boss_notes/99_Master_Formula_Sheet.md

6.8 KB · commit d2f3b6e

📐 Master Formula Sheet — ECE 2207

Quick reference for all formulas across both sections. Each entry links back to the boss note where it is derived and explained. Print this. Take it into revision. Know every formula cold.

Section A — Transformer

EMF Equation

$E = 4.44 f N Φ_{m} (volts)$

T-01: Transformer Fundamentals

Turns Ratio

$a = \frac{N _{1}}{N _{2}} = \frac{E _{1}}{E _{2}} = \frac{V _{1}}{V _{2}} = \frac{I _{2}}{I _{1}}$

Equivalent Circuit (Referred to Primary)

$R_{01} = R_{1} + a^{2} R_{2}, X_{01} = X_{1} + a^{2} X_{2}, Z_{01} = R_{01}^{2} + X_{01}^{2}$

T-04: Equivalent Circuit

Voltage Regulation

$% VR = \frac{E _{2} - V _{2}}{V _{2}} \times 100 \approx \frac{I _{2} ( R _{02} cos ϕ \pm X _{02} sin ϕ )}{V _{2}} \times 100$

(+ for lagging, - for leading)

T-05: Voltage Regulation

Condition for Zero VR

$tan ϕ = \frac{R _{02}}{X _{02}} (leading pf)$

Losses and Efficiency

$η = \frac{x S cos ϕ}{x S cos ϕ + P _{i} + x ^{2} P _{C u}} \times 100%$

where $x$ = fraction of full load, $S$ = VA rating, $P_{i}$ = iron loss, $P_{C u}$ = full-load Cu loss.

Condition for Maximum Efficiency

$P_{i} = x^{2} P_{C u} ⟹ x = \frac{P _{i}}{P _{C u}}$

T-06c: Efficiency

OC Test (Gives $R_{c}$ , $X_{m}$ , $P_{i}$ )

$cos ϕ_{0} = \frac{P _{0}}{V _{1} I _{0}}, I_{c} = I_{0} cos ϕ_{0}, I_{m} = I_{0} sin ϕ_{0}$

$R_{c} = V_{1} / I_{c}, X_{m} = V_{1} / I_{m}, P_{i} = P_{0}$

SC Test (Gives $R_{01}$ , $X_{01}$ , $P_{C u}$ )

$Z_{01} = V_{sc} / I_{sc}, R_{01} = P_{sc} / I_{sc}^{2}, X_{01} = Z_{01}^{2} - R_{01}^{2}$

T-06a: OC Test | T-06b: SC Test

$S_{A} = S \cdot \frac{Z _{B}}{Z _{A} + Z _{B}}, S_{B} = S \cdot \frac{Z _{A}}{Z _{A} + Z _{B}}$

T-09: Vector Groups & Parallel Operation

Auto-Transformer Savings

$Saving = (1 - \frac{1}{a}) \times 100%$

$VA_{induction} = (1 - \frac{1}{a}) \times VA_{conduction}$

T-10: Auto-Transformer

3-Phase Transformer Connections

Connection	$V_{2}$ (ratio)	Phase Shift
Y-Y	$V_{L} / a$	0°
$Δ$ - $Δ$	$V_{L} / a$	0°
Y- $Δ$	$V_{L} / (a 3)$	-30°
$Δ$ -Y	$V_{L} 3 / a$	+30°

T-07a: Three-Phase Connections

Scott Connection

$V_{teaser} = V \times (3 /2) \times (N_{2} / N_{1})$

$V_{main} = V \times (N_{2} / N_{1})$

Open-Delta Power Rating

$S_{open- Δ} = \frac{1}{3} \times S_{closed- Δ} = 57.7% of V-V$

T-07b: Open-Delta | T-08: Scott Connection

Section B — Induction Motor

Synchronous Speed

$N_{s} = \frac{120 f}{P} rpm$

T-12: Rotating Magnetic Field

3-Phase RMF Magnitude

$Φ_{r} = 1.5 Φ_{m} = \frac{3}{2} Φ_{m} (constant, rotating at N_{s})$

2-Phase RMF Magnitude

$Φ_{r} = Φ_{m} (constant, rotating at N_{s})$

Slip

$s = \frac{N _{s} - N}{N _{s}}, N = N_{s} (1 - s)$

T-13: Slip & Basics

Rotor Frequency

$f_{r} = s f$

Rotor Quantities at Slip $s$

$E_{2 s} = s E_{2}, X_{2 s} = s X_{2}, R_{2} = unchanged$

$I_{2} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

Equivalent Circuit: Load Resistance

$\frac{R _{2}}{s} = R_{2} + R_{L} (mech. load) R_{2} \frac{( 1 - s )}{s}$

T-14: IM Equivalent Circuit

Running Torque

$T = \frac{k s E _{2}^{2} R _{2}}{R _{2}^{2} + s ^{2} X _{2}^{2}}, k = \frac{3}{2 π n _{s}}$

