āš” DC Machines Quick Reference

Essential Formulas, Concepts & Equations for Electrical Engineers

Standard Notations

\[\begin{aligned} P &= \text{Number of Poles} \\ \Phi &= \text{Flux/pole (Wb)} \\ N &= \text{Speed of Armature (RPM)}\\ Z &= \text{Number of armature conductors} \\ A &= \text{Number of parallel paths} \\ \omega &= \text{Angular speed (rad/sec)} \\ V &= \text{Terminal Voltage (V)} \\ I &= \text{Terminal Current (A)} \\ E &= \text{Induced EMF (V)} \\ I_a &= \text{Armature Current (A)} \\ I_{sh} &= \text{Shunt Field Current (A)} \\ I_{se} &= \text{Series Field Current (A)} \\ R_a &= \text{Armature Resistance } (\Omega) \\ R_{se} &= \text{Series Field Resistance } (\Omega) \\ R_{sh} &= \text{Shunt Field Resistance } (\Omega) \\ T &= \text{Torque (Nm)} \\ \end{aligned}\]

Fundamental EMF Equation

\[E = \dfrac{\Phi P N Z}{60 A} = \dfrac{\Phi P N}{60} \times \dfrac{Z}{A}\]
Note: For lap winding A = P, for wave winding A = 2

Types of DC Machines

šŸ”Œ Shunt-Wound

Field windings connected in parallel with armature. Provides constant speed characteristics.

šŸ”— Series-Wound

Field windings in series with armature. High starting torque, variable speed.

āš” Compound-Wound

Combination of series and shunt windings. Combines advantages of both types.

šŸ”‹ Separately Excited

Field winding powered by separate source. Independent field control.

Voltage & Current Equations

Shunt DC Machine

Generator Motor
\[\begin{aligned} E &= V + I_a R_a + V_{brush}\\ I_a &= I_{sh} + I\\ I_{sh} &= \dfrac{V}{R_{sh}} \end{aligned}\] \[\begin{aligned} V &= E + I_a R_a + V_{brush}\\ I &= I_{sh} + I_a\\ I_{sh} &= \dfrac{V}{R_{sh}} \end{aligned}\]

Series Wound DC Machine

Generator Motor
\[\begin{aligned} E &= V + I_a (R_a + R_{se}) + V_{brush} \\ I_a &= I \end{aligned}\] \[\begin{aligned} V &= E + I_a (R_a + R_{se}) + V_{brush} \\ I_a &= I \end{aligned}\]

Short-Shunt Compound DC Machine

Generator Motor
\[\begin{aligned} E &= V + I_a R_a + I R_{se} + V_{brush} \\ I_a &= I + I_{sh}\\ I_{sh} &= \dfrac{V + I \cdot R_{se}}{R_{sh}} \end{aligned}\] \[\begin{aligned} V &= E + I_a R_a + I R_{se} + V_{brush} \\ I &= I_a + I_{sh}\\ I_{sh} &= \dfrac{V - I \cdot R_{se}}{R_{sh}} \end{aligned}\]

Long-Shunt Compound DC Machine

Generator Motor
\[\begin{aligned} E &= V + I_a (R_a + R_{se}) + V_{brush} \\ I_a &= I + I_{sh} \\ I_{sh} &= \dfrac{V}{R_{sh}} \end{aligned}\] \[\begin{aligned} V &= E + I_a (R_a + R_{se}) + V_{brush} \\ I &= I_a + I_{sh} \\ I_{sh} &= \dfrac{V}{R_{sh}} \end{aligned}\]

Torque Equations

Electromagnetic Torque: \[T = \dfrac{P Z \Phi I_a}{2 \pi A}\]
Simplified Form: \[T \propto \Phi I_a\]
Torque-Speed Relationship: \[P_{\text{mech}} = T \omega = T \cdot \dfrac{2 \pi N}{60}\]

Speed Equations for DC Motors

From EMF Equation: \[N = \dfrac{60 A E}{P Z \Phi} = \dfrac{E}{K \Phi}\] where \(K = \dfrac{PZ}{60A}\)
DC Shunt Motor: \[N = \dfrac{V - I_a R_a}{\Phi} \times K\]

Speed nearly constant (varies slightly with load)

DC Series Motor: \[N \propto \dfrac{V - I_a(R_a + R_{se})}{I_a} = \dfrac{V}{I_a} - (R_a + R_{se})\]

Speed varies inversely with load

Speed Control Methods

For Shunt Motors

Flux Control Method
\[N \propto \dfrac{1}{\Phi}\]

Control field current using field rheostat. Speed above base speed.

