flow
Distance Relay Characteristics:
- Impedance relay: Circular
- Reactance relay: Straight line
- Mho relay: Circular through origin
CT (Current Transformer):
Turns ratio: N_p/N_s = I_s/I_p
Burden = VA rating
PT (Potential Transformer): Turns ratio: N_p/N_s = V_p/V_s
Buchholz Relay:
Transformer protection
Detects: Internal faults, oil level low
Lightning Arrester: Protects against overvoltages
Surge Absorber: Limits surge voltage magnitude
Ideal Op-Amp Characteristics:
- Infinite open-loop gain (A_ol = ∞)
- Infinite input impedance (R_in = ∞)
- Zero output impedance (R_out = 0)
- Infinite bandwidth
- Zero offset voltage
Virtual Short: V+ = V- (negative feedback) Virtual Ground: V- = 0 (when V+ grounded)
Inverting Amplifier:
V_o = -(R_f/R₁) × V_i
Gain = -R_f/R₁
Non-Inverting Amplifier:
V_o = (1 + R_f/R₁) × V_i
Gain = 1 + R_f/R₁
Voltage Follower (Buffer):
V_o = V_i
Gain = 1
High input impedance, low output impedance
Summing Amplifier: V_o = -R_f(V₁/R₁ + V₂/R₂ + V₃/R₃ + ...)
Difference Amplifier:
V_o = (R_f/R₁)(V₂ - V₁)
When R_f/R₁ = R₃/R₂
Integrator:
V_o = -(1/RC) ∫V_i dt
Transfer function: -1/(sRC)
Differentiator:
V_o = -RC(dV_i/dt)
Transfer function: -sRC
Comparator:
No feedback (open loop)
V_o = +V_sat if V+ > V-
V_o = -V_sat if V+ < V-
Schmitt Trigger (Comparator with Hysteresis):
Upper Threshold: V_UT = +V_sat × R₁/(R₁+R₂)
Lower Threshold: V_LT = -V_sat × R₁/(R₁+R₂)
Hysteresis: V_H = V_UT - V_LT
Instrumentation Amplifier:
V_o = (R₄/R₃)(1 + 2R₂/R₁)(V₂ - V₁)
High CMRR, precise gain
Active Filters:
First Order Low Pass:
Cutoff: f_c = 1/(2πRC)
Gain: -R_f/R₁ (inverting)
First Order High Pass: Cutoff: f_c = 1/(2πRC)
Band Pass: Series connection of LPF and HPF
Band Stop (Notch): Parallel connection or twin-T
Barkhausen Criteria (for oscillation):
- |Aβ| = 1 (Loop gain = 1)
- ∠Aβ = 0° or 360° (Phase shift = 0°)
Where:
- A = Amplifier gain
- β = Feedback factor
RC Phase Shift Oscillator:
Frequency: f = 1/(2πRC√6)
Minimum gain required: A = 29
Phase shift per section: 60°
Wien Bridge Oscillator:
Frequency: f = 1/(2πRC)
Gain required: A = 3
Balanced bridge condition: R₁ = R₂ = R, C₁ = C₂ = C
Hartley Oscillator:
Frequency: f = 1/(2π√(LC_eq))
Where L_eq = L₁ + L₂ + 2M (mutual inductance)
Tapped inductor for feedback
Colpitts Oscillator:
Frequency: f = 1/(2π√(LC_eq))
Where C_eq = C₁C₂/(C₁+C₂)
Tapped capacitor for feedback
Crystal Oscillator:
Very stable frequency
Uses piezoelectric crystal
Relaxation Oscillator:
Uses RC or RL timing
Astable multivibrator
Current Relations:
I_E = I_C + I_B
I_C = β × I_B
α = β/(β+1)
β = α/(1-α)
Where:
- α = Common base current gain (0.95-0.99)
- β = Common emitter current gain (20-200)
Operating Regions:
- Active: EB junction forward, CB junction reverse (Amplifier)
- Saturation: Both junctions forward (Switch ON)
- Cutoff: Both junctions reverse (Switch OFF)
Fixed Bias:
I_B = (V_CC - V_BE)/R_B
I_C = β × I_B
V_CE = V_CC - I_CR_C
Voltage Divider Bias (Self-Bias):
More stable
V_B = V_CC × R₂/(R₁+R₂)
I_E = (V_B - V_BE)/R_E
I_C ≈ I_E
Emitter Follower:
Voltage gain ≈ 1
Current gain ≈ β+1
High input impedance, low output impedance
Open Loop: G(s) = Output(s)/Input(s) (without feedback)
Closed Loop: T(s) = G(s)/(1 + G(s)H(s))
For Unity Feedback (H=1): T(s) = G(s)/(1 + G(s))
Error Signal:
E(s) = R(s) - C(s)H(s)
For unity feedback: E(s) = R(s) - C(s)
Series Connection: G_eq = G₁ × G₂
Parallel Connection: G_eq = G₁ + G₂
Feedback Connection:
G_eq = G/(1 ± GH)
(+ for negative feedback, - for positive feedback)
Moving Take-off Point:
- Ahead of block: Multiply by block gain
- Behind block: Divide by block gain
Moving Summing Point:
- Ahead of block: Divide by block gain
- Behind block: Multiply by block gain
Mason's Gain Formula: T = (Σ P_k × Δ_k) / Δ
Where:
- P_k = Path gain of kth forward path
- Δ = 1 - ΣL₁ + ΣL₂ - ΣL₃ + ...