T-15b: Running & Max Torque

Starting Torque ( $s = 1$ )

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

Maximum Starting Torque: $R_{2} = X_{2}$

T-15a: Starting Torque

Slip at Maximum Torque

$s_{m T} = \frac{R _{2}}{X _{2}}$

Maximum Torque (Independent of $R_{2}$ !)

$T_{m a x} = \frac{k E _{2}^{2}}{2 X _{2}}$

$T_{f} / T_{m a x}$ Ratio

$\frac{T _{f}}{T _{m a x}} = \frac{2 a s _{f}}{a ^{2} + s _{f}^{2}}, a = s_{m T} = \frac{R _{2}}{X _{2}}$

Torque-Speed Approximations

Region	Condition	Approximation
Low slip	$s X_{2} ≪ R_{2}$	$T \propto s$ (linear)
High slip	$s X_{2} ≫ R_{2}$	$T \propto 1/ s$ (inversely)

T-15c: Torque-Speed Curves

Power Flow (The Golden Ratio)

$P_{g} : P_{C u, r} : P_{m} = 1 : s : (1 - s)$

$P_{C u, r} = s P_{g}, P_{m} = (1 - s) P_{g}$

$η_{rotor} = (1 - s)$

T-16: Power Flow

No-Load Test (Gives $R_{c}$ , $X_{m}$ )

$cos ϕ_{0} = \frac{P _{0}}{3 V _{0} I _{0}}$

$R_{c} = V_{ϕ} / I_{c}, X_{m} = V_{ϕ} / I_{m}$

T-17a: No-Load Test

Blocked Rotor Test (Gives $R_{01}$ , $X_{01}$ )

$Z_{01} = \frac{V _{sc} / 3}{I _{sc}}, R_{01} = \frac{P _{sc}}{3 I _{sc}^{2}}, X_{01} = Z_{01}^{2} - R_{01}^{2}$

$R_{2}^{'} = R_{01} - R_{1}, X_{1} = X_{2}^{'} = X_{01} /2$

T-17b: Blocked Rotor Test

Star-Delta Starter

$I_{L, Y} / I_{L, Δ} = 1/3, T_{s t, Y} / T_{s t, Δ} = 1/3$

Equivalent auto-transformer ratio: $x = 1/ 3 = 57.7%$

T-19: Starting Methods

Rotor Resistance Speed Control (Constant Torque)

$\frac{s _{1}}{R _{2}} = \frac{s _{2}}{R _{2} + R _{ext}}$

T-20: Speed Control & Braking

Induction Generator: Engine Speed

$N_{engine} = N_{s} (1 + ∣ s ∣)$

SEIG Capacitance (Delta)

$Q = 3 V_{L} I_{L} sin ϕ, X_{C} = V_{L}^{2} / (Q /3), C = 1/ (2 π f X_{C})$

T-21: Induction Generator

DFRT: Pulsating Field Decomposition

$Φ = \frac{Φ _{m}}{2} sin (ω t - θ) + \frac{Φ _{m}}{2} sin (ω t + θ)$

Slip for forward field: $s_{f} = s$ . Slip for backward field: $s_{b} = 2 - s$ .

T-22: DFRT & 1-Phase IM

Capacitor-Start: Max Starting Torque

$X_{C} = X_{a} + R_{a} tan (90° - ϕ_{m}), C = \frac{1}{2 π f X _{C}}$

T-23: 1-Phase IM Starting Methods

Quick Number Reference

Quantity	Typical Range
Full-load slip	2-5%
No-load current	25-40% of rated
Starting current (DOL)	5-8x rated
Power factor (full load)	0.8-0.88 lagging
Power factor (no load)	0.08-0.2 lagging
Efficiency (large motor)	85-95%
Starting torque (squirrel-cage)	1.5-2x FL
Starting torque (cap-start)	2-4x FL
Breakdown torque	2-3x FL

🏠 Index

boss_notes/99_Master_Formula_Sheet.md

6.8 KB · commit d2f3b6e

📐 Master Formula Sheet — ECE 2207

Quick reference for all formulas across both sections. Each entry links back to the boss note where it is derived and explained. Print this. Take it into revision. Know every formula cold.