Armature Voltage Control
\[N \propto V\]

Vary applied voltage. Speed below base speed. Best method.

Armature Resistance Control
\[N \propto (V - I_a R_a)\]

Add external resistance in series with armature. Poor efficiency.

For Series Motors

  • Flux Control: Diverter across series field or field taps
  • Variable Resistance: Series resistance with armature
  • Voltage Control: Variable voltage supply

Power Equations

Electrical Power Input: \[P_{\text{in}} = V I_a\]
Power Developed in Armature: \[P_a = E I_a\]
Mechanical Power Output: \[P_{\text{out}} = T \omega = T \cdot \dfrac{2 \pi N}{60}\]
Power Flow: Electrical Input ā†’ Armature Developed Power ā†’ Mechanical Output

Losses in DC Machines

1. Copper Losses (IĀ²R losses)

  • Armature copper loss: \(P_{\text{cu,a}} = I_a^2 R_a\)
  • Shunt field copper loss: \(P_{\text{cu,sh}} = \dfrac{V^2}{R_{sh}} = I_{sh}^2 R_{sh}\)
  • Series field copper loss: \(P_{\text{cu,se}} = I_a^2 R_{se}\)
  • Brush contact loss: \(P_{\text{brush}} = V_{\text{brush}} \times I_a\)

2. Iron (Core) Losses

  • Hysteresis Loss: \(P_h = K_h B_{\text{max}}^{1.6} f V\)
    Proportional to frequency and volume
  • Eddy Current Loss: \(P_e = K_e B_{\text{max}}^2 f^2 t^2 V\)
    Proportional to square of frequency and lamination thickness
Total Iron Loss: \(P_{\text{iron}} = P_h + P_e\) (approximately constant at rated speed)

3. Mechanical Losses

  • Friction Loss: Bearing friction, brush friction
  • Windage Loss: Air friction on rotating parts
Stray Losses: Miscellaneous losses (typically 1% of output). Iron losses + Mechanical losses = Rotational losses (constant at rated speed)

Efficiency

General Formula: \[\eta = \dfrac{P_{\text{out}}}{P_{\text{in}}} = \dfrac{P_{\text{out}}}{P_{\text{out}} + \text{Total Losses}}\]
Generator Efficiency
\[\eta_{\text{gen}} = \dfrac{V I}{E I_a} = \dfrac{\text{Output}}{\text{Armature Power}}\]
Motor Efficiency
\[\eta_{\text{mot}} = \dfrac{T \omega}{V I_a} = \dfrac{\text{Mech. Output}}{\text{Electrical Input}}\]

Condition for Maximum Efficiency

\[\text{Variable Losses} = \text{Constant Losses}\] \[I_a^2 R_a = P_{\text{iron}} + P_{\text{mech}} + P_{\text{shunt field}}\]

Construction & Winding Concepts

Pole Pitch

\[\text{Pole Pitch (conductors)} = \dfrac{Z}{P}\] \[\text{Pole Pitch (slots)} = \dfrac{\text{Total Slots}}{P}\]

Distance between centers of adjacent poles.

Commutator Pitch (Yc)

  • Lap Winding: \(Y_c = 1\) (or \(\pm 1\) for progressive/retrogressive)
  • Wave Winding: \(Y_c = \dfrac{C \pm 1}{P}\) where C = number of commutator segments

Coil Span (Ycs)

  • Full-Pitch Winding: \(Y_{cs} = \text{Pole Pitch}\)
  • Short-Pitch Winding: \(Y_{cs} < \text{Pole Pitch}\)

Back Pitch (Yb) and Front Pitch (Yf)

  • Back Pitch: \(Y_b = \dfrac{Z}{P}\) (approximately)
  • Relation: \(Y_b = Y_f \pm 2\) (for lap winding)
  • Average Pitch: \(Y_a = \dfrac{Y_b + Y_f}{2}\)

Number of Parallel Paths (A)

  • Simplex Lap Winding: \(A = P\)
  • Duplex Lap Winding: \(A = 2P\)
  • Simplex Wave Winding: \(A = 2\)
  • Duplex Wave Winding: \(A = 4\)

Electrical & Mechanical Degrees

\[\theta_e = \dfrac{P}{2} \times \theta_m\]
Example: For a 4-pole machine, 180Ā° mechanical = 360Ā° electrical

Armature Reaction

Definition: The effect of armature MMF on main field flux distribution. It causes flux distortion and reduces net flux.

