DC generator and motor characteristics

Wire the same DC machine's field winding one way and it makes a nearly constant voltage source; wire it another way and it becomes a machine that can start a loaded crane from standstill but must never run unloaded. The difference is how the field winding — shunt (parallel, many turns, high resistance) or series (in line, few turns, low resistance) — is connected relative to the armature. That wiring choice, plus whether the machine is driven mechanically (generator) or electrically (motor), fixes a whole family of curves: how terminal voltage sags with load current, and how speed and torque respond to armature current. Reading these curves is how an engineer picks the right DC machine for lighting, traction, cranes or a bench supply, and it all traces back to one relation, the back-EMF equation $E_b = V - I_aR_a$ .

The core insight

A DC machine's field connection — shunt, series, or compound — together with the back-EMF relation $E_b = V - I_aR_a$ and $N \propto E_b/\Phi$ , $T \propto \Phi I_a$ , fixes whether it behaves as a nearly constant-voltage generator or constant-speed motor (shunt), a rising-then-drooping generator or high-torque but dangerously overspeeding motor (series), or a compromise between the two (compound).

The core idea

Field connections set how flux responds to load. In a shunt machine the field winding sits in parallel with the armature, so field current $I_f = V/R_{\text{field}}$ is essentially independent of the load — flux $\Phi$ stays nearly constant. In a series machine the field winding carries the same current as the armature and the load, so flux $\Phi$ rises with load current until magnetic saturation flattens it out. A compound machine carries both a shunt field and a series field, either cumulative (aiding, fluxes add) or differential (opposing) — cumulative compounding is the practical case worth remembering, since it blends shunt steadiness with series boost.

Generators: terminal voltage vs load current, driven at constant speed. Two effects fight the terminal voltage as load grows: the armature-resistance drop $I_aR_a$ always pulls it down, while any change in field current changes $\Phi$ and hence the EMF. In a shunt generator, field current falls slightly as terminal voltage falls, adding a small extra droop to the $I_aR_a$ drop — voltage sags moderately but stays close to constant. In a series generator, more load current means more field current means higher flux and higher EMF, so voltage first rises with load, then droops once saturation and the $I_aR_a$ drop take over — unsuited to a constant-voltage supply. Cumulative compounding lets the series field's boost offset the shunt-like droop, giving a flatter (or deliberately over-compounded, rising) characteristic — the standard choice where voltage regulation matters.

Motors: speed and torque vs armature current, at constant supply voltage. Every motor characteristic follows from one equation, the back-EMF relation:

E_b = V - I_aR_a

where $V$ is the constant supply voltage, $I_a$ the armature current drawn at that load, $R_a$ the armature resistance, and $E_b$ the back-EMF — the voltage the rotating armature generates in opposition to $V$ (Lenz's law), proportional to both flux and speed via $E_b = k\Phi N$ . Rearranging gives the two design relations used throughout: speed $N \propto E_b/\Phi$ , and torque $T \propto \Phi I_a$ .

Shunt motor — $\Phi$ nearly constant, so as load current $I_a$ rises, $E_b$ drops only by the small $I_aR_a$ term, and speed droops only slightly: approximately constant-speed. Torque rises roughly linearly with $I_a$ .
Series motor — $\Phi$ grows with $I_a$ (pre-saturation $T \propto \Phi I_a \propto I_a^2$ , so torque climbs faster than linearly, giving very high starting torque), while speed $N \propto E_b/\Phi$ falls sharply as load rises and would climb dangerously high at no load, since $\Phi \to 0$ removes the denominator's brake on speed.
Compound motor — cumulative compounding lands between the two: a moderate speed droop with load and better starting torque than plain shunt, without a series motor's unsafe no-load overspeed.