- L₁ = Individual loop gains
- L₂ = Gain products of two non-touching loops
- Δ_k = Δ for part of graph not touching kth forward path
Stable System:
All poles in Left Half Plane (LHP)
Impulse response decays to zero
Unstable System:
Any pole in Right Half Plane (RHP)
Impulse response grows without bound
Marginally Stable:
Poles on imaginary axis (simple)
Sustained oscillations
Routh-Hurwitz Criterion:
Construct Routh array
Count sign changes in first column = Number of RHP poles
If any element in first column is zero: System is marginally stable or unstable
Special Cases:
- Zero in first column: Replace with ε (small positive), proceed
- Entire row zero: Form auxiliary polynomial, take derivative, continue
Angle Condition: ∠G(s)H(s) = ±180°(2k+1)
Magnitude Condition: |G(s)H(s)| = 1
Rules:
- Number of branches = Number of poles
- Locus starts at open-loop poles (K=0), ends at zeros (K=∞) or infinity
- Real axis segments: To the left of odd number of poles+zeros
- Asymptotes angles: (2k+1)180°/(P-Z)
- Centroid: (Σpoles - Σzeros)/(P-Z)
- Breakaway points: dK/ds = 0
- Imaginary axis crossing: Routh criterion
Nyquist Criterion: N = Z - P Where:
- N = Number of encirclements of -1+j0 point
- Z = Number of closed-loop poles in RHP
- P = Number of open-loop poles in RHP
For stability: Z = 0, so N = -P
Gain Margin:
GM = 1/|G(jω)H(jω)| at phase crossover frequency
In dB: GM(dB) = -20log|G(jω_pc)H(jω_pc)|
Phase Margin:
PM = 180° + ∠G(jω_gc)H(jω_gc)
At gain crossover frequency (where |GH| = 1)
Bode Plot:
- Magnitude plot: 20log|G(jω)| vs log(ω)
- Phase plot: ∠G(jω) vs log(ω)
Corner Frequency: ω_c = 1/τ for first-order system
Slope Changes:
- Pole at origin: -20 dB/decade
- Zero at origin: +20 dB/decade
- Real pole: -20 dB/decade (starting at ω_c)
- Real zero: +20 dB/decade (starting at ω_c)
First Order System: Transfer function: C(s)/R(s) = 1/(1+τs)
Step response: c(t) = 1 - e^(-t/τ) Time constant: τ Settling time (2%): 4τ Settling time (5%): 3τ
Second Order System: Transfer function: ω_n²/(s² + 2ζω_ns + ω_n²)
Where:
- ω_n = Natural frequency (rad/s)
- ζ = Damping ratio
Characteristics:
Rise time (10% to 90%): t_r ≈ (π - θ)/(ω_n√(1-ζ²)) where θ = cos⁻¹(ζ)
Peak time: t_p = π/(ω_n√(1-ζ²))
Peak overshoot: M_p = e^(-ζπ/√(1-ζ²)) × 100%
Settling time (2% criterion): t_s = 4/(ζω_n)
Settling time (5% criterion): t_s = 3/(ζω_n)
Damping Cases:
- ζ = 0: Undamped (sustained oscillations)
- 0 < ζ < 1: Underdamped (decaying oscillations)
- ζ = 1: Critically damped (fastest without overshoot)
- ζ > 1: Overdamped (no overshoot, slow)
Error:
e(t) = r(t) - c(t)