Demagnetizing AT per Pole

\[AT_{\text{dem}} = \dfrac{Z I_a}{P} \times \dfrac{\theta_m}{360Ā°}\]

where Īøm = brush shift angle in mechanical degrees

Cross-Magnetizing AT per Pole

\[AT_{\text{cross}} = \dfrac{Z I_a}{P} \left(\dfrac{1}{2} - \dfrac{\theta_m}{360Ā°}\right)\]

Compensating Winding

\[AT_{\text{comp}} = \dfrac{Z I_a}{2P}\]

Neutralizes armature reaction by producing opposing MMF.

Effects of Armature Reaction

  • Distortion of main field flux
  • Reduction in generated EMF
  • Sparking at brushes
  • Shifted magnetic neutral axis

Characteristics of DC Machines

DC Generator Characteristics

Open Circuit Characteristic (OCC)

Also called: Magnetization curve or No-load saturation curve

Plot of E vs If at constant speed with no load

Load Characteristic

Plot of terminal voltage V vs load current I

  • Shunt: Slight drooping
  • Series: Rising then falling
  • Compound: Can be level, over, or under compounded

DC Motor Characteristics

Three main characteristics:
  • Torque vs Armature Current (T vs Ia): Electrical characteristic
  • Speed vs Armature Current (N vs Ia): Speed characteristic
  • Speed vs Torque (N vs T): Mechanical characteristic

Motor Characteristic Summary

Motor Type Speed-Load Torque-Current Applications
Shunt Motor Nearly constant speed \(T \propto I_a\) Lathes, pumps, fans, constant speed applications
Series Motor Variable speed (high at low load) \(T \propto I_a^2\) Traction, cranes, hoists (high starting torque)
Compound Motor Between shunt and series Between linear and square Elevators, rolling mills, presses

Starting of DC Motors

Why Starter Required? At starting, N = 0, so back EMF E = 0. Therefore, starting current \(I_a = \dfrac{V}{R_a}\) is very high (10-15 times rated current) since Ra is small.

Types of Starters

3-Point Starter
  • For shunt and compound motors
  • No-volt coil (NVC) protection
  • Overload release (OLR) protection
  • Field current through NVC
4-Point Starter
  • For shunt motors with field regulator
  • NVC independent of field current
  • Better for field control applications
  • Fourth terminal for separate NVC supply

Starting Methods

  • Direct-on-Line (DOL): Only for small motors (< 1 HP)
  • Series Resistance Starting: Most common method
  • Reduced Voltage Starting: Using auto-transformer or rheostat
Series Motor Starting: Never start under no-load condition! May reach dangerously high speeds. Always start with load connected.

Braking of DC Motors

Regenerative Braking

Condition: E > V

Motor acts as generator, returns energy to supply. Most efficient method. Used in electric vehicles and elevators.

\[I_a = \dfrac{E - V}{R_a}\]
Dynamic (Rheostatic) Braking

Armature disconnected from supply and connected across a resistor. Field remains excited.

Energy dissipated as heat in resistor. Good for rapid stopping.

\[I_a = \dfrac{E}{R_a + R_{\text{ext}}}\]
Plugging (Reverse Current)

Supply polarity reversed while motor is running. Very rapid braking with high braking torque.

Caution: High current, need to add series resistance.

Testing of DC Machines

Swinburne's Test (No-Load Test)

Type: Indirect, no-load test for shunt motors/generators
Procedure: Motor run at no-load at rated voltage and speed
  • Measure: V, Iā‚€, N (no-load speed)
  • Constant losses: \(W_c = V I_0 - I_0^2 R_a\)
  • For generator: \(\eta_g = \dfrac{VI}{VI + I_a^2 R_a + W_c}\)
  • For motor: \(\eta_m = \dfrac{VI - I_a^2 R_a - W_c}{VI}\)
Advantage: Economical, low power consumption
Disadvantage: Cannot account for change in iron losses with load

Brake Test (Direct Method)

Measures actual output power using mechanical brake \[T = (W_1 - W_2) \times R \text{ (Nm)}\] \[P_{\text{out}} = \dfrac{2\pi NT}{60}\] \[\eta = \dfrac{P_{\text{out}}}{VI}\]

where Wā‚, Wā‚‚ are spring balance readings, R is brake drum radius

Retardation Test (Running Down Test)

Purpose: To find stray losses (iron + mechanical losses)
Motor allowed to decelerate. Time taken to slow down from Nā‚ to Nā‚‚ is measured. \[W_s = \dfrac{J \omega_1 (N_1 - N_2)}{60t}\] where J = moment of inertia

Hopkinson's Test (Back-to-Back Test)

Type: Regenerative test for two identical shunt machines
Setup: Two machines mechanically coupled, one as motor, one as generator
  • Generator output feeds motor input
  • Only losses drawn from supply
  • Very efficient for large machines
  • Both machines tested simultaneously

Field's Test (For Series Motors)

Two identical series motors connected in series electrically and in opposition mechanically. Similar to Hopkinson's test but for series machines.

