# DC Machines and Equivalent Circuits

## Separately Excited \( I_A \neq I_F \)

Separate supplies for field and armature

$$ T = K_T \cdot I_F \cdot I_A $$

$$ I_F = \frac{V_F}{R_F} $$

$$ E_{BackEMF} = K_E \cdot \omega \cdot I_F $$

$$ I_A = \frac{(V_A - E_{BackEMF})}{R_A} $$

$$ P_{ElectricalInput} = V_A I_A + V_F I_F $$

$$ P_{InternalMechanicalOutput} = T \omega = E_{BackEMF} I_A $$

Where \( T \) is torque. \( I_F \) and \( I_A \) are field and armature currents respectively. \( K_T \) and \( K_E \) are relevant constants. \( \omega \) is angular velocity. \( P_{ElectricalInput} \) is the combined power input of the field and armature supplies. \( P_{InternalMechanicalOutput} \) is the total power output of the motor movement.

## Series Excited \( I_F = I_A \)

Field and armature circuits are in series with a single supply

$$ T = K_T I^2 $$

$$ I = \frac{(V - E_{BackEMF})}{R_F + R_A} $$

$$ E_{BackEMF} = K_E \cdot \omega \cdot I $$

$$ P_{ElectricalInput} = VI $$

$$ P_{InternalMechanicalOutput} = T \omega = E_{BackEMF} \cdot I $$

## Series Excited with Field Weakening \( I_A = I_F + I_W \)

$$ T = K_T I^2 (\frac{R_W}{R_F + R_W}) $$

$$ I_F = I_A (\frac{R_W}{R_F + R_W}) $$

$$ I = \frac{(V - E_{BackEMF})}{(R_F \parallel R_W + R_A)} $$

$$ R_F \parallel R_W = \frac{R_F \cdot R_W}{(R_F + R_W)} $$

$$ E_{BackEMF} = K_E \cdot \omega \cdot I_F $$

$$ P_{ElectricalInput} = VI = V \cdot I_A $$

$$ P_{InternalMechanicalOutput} = T \omega = E_{BackEMF} \cdot I_A $$

## Control Sequence

### Starting

Two DC motors are arranged in series, splitting the voltage to half of the supply. The variable resistor \( R_{F\_Bypass} \) is set to open circuit so that all current flows through the normal field winding. \( R_S \) is set to a relatively high value to limit the initial current in the actual motor.

Assuming the motors are starting from standstill, \( \omega \) is zero or very low, therefore \( E_{BackEMF} \) is also zero or very low. The current in the motor is limited only by the three series resistors, \( R_S \), \( R_F \) and \( R_A \).

### Accelerating

As the starting circuit reaches its relatively low maximum equilibrium velocity, the variable resistor in series, \( R_S \) is gradually decreased to allow for more current. Since \( \omega \) is increasing, \( E_{BackEMF} \) will increase proportionally.

As \( R_S \) reaches zero, the series circuit layout reaches its maximum output without field weakening.

### Switching to Parallel

By switching to parallel, each motor now has the entire source voltage drawn across it. Since this would be a large jump in power output, the series resistor \( R_S \) is again set to a large value to limit the current in the motors.

### Parallel Acceleration

\( R_S \) is once again decreased to allow for a gradual increase in current and velocity.

### Field Weakening

As the previous circuit reaches its maximum velocity, if the load force is relatively low (air resistance, friction), the field current can be weakened. This limits \( E_{BackEMF} \) by counteracting \( \omega \). At this point, \( R_S \) should be zero.

This control scheme accumulates a large power loss in resistors. Of which, variable resistors are actually composed of quantised steps which need to be triggered via electromechanical relays. Regenerative braking and other power recycling systems are difficult to implement.