Section A — Transformer

EMF Equation

$E = 4.44 f N Φ_{m} (volts)$

T-01: Transformer Fundamentals

Turns Ratio

$a = \frac{N _{1}}{N _{2}} = \frac{E _{1}}{E _{2}} = \frac{V _{1}}{V _{2}} = \frac{I _{2}}{I _{1}}$

Equivalent Circuit (Referred to Primary)

$R_{01} = R_{1} + a^{2} R_{2}, X_{01} = X_{1} + a^{2} X_{2}, Z_{01} = R_{01}^{2} + X_{01}^{2}$

T-04: Equivalent Circuit

Voltage Regulation

$% VR = \frac{E _{2} - V _{2}}{V _{2}} \times 100 \approx \frac{I _{2} ( R _{02} cos ϕ \pm X _{02} sin ϕ )}{V _{2}} \times 100$

(+ for lagging, - for leading)

T-05: Voltage Regulation

Condition for Zero VR

$tan ϕ = \frac{R _{02}}{X _{02}} (leading pf)$

Losses and Efficiency

$η = \frac{x S cos ϕ}{x S cos ϕ + P _{i} + x ^{2} P _{C u}} \times 100%$

where $x$ = fraction of full load, $S$ = VA rating, $P_{i}$ = iron loss, $P_{C u}$ = full-load Cu loss.

Condition for Maximum Efficiency

$P_{i} = x^{2} P_{C u} ⟹ x = \frac{P _{i}}{P _{C u}}$

T-06c: Efficiency

OC Test (Gives $R_{c}$ , $X_{m}$ , $P_{i}$ )

$cos ϕ_{0} = \frac{P _{0}}{V _{1} I _{0}}, I_{c} = I_{0} cos ϕ_{0}, I_{m} = I_{0} sin ϕ_{0}$

$R_{c} = V_{1} / I_{c}, X_{m} = V_{1} / I_{m}, P_{i} = P_{0}$

SC Test (Gives $R_{01}$ , $X_{01}$ , $P_{C u}$ )

$Z_{01} = V_{sc} / I_{sc}, R_{01} = P_{sc} / I_{sc}^{2}, X_{01} = Z_{01}^{2} - R_{01}^{2}$

T-06a: OC Test | T-06b: SC Test

$S_{A} = S \cdot \frac{Z _{B}}{Z _{A} + Z _{B}}, S_{B} = S \cdot \frac{Z _{A}}{Z _{A} + Z _{B}}$

T-09: Vector Groups & Parallel Operation

Auto-Transformer Savings

$Saving = (1 - \frac{1}{a}) \times 100%$

$VA_{induction} = (1 - \frac{1}{a}) \times VA_{conduction}$

T-10: Auto-Transformer

3-Phase Transformer Connections

Connection	$V_{2}$ (ratio)	Phase Shift
Y-Y	$V_{L} / a$	0°
$Δ$ - $Δ$	$V_{L} / a$	0°
Y- $Δ$	$V_{L} / (a 3)$	-30°
$Δ$ -Y	$V_{L} 3 / a$	+30°

