DC Machine Power

Below we have the equivalent circuit diagrams of a DC generator and a DC motor.

In the diagrams above:

$V_{a}$ is the armature terminal voltage
$R_{a}$ is the armature resistance
$L_{a}$ is the armature inductance
$V_{f}$ is the field winding terminal voltage
$R_{f}$ is the field resistance
$L_{f}$ is the field inductance
$I_{a}$ is the armature current
$I_{f}$ is the field current,
$τ_{m}$ is the developed toque in motor mode and the applied prime mover torque in generator mode
$τ_{L}$ is the load toque in motor mode
$τ_{c}$ is the counter toque in generator mode
$E_{gen}$ is the generated voltage in generator mode
$E_{c}$ is the counter emf in motor mode

At DC steady state, the inductance behaves as a short circuit. The equivalent circuit diagrams above simplify to:

We have:

I_{f} ϕ E_{gen} I_{a} τ_{c} = \frac{V _{f}}{R _{f}} = \frac{N _{f} I _{f}}{R} = \frac{ZP}{2 πa} ϕ ω_{m} = \frac{ZP}{60 a} ϕ n = \frac{E _{gen}}{R _{a} + R _{L}} = \frac{ZP}{2 πa} ϕ I_{a}

Note that $N_{f}$ is the number of turns of field winding.

Power Flow of DC Generator

We have:

I_{f} ϕ E_{gen} I_{a} τ_{c} = \frac{V _{f}}{R _{f}} = \frac{N _{f} I _{f}}{R} = \frac{ZP}{2 πa} ϕ ω_{m} = \frac{ZP}{60 a} ϕ n = \frac{E _{gen}}{R _{a} + R _{L}} = \frac{ZP}{2 πa} ϕ I_{a}

Thus, we have the following power quantities:

P_{f} P_{in} P_{core loss} P_{mech loss} P_{misc. loss} P_{conv.} P_{a} P_{out} P_{gen} P_{out} = I_{f}^{2} R_{f} (Field copper loss) = P_{m} = ω_{m} τ_{m} (Mechanical power of prime mover) = P_{hysteresis} + P_{eddy current} (Core loss) = P_{friction loss} + P_{windage loss} \approx 1% of full load power = E_{gen} I_{a} = ω_{m} τ_{c} (Power converted from mechanical to electrical) = I_{a}^{2} R_{a} (Armature copper loss) = V_{a} I_{a} = I_{a}^{2} R_{L} = \frac{V _{a}^{2}}{R _{L}} = P_{in} - P_{mech loss} - P_{core loss} - P_{misc. loss} = P_{gen} - P_{a}

The above can be summarized by the diagram below:

The efficiency of the DC generator can be found based on the relations given above:

η = \frac{P _{out}}{P _{in}} \times 100% = \frac{P _{in} - P _{loss}}{P _{in}} \times 100% = \frac{P _{out}}{P _{out} + P _{loss}} \times 100%

We have:

I_{f} ϕ E_{c} I_{a} τ_{m} = \frac{V _{f}}{R _{f}} = \frac{N _{f} I _{f}}{R} = \frac{ZP}{2 πa} ϕ ω_{m} = \frac{ZP}{60 a} ϕ n = \frac{E _{gen}}{R _{a} + R _{L}} = \frac{ZP}{2 πa} ϕ I_{a}

Then, we have the following power quantities:

P_{f} P_{in} P_{a} P_{conv.} P_{conv.} P_{core loss} P_{mech loss} P_{misc. loss} P_{out} = I_{f}^{2} R_{f} (Field copper loss) = V_{a} I_{a} (Input electric power) = I_{a}^{2} R_{a} (Armature copper loss) = E_{gen} I_{a} = ω_{m} τ_{c} (Power converted from electrical to mechanical) = P_{in} - P_{a} = P_{hysteresis} + P_{eddy current} = P_{friction loss} + P_{windage loss} \approx 1% of full load power = P_{conv.} - P_{core loss} - P_{mech loss} - P_{misc. loss}

This is summarized by the power flow diagram shown

The efficiency of the DC generator can be found based on the relations given above:

η = \frac{P _{out}}{P _{in}} \times 100% = \frac{P _{in} - P _{loss}}{P _{in}} \times 100% = \frac{P _{out}}{P _{out} + P _{loss}} \times 100%

To improve efficiency, DC machines are designed such that:

Copper losses ( $P_{f} = I_{f} R_{f}^{2}$ and $P_{a} = I_{a}^{2} R_{a}$ ) are minimized by reducing $R_{a}$ and $R_{f}$
Core losses are minimized by using high-quality soft ferromagnetic materials (except in permanent magnet machines)
Mechanical losses are reduced through good mechanical design
Using a small $R_{a}$ reduces the voltage drop in the armature circuit.

In electric machines, the mechanical power is usually given in horsepower (hp), where $1 hp = 765 W$ .