E(s) = R(s) - C(s)
Final Value Theorem: e_ss = lim(s→0) sE(s) = lim(s→0) sR(s)/(1+G(s)H(s))
Error Constants:
Step Input (R(s) = 1/s):
Position error constant: K_p = lim(s→0) G(s)H(s)
e_ss = 1/(1+K_p)
Ramp Input (R(s) = 1/s²):
Velocity error constant: K_v = lim(s→0) sG(s)H(s)
e_ss = 1/K_v
Parabolic Input (R(s) = 1/s³):
Acceleration error constant: K_a = lim(s→0) s²G(s)H(s)
e_ss = 1/K_a
System Type vs Steady State Error:
Type 0: Finite e_ss for step, ∞ for ramp and parabolic Type 1: Zero e_ss for step, finite for ramp, ∞ for parabolic Type 2: Zero e_ss for step and ramp, finite for parabolic
Unit Step: u(t) = 1 for t ≥ 0, 0 for t < 0
Unit Impulse:
δ(t) = du(t)/dt
∫δ(t)dt = 1
δ(at) = (1/|a|)δ(t)
Unit Ramp: r(t) = t for t ≥ 0, 0 for t < 0
Rectangular Pulse: Π(t/τ) = 1 for |t| ≤ τ/2, 0 otherwise
Triangular Pulse: Λ(t/τ) = 1 - |t|/τ for |t| ≤ τ, 0 otherwise
Sinc Function: sinc(t) = sin(πt)/(πt)
Time Shifting: x(t - t₀): Delay by t₀ (right shift) x(t + t₀): Advance by t₀ (left shift)
Time Scaling: x(at): Compress by factor a if a > 1, expand if 0 < a < 1 x(-t): Time reversal
Amplitude Scaling: Ax(t): Multiply amplitude by A
Linearity: T[a₁x₁(t) + a₂x₂(t)] = a₁T[x₁(t)] + a₂T[x₂(t)]
Time Invariance: If y(t) = T[x(t)], then y(t-t₀) = T[x(t-t₀)]
Causality: Output at t depends only on present and past inputs h(t) = 0 for t < 0
Stability (BIBO): Every bounded input produces bounded output ∫|h(t)|dt < ∞ (for continuous time)
Memory: System with memory: Output depends on past/future inputs Memoryless: Output depends only on present input
Invertibility: Distinct inputs produce distinct outputs
Continuous Time: y(t) = x(t) * h(t) = ∫x(τ)h(t-τ)dτ
Properties:
- Commutative: x * h = h * x
- Associative: (x * h₁) * h₂ = x * (h₁ * h₂)
- Distributive: x * (h₁ + h₂) = x * h₁ + x * h₂
Discrete Time: y[n] = x[n] * h[n] = Σx[k]h[n-k]
Definition: X(s) = ∫x(t)e^(-st)dt (from 0⁻ to ∞)
Region of Convergence (ROC): Values of s for which transform exists
Common Transforms:
| x(t) | X(s) | ROC |
|---|---|---|
| δ(t) | 1 | All s |
| u(t) | 1/s | Re(s) > 0 |
| t | 1/s² | Re(s) > 0 |
| tⁿ | n!/sⁿ⁺¹ | Re(s) > 0 |
| e^(-at) | 1/(s+a) | Re(s) > -a |
| te^(-at) | 1/(s+a)² | Re(s) > -a |
| sin(ωt) | ω/(s²+ω²) | Re(s) > 0 |
| cos(ωt) | s/(s²+ω²) | Re(s) > 0 |
| e^(-at)sin(ωt) | ω/((s+a)²+ω²) | Re(s) > -a |
| e^(-at)cos(ωt) | (s+a)/((s+a)²+ω²) | Re(s) > -a |
Properties:
Linearity: a₁x₁(t) + a₂x₂(t) ↔ a₁X₁(s) + a₂X₂(s)
Time Shifting: x(t-t₀)u(t-t₀) ↔ e^(-st₀)X(s)
Frequency Shifting: e^(s₀t)x(t) ↔ X(s-s₀)
Time Scaling: x(at) ↔ (1/|a|)X(s/a)
Time Differentiation: dx/dt ↔ sX(s) - x(0⁻)
