EE-361

A Brief Review of DC Machine Models

Ozan Keysan

Office: C-113 • Tel: 210 7586

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Model of DC Motors

Drawing

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Simplest Equivalent Circuit of a DC MachineA voltage source (proportional to speed) connected in series with a resistance (Armature resistance)3 / 16

Simplest Equivalent Circuit of a DC Machine

A voltage source (proportional to speed) connected in series with a resistance (Armature resistance)

Drawing

Under constant flux (as in permanent magnet DC machine)

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Induced Voltage in Armature4 / 16

Induced Voltage in ArmatureEa=KaωmΦpp4 / 16

Induced Voltage in ArmatureEa=KaωmΦppEa: Induced armature voltage
Ka: Armature Constant
ωm: Mechanical Speed (rad/s)
Φpp: Flux per-pole
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DC Machines5 / 16

DC MachinesMotor Action: Electrical Energy is converted to Mechanical Energy5 / 16

DC MachinesMotor Action: Electrical Energy is converted to Mechanical EnergyGenerator Action: Mechanical Energy is converted to Electrical Energy5 / 16

DC Machines

Motor Action: Electrical Energy is converted to Mechanical Energy

Generator Action: Mechanical Energy is converted to Electrical Energy

No difference between a DC motor and generator

Just a mode of operating point

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Equivalent Circuit of a DC MotorIf the flux is generated by another coil (Field Winding)6 / 16

Equivalent Circuit of a DC Motor

If the flux is generated by another coil (Field Winding)

Drawing

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Equivalent Circuit of a DC Motor

Drawing

If $V_a > E_a$ motoring action

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Equivalent Circuit of a DC Motor

Drawing

If $V_a > E_a$ motoring action

If $V_a < E_a$ generating action

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Electromechanical Power

Drawing

Electromechanical Power = Armature Power - Armature Losses

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Electromechanical Power

Drawing

Electromechanical Power = Armature Power - Armature Losses

$P_{out} = V_a I_a - I_a R_a = E_a I_a$

(for the motoring mode)

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Torque Relations

Drawing

$P_{mech} =$

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Torque Relations

Drawing

$P_{mech} =$ $T \omega = E_a I_a$

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Torque Relations

Drawing

$P_{mech} =$ $T \omega = E_a I_a$

$T \omega = K_a \omega \Phi_{pp} I_a$

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Torque Relations

Drawing

$P_{mech} =$ $T \omega = E_a I_a$

$T \omega = K_a \omega \Phi_{pp} I_a$

$T =K_a \Phi_{pp} I_a$

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Torque is Proportional to CurrentT=KaΦppIa10 / 16

What happens to DC motor at start-up?11 / 16

What happens to DC motor at start-up?Rotor speed is zero: ω=0Induced armature voltage:11 / 16

What happens to DC motor at start-up?Rotor speed is zero: ω=0Induced armature voltage:Ea=KaωmΦpp=011 / 16

What happens to DC motor at start-up?Rotor speed is zero: ω=0Induced armature voltage:Ea=KaωmΦpp=0Armature Current: Ia=(Vt−Ea)/Ra11 / 16

What happens to DC motor at start-up?Rotor speed is zero: ω=0Induced armature voltage:Ea=KaωmΦpp=0Armature Current: Ia=(Vt−Ea)/RaMaximum Current and Torque at Startup11 / 16

Maximum Torque at Startup12 / 16

Maximum Torque at Startup

Result: Never attempt a drag race with an electric car.

Drawing

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You need to wait until EE362, but

Electric Cars: Tesla Model S Induction Motor

13 / 16

Tesla vs Bugatti Veyron

Drawing

Tesla Acceleration Reactions, Tesla trolls muscle cars

More info: What's inside Tesla Model 3?, Tesla Model 3's motor animation, Tesla motor assembly

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SummaryInduced Voltage Proportional to Speed
Ea=KaωmΦpp
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SummaryInduced Voltage Proportional to Speed
Ea=KaωmΦpp
Torque Proportional to Armature Current
T=KaΦppIa
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EE-361

A Brief Review of DC Machine Models

Ozan Keysan

Model of DC Motors

Simplest Equivalent Circuit of a DC Machine

A voltage source (proportional to speed) connected in series with a resistance (Armature resistance)

Simplest Equivalent Circuit of a DC Machine

A voltage source (proportional to speed) connected in series with a resistance (Armature resistance)

Under constant flux (as in permanent magnet DC machine)

Induced Voltage in Armature

Induced Voltage in Armature

Ea=KaωmΦppE_a = K_a \omega_m \Phi_{pp}

Induced Voltage in Armature

Ea=KaωmΦppE_a = K_a \omega_m \Phi_{pp}

EaE_a: Induced armature voltage

KaK_a: Armature Constant

ωm\omega_m: Mechanical Speed (rad/s)