T-07a: Three-Phase Connections

Scott Connection

$V_{teaser} = V \times (3 /2) \times (N_{2} / N_{1})$

$V_{main} = V \times (N_{2} / N_{1})$

Open-Delta Power Rating

$S_{open- Δ} = \frac{1}{3} \times S_{closed- Δ} = 57.7% of V-V$

T-07b: Open-Delta | T-08: Scott Connection

Section B — Induction Motor

Synchronous Speed

$N_{s} = \frac{120 f}{P} rpm$

T-12: Rotating Magnetic Field

3-Phase RMF Magnitude

$Φ_{r} = 1.5 Φ_{m} = \frac{3}{2} Φ_{m} (constant, rotating at N_{s})$

2-Phase RMF Magnitude

$Φ_{r} = Φ_{m} (constant, rotating at N_{s})$

Slip

$s = \frac{N _{s} - N}{N _{s}}, N = N_{s} (1 - s)$

T-13: Slip & Basics

Rotor Frequency

$f_{r} = s f$

Rotor Quantities at Slip $s$

$E_{2 s} = s E_{2}, X_{2 s} = s X_{2}, R_{2} = unchanged$

$I_{2} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

Equivalent Circuit: Load Resistance

$\frac{R _{2}}{s} = R_{2} + R_{L} (mech. load) R_{2} \frac{( 1 - s )}{s}$

T-14: IM Equivalent Circuit

Running Torque

$T = \frac{k s E _{2}^{2} R _{2}}{R _{2}^{2} + s ^{2} X _{2}^{2}}, k = \frac{3}{2 π n _{s}}$

T-15b: Running & Max Torque

Starting Torque ( $s = 1$ )

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

Maximum Starting Torque: $R_{2} = X_{2}$

T-15a: Starting Torque

Slip at Maximum Torque

$s_{m T} = \frac{R _{2}}{X _{2}}$

Maximum Torque (Independent of $R_{2}$ !)

$T_{m a x} = \frac{k E _{2}^{2}}{2 X _{2}}$

$T_{f} / T_{m a x}$ Ratio

$\frac{T _{f}}{T _{m a x}} = \frac{2 a s _{f}}{a ^{2} + s _{f}^{2}}, a = s_{m T} = \frac{R _{2}}{X _{2}}$

Torque-Speed Approximations

Region	Condition	Approximation
Low slip	$s X_{2} ≪ R_{2}$	$T \propto s$ (linear)
High slip	$s X_{2} ≫ R_{2}$	$T \propto 1/ s$ (inversely)

T-15c: Torque-Speed Curves

Power Flow (The Golden Ratio)

$P_{g} : P_{C u, r} : P_{m} = 1 : s : (1 - s)$

$P_{C u, r} = s P_{g}, P_{m} = (1 - s) P_{g}$

$η_{rotor} = (1 - s)$

T-16: Power Flow

No-Load Test (Gives $R_{c}$ , $X_{m}$ )

$cos ϕ_{0} = \frac{P _{0}}{3 V _{0} I _{0}}$

$R_{c} = V_{ϕ} / I_{c}, X_{m} = V_{ϕ} / I_{m}$

T-17a: No-Load Test

Blocked Rotor Test (Gives $R_{01}$ , $X_{01}$ )

$Z_{01} = \frac{V _{sc} / 3}{I _{sc}}, R_{01} = \frac{P _{sc}}{3 I _{sc}^{2}}, X_{01} = Z_{01}^{2} - R_{01}^{2}$

$R_{2}^{'} = R_{01} - R_{1}, X_{1} = X_{2}^{'} = X_{01} /2$

T-17b: Blocked Rotor Test

Star-Delta Starter

$I_{L, Y} / I_{L, Δ} = 1/3, T_{s t, Y} / T_{s t, Δ} = 1/3$

Equivalent auto-transformer ratio: $x = 1/ 3 = 57.7%$

T-19: Starting Methods

Rotor Resistance Speed Control (Constant Torque)

$\frac{s _{1}}{R _{2}} = \frac{s _{2}}{R _{2} + R _{ext}}$

T-20: Speed Control & Braking

Induction Generator: Engine Speed

$N_{engine} = N_{s} (1 + ∣ s ∣)$

SEIG Capacitance (Delta)

$Q = 3 V_{L} I_{L} sin ϕ, X_{C} = V_{L}^{2} / (Q /3), C = 1/ (2 π f X_{C})$

T-21: Induction Generator

DFRT: Pulsating Field Decomposition

$Φ = \frac{Φ _{m}}{2} sin (ω t - θ) + \frac{Φ _{m}}{2} sin (ω t + θ)$

Slip for forward field: $s_{f} = s$ . Slip for backward field: $s_{b} = 2 - s$ .