Time Integration: ∫x(τ)dτ ↔ X(s)/s
Initial Value Theorem: x(0⁺) = lim(s→∞) sX(s)
Final Value Theorem: lim(t→∞) x(t) = lim(s→0) sX(s) (Valid only if poles in LHP or single pole at origin)
Convolution: x(t)*h(t) ↔ X(s)H(s)
Multiplication: x(t)h(t) ↔ (1/2πj)X(s)*H(s)
For Periodic Signal x(t) with period T:
Exponential Form: x(t) = Σ C_n e^(jnω₀t)
Where:
- ω₀ = 2π/T (fundamental frequency)
- C_n = (1/T)∫x(t)e^(-jnω₀t)dt over one period
Trigonometric Form: x(t) = a₀ + Σ[a_n cos(nω₀t) + b_n sin(nω₀t)]
Where:
- a₀ = (1/T)∫x(t)dt (DC component)
- a_n = (2/T)∫x(t)cos(nω₀t)dt
- b_n = (2/T)∫x(t)sin(nω₀t)dt
Relations:
C₀ = a₀
C_n = (a_n - jb_n)/2
C_(-n) = (a_n + jb_n)/2
Symmetry Properties:
- Even symmetry [x(t) = x(-t)]: b_n = 0
- Odd symmetry [x(t) = -x(-t)]: a₀ = a_n = 0
- Half-wave symmetry [x(t) = -x(t±T/2)]: Only odd harmonics
Definition: X(ω) = ∫x(t)e^(-jωt)dt
Inverse: x(t) = (1/2π)∫X(ω)e^(jωt)dω
Common Transforms:
| x(t) | X(ω) |
|---|---|
| δ(t) | 1 |
| 1 | 2πδ(ω) |
| u(t) | πδ(ω) + 1/jω |
| e^(jω₀t) | 2πδ(ω-ω₀) |
| cos(ω₀t) | π[δ(ω-ω₀) + δ(ω+ω₀)] |
| sin(ω₀t) | (π/j)[δ(ω-ω₀) - δ(ω+ω₀)] |
| e^(-at)u(t) | 1/(a+jω) |
| te^(-at)u(t) | 1/(a+jω)² |
| rect(t/τ) | τsinc(ωτ/2) |
| sinc(t) | rect(ω/2π) |
Properties:
Linearity, Time Shifting, Frequency Shifting, Time Scaling, Duality, Time Differentiation, Time Integration, Convolution, Parseval's Theorem
Thevenin's Theorem: Any linear network can be replaced by:
- V_th = Open circuit voltage
- R_th = Equivalent resistance with all sources deactivated
Norton's Theorem: Any linear network can be replaced by:
- I_N = Short circuit current
- R_N = Equivalent resistance (same as R_th)
Relation: V_th = I_N × R_th
Superposition Theorem: Response in any element = Sum of responses due to each source acting alone
Maximum Power Transfer: Maximum power transferred when: R_L = R_th P_max = V_th²/(4R_th)
Millman's Theorem: For parallel voltage sources: V = (ΣV_k/R_k)/(Σ1/R_k)
Tellegen's Theorem: For any network: Σv_k × i_k = 0 Conservation of power
Reciprocity Theorem: In a linear bilateral network, ratio of response to excitation is same if positions are interchanged
Impedance (Z) Parameters:
V₁ = Z₁₁I₁ + Z₁₂I₂
V₂ = Z₂₁I₁ + Z₂₂I₂
For reciprocal network: Z₁₂ = Z₂₁
Admittance (Y) Parameters:
I₁ = Y₁₁V₁ + Y₁₂V₂
I₂ = Y₂₁V₁ + Y₂₂V₂
For reciprocal network: Y₁₂ = Y₂₁
Hybrid (h) Parameters:
V₁ = h₁₁I₁ + h₁₂V₂
I₂ = h₂₁I₁ + h₂₂V₂
Inverse Hybrid (g) Parameters:
I₁ = g₁₁V₁ + g₁₂I₂
V₂ = g₂₁V₁ + g₂₂I₂
Transmission (ABCD) Parameters:
V₁ = AV₂ - BI₂
I₁ = CV₂ - DI₂
Relations between Parameters: Z = Y⁻¹, h and g are inverses, etc.