DC Generator Build-up Conditions

Conditions for Voltage Build-up (Self-Excited Generator)

  • There must be residual magnetism in poles
  • Field winding connections must produce flux in same direction as residual flux
  • Total field circuit resistance must be less than critical resistance
  • Speed must be greater than critical speed

Critical Field Resistance

Definition: Maximum field resistance above which generator fails to build up voltage
Graphically: Tangent to initial linear portion of OCC (magnetization curve) from origin gives critical resistance slope.

Critical Speed

Minimum speed below which voltage doesn't build up at given field resistance. \[N_c = N \times \dfrac{R_{\text{field}}}{R_{\text{critical}}}\]

Parallel Operation of DC Generators

Conditions for Parallel Operation

  • Terminal voltages must be equal
  • Polarities must be same (positive to positive, negative to negative)
  • External load characteristics should be drooping
  • Generated EMFs should be equal

Load Sharing

For generators with same external characteristics: \[\dfrac{I_1}{I_2} = \dfrac{E_1 - V}{E_2 - V}\]
Series Generators: Cannot be operated in parallel (rising characteristic causes instability)
Shunt Generators: Can be paralleled with equalizer bar to prevent circulating current

Applications of DC Machines

DC Generators

Separately Excited

Ward-Leonard speed control systems, electroplating, precision voltage control

Shunt Generator

Battery charging, excitation for alternators, general-purpose constant voltage supply

Series Generator

Boosters, arc welding (constant current), voltage compensators

Compound Generator

DC supply for lighting, power supply systems where constant voltage required despite load variations

DC Motors

Shunt Motor

Constant Speed: Lathes, centrifugal pumps, fans, blowers, conveyors, machine tools

Series Motor

High Starting Torque: Electric traction (trains, trams), cranes, hoists, elevators, electric cars

Compound Motor

Variable Torque & Speed: Rolling mills, shears, punches, presses, conveyors with heavy starting load

Important Relationships & Ratios

EMF and Speed Relationship: \[\dfrac{E_1}{E_2} = \dfrac{\Phi_1 N_1}{\Phi_2 N_2}\]
Torque Ratio: \[\dfrac{T_1}{T_2} = \dfrac{\Phi_1 I_{a1}}{\Phi_2 I_{a2}}\]
Power and Torque: \[P = T \times \omega = \dfrac{2\pi NT}{60}\] \[T = \dfrac{9.55 \times P}{N} \text{ (P in watts, N in RPM)}\]
Voltage Regulation (Generator): \[\text{Voltage Regulation} = \dfrac{E - V}{V} \times 100\%\]
Speed Regulation (Motor): \[\text{Speed Regulation} = \dfrac{N_0 - N_{fl}}{N_{fl}} \times 100\%\]

Quick Formula Reference Card

āš” Core Equations

\[E = \dfrac{\Phi PNZ}{60A}\] \[T = \dfrac{PZ\Phi I_a}{2\pi A}\] \[N = \dfrac{E}{K\Phi}\] \[P = T\omega = EI_a\]

šŸ“Š Efficiency

\[\eta = \dfrac{P_{out}}{P_{in}}\] \[\text{Max } \eta: I_a^2R_a = W_c\] \[\text{Losses} = W_{cu} + W_c\]

šŸ”„ Motor Equations

\[V = E + I_aR_a\] \[E_b = V - I_aR_a\] \[I_a = \dfrac{V-E}{R_a}\]

āš™ļø Generator Equations

\[E = V + I_aR_a\] \[V = E - I_aR_a\] \[\text{Reg} = \dfrac{E-V}{V} \times 100\%\]

Exam Tips & Common Mistakes

āš ļø Common Mistakes to Avoid:
  • Confusing generator and motor voltage equations (signs of IaRa term)
  • Forgetting brush drop voltage (typically 2V for carbon brushes)
  • Wrong parallel paths: Lap (A=P), Wave (A=2)
  • Not converting RPM to rad/s when using Ļ‰: \(\omega = \dfrac{2\pi N}{60}\)
  • Mixing up field resistances in compound machines
  • Forgetting to account for shunt field current in total current calculations
āœ… Quick Checks:
  • For motors: V > E (supply voltage opposes back EMF)
  • For generators: E > V (generated EMF higher than terminal)
  • Efficiency always < 100%
  • Speed regulation should be positive for normal operation
  • Power developed in armature: Pa = EIa (always)

šŸ“š DC Machines Quick Reference Guide

Essential formulas and concepts for Electrical Engineering