Φpp\Phi_{pp}: Flux per-pole

DC Machines

DC Machines

Motor Action: Electrical Energy is converted to Mechanical Energy

DC Machines

Motor Action: Electrical Energy is converted to Mechanical Energy

Generator Action: Mechanical Energy is converted to Electrical Energy

DC Machines

Motor Action: Electrical Energy is converted to Mechanical Energy

Generator Action: Mechanical Energy is converted to Electrical Energy

No difference between a DC motor and generator

Just a mode of operating point

Equivalent Circuit of a DC Motor

If the flux is generated by another coil (Field Winding)

Equivalent Circuit of a DC Motor

If the flux is generated by another coil (Field Winding)

Equivalent Circuit of a DC Motor

If Va>EaV_a > E_a motoring action

Equivalent Circuit of a DC Motor

If Va>EaV_a > E_a motoring action

If Va<EaV_a < E_a generating action

Electromechanical Power

Electromechanical Power = Armature Power - Armature Losses

Electromechanical Power

Electromechanical Power = Armature Power - Armature Losses

Pout=VaIa−IaRa=EaIaP_{out} = V_a I_a - I_a R_a = E_a I_a

(for the motoring mode)

Torque Relations

Pmech=P_{mech} =

Torque Relations

Pmech=P_{mech} = Tω=EaIa T \omega = E_a I_a

Torque Relations

Pmech=P_{mech} = Tω=EaIa T \omega = E_a I_a

Tω=KaωΦppIa T \omega = K_a \omega \Phi_{pp} I_a

Torque Relations

Pmech=P_{mech} = Tω=EaIa T \omega = E_a I_a

Tω=KaωΦppIa T \omega = K_a \omega \Phi_{pp} I_a

T=KaΦppIaT =K_a \Phi_{pp} I_a

Torque is Proportional to Current

T=KaΦppIaT =K_a \Phi_{pp} I_a

What happens to DC motor at start-up?

What happens to DC motor at start-up?

Rotor speed is zero: ω=0\omega =0

Induced armature voltage:

What happens to DC motor at start-up?

Rotor speed is zero: ω=0\omega =0

Induced armature voltage:Ea=KaωmΦpp=0E_a = K_a \omega_m \Phi_{pp}=0

What happens to DC motor at start-up?

Rotor speed is zero: ω=0\omega =0

Induced armature voltage:Ea=KaωmΦpp=0E_a = K_a \omega_m \Phi_{pp}=0

Armature Current: Ia=(Vt−Ea)/RaI_a = (V_t -E_a)/R_a

What happens to DC motor at start-up?

Rotor speed is zero: ω=0\omega =0

Induced armature voltage:Ea=KaωmΦpp=0E_a = K_a \omega_m \Phi_{pp}=0

Armature Current: Ia=(Vt−Ea)/RaI_a = (V_t -E_a)/R_a

Maximum Current and Torque at Startup

Maximum Torque at Startup

Maximum Torque at Startup

Result: Never attempt a drag race with an electric car.

You need to wait until EE362, but

Electric Cars: Tesla Model S Induction Motor

Tesla vs Bugatti Veyron

Tesla Acceleration Reactions, Tesla trolls muscle cars

$E_a = K_a \omega_m \Phi_{pp}$

$E_a = K_a \omega_m \Phi_{pp}$

$E_a$ : Induced armature voltage

$K_a$ : Armature Constant

$\omega_m$ : Mechanical Speed (rad/s)

$\Phi_{pp}$ : Flux per-pole

If $V_a > E_a$ motoring action

If $V_a > E_a$ motoring action

If $V_a < E_a$ generating action

$P_{out} = V_a I_a - I_a R_a = E_a I_a$

$P_{mech} =$

$P_{mech} =$ $T \omega = E_a I_a$

$P_{mech} =$ $T \omega = E_a I_a$

$T \omega = K_a \omega \Phi_{pp} I_a$

$P_{mech} =$ $T \omega = E_a I_a$

$T \omega = K_a \omega \Phi_{pp} I_a$

$T =K_a \Phi_{pp} I_a$

$T =K_a \Phi_{pp} I_a$

Rotor speed is zero: $\omega =0$

Rotor speed is zero: $\omega =0$

Induced armature voltage: $E_a = K_a \omega_m \Phi_{pp}=0$

Rotor speed is zero: $\omega =0$

Induced armature voltage: $E_a = K_a \omega_m \Phi_{pp}=0$

Armature Current: $I_a = (V_t -E_a)/R_a$

Rotor speed is zero: $\omega =0$

Induced armature voltage: $E_a = K_a \omega_m \Phi_{pp}=0$

Armature Current: $I_a = (V_t -E_a)/R_a$

$E_a = K_a \omega_m \Phi_{pp}$

$E_a = K_a \omega_m \Phi_{pp}$

$T =K_a \Phi_{pp} I_a$