Image Impedance:
Z_i1 = √(Z_oc1 × Z_sc1)
Z_i2 = √(Z_oc2 × Z_sc2)
Characteristic Impedance: For symmetrical network: Z₀ = √(Z_oc × Z_sc)
Series RLC: Resonant frequency: f₀ = 1/(2π√(LC)) Quality factor: Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C) Bandwidth: BW = f₀/Q = R/(2πL)
At resonance:
- Z = R (minimum)
- I = V/R (maximum)
- Power factor = 1
- V_L = V_C = QV
Parallel RLC: Resonant frequency: f₀ = 1/(2π√(LC)) Quality factor: Q = ω₀CR = R/(ω₀L) = R√(C/L) Bandwidth: BW = f₀/Q = 1/(2πCR)
At resonance:
- Z = R (maximum for ideal)
- I = V/R (minimum)
- Power factor = 1
- I_L = I_C = QI
Types:
- Low Pass (LPF): Passes low frequencies
- High Pass (HPF): Passes high frequencies
- Band Pass (BPF): Passes band of frequencies
- Band Stop (BSF): Rejects band of frequencies
Cutoff Frequency: ω_c = 1/(RC) for simple RC filters
First Order LPF:
Transfer function: 1/(1 + sRC)
Cutoff: f_c = 1/(2πRC)
Roll-off: -20 dB/decade
First Order HPF:
Transfer function: sRC/(1 + sRC)
Cutoff: f_c = 1/(2πRC)
Roll-off: +20 dB/decade
Second Order Filters: Roll-off: ±40 dB/decade
Butterworth Filter: Maximally flat response in passband
Chebyshev Filter: Steeper roll-off but ripple in passband
Bessel Filter: Linear phase response
Tree: Connected subgraph containing all nodes but no loops
Twigs: Branches in tree = n - 1 (where n = number of nodes)
Links: Branches not in tree = b - (n - 1) (where b = total branches)
Fundamental Cutset: One twig + minimum links to form cutset
Fundamental Loop: One link + unique path in tree
Incidence Matrix (A):
Dimensions: (n-1) × b
Elements: +1, -1, 0
Fundamental Cutset Matrix (Q): Dimensions: (n-1) × b
Fundamental Loop Matrix (B): Dimensions: (b-n+1) × b
Relation: BQ^T = 0 or QB^T = 0
Wheatstone Bridge: Measures: Medium resistances (1Ω to 100kΩ) Balance condition: R₁/R₂ = R₃/R₄ or R₁R₄ = R₂R₃
Kelvin Double Bridge: Measures: Low resistances (< 1Ω) Eliminates: Lead and contact resistances Balance: R₁/R₂ = R₃/R₄ + R_g (guard wire)
Maxwell Bridge:
Measures: Inductance
Balance: L = R₁R₂C₃
Q-factor: Q = ωL/R
Hay Bridge: Measures: High Q inductors Balance: Different from Maxwell for high Q
Anderson Bridge: Measures: Inductance using standard capacitor and resistors
Schering Bridge:
Measures: Capacitance and dissipation factor
Balance: C₁/C₂ = R₃/R₄
Dissipation factor: D = tanδ = ωC₄R₄
De Sauty Bridge: Measures: Capacitance Balance: C₁/C₂ = R₄/R₃
Wien Bridge: Measures: Frequency Balance: f = 1/(2π√(R₁R₂C₁C₂))
PMMC (Permanent Magnet Moving Coil):
Measures: DC only
Principle: Force on current carrying coil in magnetic field
Deflecting torque: T_d = NBAI
Controlling torque: T_c = Kθ
At equilibrium: T_d = T_c
θ ∝ I
Advantages: High sensitivity, uniform scale, no hysteresis Disadvantages: Only DC, delicate, costly
Moving Iron: Measures: AC and DC Principle: Force on iron piece due to magnetic field Attraction type or repulsion type Scale: Non-uniform (cramped at beginning)
Electrodynamometer: Measures: AC and DC (power, voltage, current) Principle: Force between fixed and moving coils Air cored coils (no hysteresis) Can be used as wattmeter, voltmeter, ammeter
Induction Type: Measures: AC energy (energy meters) Principle: Rotating magnetic field induces eddy currents Aluminum disc rotates proportional to power
Hot Wire: Measures: RMS value (AC and DC) Principle: Heating effect (I²R)
Thermocouple: Measures: High frequency AC, RF Principle: Seebeck effect
Electrostatic: Measures: High voltages Principle: Force between charged plates
Wattmeter (Electrodynamometer): Current coil: Series with load (carries load current) Pressure coil: Parallel with load (measures voltage) Reading: P = VI cosφ
Three-Phase Power:
Two-Wattmeter Method:
W₁ = V_L I_L cos(30° - φ)
W₂ = V_L I_L cos(30° + φ)
Total power: W₁ + W₂ = √3 V_L I_L cosφ
If φ < 60°: Both readings positive If φ = 60°: One reading zero If φ > 60°: One reading negative
Reactive power: √3(W₁ - W₂) = √3 V_L I_L sinφ Power factor: tanφ = √3(W₁ - W₂)/(W₁ + W₂)
Types of Errors:
Gross Errors: Human mistakes (reading, recording, calculation)
Systematic Errors:
- Instrumental: Due to instrument defects
- Environmental: Temperature, humidity, etc.