T-22: DFRT & 1-Phase IM

Capacitor-Start: Max Starting Torque

$X_{C} = X_{a} + R_{a} tan (90° - ϕ_{m}), C = \frac{1}{2 π f X _{C}}$

T-23: 1-Phase IM Starting Methods

Quick Number Reference

Quantity	Typical Range
Full-load slip	2-5%
No-load current	25-40% of rated
Starting current (DOL)	5-8x rated
Power factor (full load)	0.8-0.88 lagging
Power factor (no load)	0.08-0.2 lagging
Efficiency (large motor)	85-95%
Starting torque (squirrel-cage)	1.5-2x FL
Starting torque (cap-start)	2-4x FL
Breakdown torque	2-3x FL

🏠 Index

📐 Master Formula Sheet — ECE 2207

Section A — Transformer

EMF Equation

Turns Ratio

Equivalent Circuit (Referred to Primary)

Voltage Regulation

Condition for Zero VR

Losses and Efficiency

Condition for Maximum Efficiency

OC Test (Gives Rc​, Xm​, Pi​)

SC Test (Gives R01​, X01​, PCu​)

Parallel Operation (Load Sharing)

Auto-Transformer Savings

3-Phase Transformer Connections

Scott Connection

Open-Delta Power Rating

Section B — Induction Motor

Synchronous Speed

3-Phase RMF Magnitude

2-Phase RMF Magnitude

Slip

Rotor Frequency

Rotor Quantities at Slip s

Equivalent Circuit: Load Resistance

Running Torque

Starting Torque (s=1)

Maximum Starting Torque: R2​=X2​

Slip at Maximum Torque

Maximum Torque (Independent of R2​!)

Tf​/Tmax​ Ratio

Torque-Speed Approximations

Power Flow (The Golden Ratio)

No-Load Test (Gives Rc​, Xm​)

Blocked Rotor Test (Gives R01​, X01​)

Star-Delta Starter

Rotor Resistance Speed Control (Constant Torque)

Induction Generator: Engine Speed

SEIG Capacitance (Delta)

DFRT: Pulsating Field Decomposition

Capacitor-Start: Max Starting Torque

Quick Number Reference

📐 Master Formula Sheet — ECE 2207

Section A — Transformer

EMF Equation

Turns Ratio

Equivalent Circuit (Referred to Primary)

Voltage Regulation

Condition for Zero VR

Losses and Efficiency

Condition for Maximum Efficiency

OC Test (Gives Rc​, Xm​, Pi​)

SC Test (Gives R01​, X01​, PCu​)

Parallel Operation (Load Sharing)

Auto-Transformer Savings

3-Phase Transformer Connections

Scott Connection

Open-Delta Power Rating

Section B — Induction Motor

Synchronous Speed

3-Phase RMF Magnitude

2-Phase RMF Magnitude

Slip

Rotor Frequency

Rotor Quantities at Slip s

Equivalent Circuit: Load Resistance

Running Torque

Starting Torque (s=1)

Maximum Starting Torque: R2​=X2​

Slip at Maximum Torque

Maximum Torque (Independent of R2​!)

Tf​/Tmax​ Ratio

Torque-Speed Approximations

Power Flow (The Golden Ratio)

No-Load Test (Gives Rc​, Xm​)

Blocked Rotor Test (Gives R01​, X01​)

Star-Delta Starter

Rotor Resistance Speed Control (Constant Torque)

Induction Generator: Engine Speed

SEIG Capacitance (Delta)

DFRT: Pulsating Field Decomposition

OC Test (Gives $R_{c}$ , $X_{m}$ , $P_{i}$ )

SC Test (Gives $R_{01}$ , $X_{01}$ , $P_{C u}$ )

Rotor Quantities at Slip $s$

Starting Torque ( $s = 1$ )

Maximum Starting Torque: $R_{2} = X_{2}$

Maximum Torque (Independent of $R_{2}$ !)

$T_{f} / T_{m a x}$ Ratio

No-Load Test (Gives $R_{c}$ , $X_{m}$ )

Blocked Rotor Test (Gives $R_{01}$ , $X_{01}$ )

OC Test (Gives $R_{c}$ , $X_{m}$ , $P_{i}$ )

SC Test (Gives $R_{01}$ , $X_{01}$ , $P_{C u}$ )

Rotor Quantities at Slip $s$

Starting Torque ( $s = 1$ )

Maximum Starting Torque: $R_{2} = X_{2}$

Maximum Torque (Independent of $R_{2}$ !)

$T_{f} / T_{m a x}$ Ratio

No-Load Test (Gives $R_{c}$ , $X_{m}$ )

Blocked Rotor Test (Gives $R_{01}$ , $X_{01}$ )