- Observational: Parallax, etc.
Random Errors: Unpredictable variations
Error Terms:
Absolute Error: ε = X_m - X_t
(Where X_m = measured value, X_t = true value)
Relative Error: ε_r = ε/X_t
Percentage Error: %Error = (ε/X_t) × 100
Accuracy: Closeness to true value Precision: Repeatability of measurements
Resolution: Smallest change detectable Sensitivity: Ratio of output change to input change
Loading Effect: Instrument draws power from circuit, affecting measurement Minimized by: High input impedance for voltmeters, low impedance for ammeters
Basic Laws:
Commutative: A+B = B+A, AB = BA Associative: A+(B+C) = (A+B)+C, A(BC) = (AB)C Distributive: A(B+C) = AB+AC, A+BC = (A+B)(A+C)
Identity: A+0 = A, A·1 = A Null: A+1 = 1, A·0 = 0 Idempotent: A+A = A, A·A = A Complement: A+A' = 1, A·A' = 0 Involution: (A')' = A Absorption: A+AB = A, A(A+B) = A De Morgan's: (A+B)' = A'B', (AB)' = A'+B'
AND: Y = AB (Output high only if all inputs high) OR: Y = A+B (Output high if any input high) NOT: Y = A' (Inverter) NAND: Y = (AB)' (Universal gate) NOR: Y = (A+B)' (Universal gate) XOR: Y = A⊕B = A'B + AB' (Output high if inputs differ) XNOR: Y = (A⊕B)' = AB + A'B' (Output high if inputs same)
Universal Gates: NAND and NOR can implement any logic function
SR Flip-Flop: S R | Q 0 0 | No change 0 1 | 0 (Reset) 1 0 | 1 (Set) 1 1 | Invalid/Forbidden
JK Flip-Flop: J K | Q 0 0 | No change 0 1 | 0 (Reset) 1 0 | 1 (Set) 1 1 | Toggle
D Flip-Flop: Q = D (Delay)
T Flip-Flop: T = 0: No change T = 1: Toggle
Characteristic Equations: SR: Q⁺ = S + R'Q (with SR = 0) JK: Q⁺ = JQ' + K'Q D: Q⁺ = D T: Q⁺ = T⊕Q
Excitation Tables: Table showing required inputs for desired state transition
Asynchronous (Ripple) Counter: Flip-flops triggered one after another Propagation delay accumulates Simple, slow
Synchronous Counter: All flip-flops triggered simultaneously Faster, more complex
Modulus: Number of distinct states Mod-n counter counts 0 to n-1
Maximum Modulus: For n flip-flops: 2ⁿ states
Decade Counter: Mod-10 counter (counts 0 to 9)
Ring Counter: One bit circulates through flip-flops Mod-n requires n flip-flops
Johnson Counter (Twisted Ring): Output of last inverted and fed to first Mod-2n requires n flip-flops
Binary to Decimal: Σb_i × 2^i
Decimal to Binary: Repeated division by 2
Octal to Binary: Each octal digit = 3 binary bits
Hexadecimal to Binary: Each hex digit = 4 binary bits
Complements:
1's Complement: Flip all bits 2's Complement: 1's complement + 1 (used for subtraction)
Binary Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 with carry 1)
Binary Subtraction (using 2's complement): A - B = A + (2's complement of B)
Percentages: % change = [(New - Old)/Old] × 100 Successive changes: a% followed by b% = (a + b + ab/100)%
Profit & Loss: Profit% = (Profit/CP) × 100 = [(SP-CP)/CP] × 100 Loss% = (Loss/CP) × 100 = [(CP-SP)/CP] × 100
Simple Interest: SI = PRT/100 Amount = P + SI = P(1 + RT/100)
Compound Interest: Amount = P(1 + R/100)^T For half-yearly: R' = R/2, T' = 2T For quarterly: R' = R/4, T' = 4T
Time & Work: If A can do work in n days, 1 day's work = 1/n If A and B together in x days, 1/x = 1/a + 1/b
Pipes & Cisterns: Inlet pipe: +ve work Outlet pipe: -ve work
Speed, Distance, Time: Speed = Distance/Time Relative speed: Same direction: S₁ - S₂, Opposite: S₁ + S₂
Trains: Time = (Length of train + Length of platform)/Speed
Boats & Streams: Downstream: Speed = B + S Upstream: Speed = B - S Where B = boat speed, S = stream speed
Probability: P(E) = n(E)/n(S) P(E') = 1 - P(E) P(A∪B) = P(A) + P(B) - P(A∩B) For independent: P(A∩B) = P(A) × P(B)
Permutations: ⁿPᵣ = n!/(n-r)!
Combinations: ⁿCᵣ = n!/[(n-r)!r!]
Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1) ∫1/x dx = ln|x| + C ∫e^x dx = e^x + C ∫a^x dx = a^x/ln(a) + C ∫sin(x) dx = -cos(x) + C ∫cos(x) dx = sin(x) + C
Differentiation: d/dx(xⁿ) = nxⁿ⁻¹ d/dx(e^x) = e^x d/dx(ln x) = 1/x d/dx(sin x) = cos x d/dx(cos x) = -sin x
Matrices: det(AB) = det(A)det(B) det(A⁻¹) = 1/det(A) det(A^T) = det(A) Eigenvalues: |A - λI| = 0 Trace = Sum of eigenvalues det = Product of eigenvalues
✓ Read question carefully ✓ Identify keywords for formula selection ✓ Write formula first, then substitute values ✓ Check units in final answer ✓ Manage time: 1.8 min per question (100 questions in 3 hours) ✓ Attempt aptitude first (easy marks) ✓ Darken circles properly ✓ Use rough sheet for calculations
✗ Don't guess randomly (negative marking: -1/3) ✗ Don't spend >3 min on any question ✗ Don't panic if you don't know answer ✗ Don't leave OMR sheet incomplete ✗ Don't forget to attempt all aptitude questions
- Start with General Aptitude (15 questions, ~25 min)
- Engineering Mathematics (10-12 questions, ~20 min)
- Your strong subjects first
- Skip uncertain questions, mark for review
- Come back to marked questions if time permits
Print/Write these 20 most important:
- Transformer: E = 4.44fNΦ
- DC Motor: E_b = V - I_aR_a
- Induction Motor: s = (N_s-N_r)/N_s
- Synchronous: P = (EV/X_s)sinδ
- Transmission: SIL = V²/Z_c
- Per Unit: Z_pu = Z_actual/Z_base
- Rectifier: V_avg = (2V_m/π)cosα
- Buck: V_o = DV_s
- Boost: V_o = V_s/(1-D)
- Inverting Op-Amp: V_o = -(R_f/R₁)V_i
- Non-Inverting: V_o = (1+R_f/R₁)V_i
- Closed Loop: T(s) = G/(1+GH)
- 2nd Order: T_p = π/(ω_n√(1-ζ²))
- Steady State: e_ss = lim(s→0) sE(s)
- Laplace: L{e^(-at)} = 1/(s+a)
- Thevenin: V_th = V_oc, R_th = V_oc/I_sc
- Resonance: f₀ = 1/(2π√(LC))
- Wheatstone: R₁/R₂ = R₃/R₄
- Slip: s = (N_s-N_r)/N_s
- Power: S = √(P²+Q²) = VI
ALL THE BEST FOR GATE EE! 🎯
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Total Formulas: 100+