Figure

Two curves side by side, terminal voltage falling gently with load current for a shunt generator and rising then falling for a series generator, next to speed falling gently with armature current for a shunt motor and falling steeply for a series motor

Derivation ASpeed of a shunt motor from the back-EMF relation

The goal is to derive, from the back-EMF equation and the assumption of constant flux, exactly why a shunt motor's speed depends on load only through the small armature-resistance drop, and to obtain the ratio $N_2/N_1 = E_{b2}/E_{b1}$ used to predict a new operating speed once the armature current changes — the algebraic tool behind every shunt-motor speed-regulation problem.

Step 1. Write the back-EMF relation at the original load, before the current change: $E_{b1} = V - I_{a1}R_a$ , where $V$ is the fixed supply voltage and $I_{a1}$ the armature current drawn at that load. This is just Kirchhoff's voltage law around the armature loop: the supply voltage splits between the resistive drop $I_{a1}R_a$ and the back-EMF opposing it.

Step 2. Write the same relation after the load changes and armature current becomes $I_{a2}$ : $E_{b2} = V - I_{a2}R_a$ . Since $V$ and $R_a$ are unchanged (same supply, same machine), the only thing that moves $E_{b2}$ away from $E_{b1}$ is the change in $I_a$ .

Step 3. Bring in the EMF equation $E_b = k\Phi N$ , where $k$ is a machine constant. For a shunt motor the field current is fixed by the constant supply voltage across a fixed field resistance, so $\Phi$ is (approximately) unchanged between the two loads. Dividing the two EMF equations cancels both $k$ and $\Phi$ : $\dfrac{E_{b2}}{E_{b1}} = \dfrac{k\Phi N_2}{k\Phi N_1} = \dfrac{N_2}{N_1}$ .

Step 4. Combine Steps 1–3 into the working formula: $N_2 = N_1\times\dfrac{V - I_{a2}R_a}{V - I_{a1}R_a}$ . Because $I_aR_a$ is small compared with $V$ for a well-designed machine, $N_2$ stays close to $N_1$ even for a large swing in $I_a$ — the algebraic reason a shunt motor is called approximately constant-speed, and the exact reason it is not perfectly constant.

Worked example

Problem. A DC shunt motor is connected to a 220 V supply with armature resistance $R_a = 0.5\ \Omega$ . At one load it draws armature current $I_{a1} = 20$ A and runs at $N_1 = 1000$ rpm. The load then increases so that armature current rises to $I_{a2} = 30$ A, with flux assumed constant. Find the back-EMF at each load and the new speed.

Step 1. Back-EMF at the first load: $E_{b1} = V - I_{a1}R_a = 220 - (20\times0.5) = 220 - 10 = 210$ V.

Step 2. Back-EMF at the second load: $E_{b2} = V - I_{a2}R_a = 220 - (30\times0.5) = 220 - 15 = 205$ V.

Step 3. Apply the derivation's speed ratio, since flux is constant: $N_2 = N_1\times\dfrac{E_{b2}}{E_{b1}} = 1000\times\dfrac{205}{210} = 976.2$ rpm.

Answer: $E_{b1} = 210$ V, $E_{b2} = 205$ V, and the new speed is $N_2 \approx 976$ rpm — a droop of under 2.5% for a 50% rise in armature current, illustrating why a shunt motor is called approximately constant-speed.

Competitive stretch

For the same $I_a = 30$ A on a series motor instead, $E_b = V - I_aR_a$ still gives 205 V — but the speed ratio of Step 3 no longer applies, because $\Phi$ has grown with $I_a$ , so $N\propto E_b/\Phi$ falls by more than the drop in $E_b$ alone suggests.

Common traps

Running a series motor unloaded or on a slipping belt drive — at no load, $I_a$ and hence $\Phi$ collapse toward zero, and since $N \propto E_b/\Phi$ , speed can rise without bound and destroy the machine. Series motors must be directly coupled to their load.
Treating a shunt motor's speed as exactly constant — it only approximately holds because of the small $I_aR_a$ term in $E_b = V - I_aR_a$ ; true constant speed is the idealised zero-resistance limit.
Mixing up generator characteristics (terminal voltage vs load current, machine driven mechanically) with motor characteristics (speed/torque vs armature current, machine driven electrically) — same machine, two different operating regimes.
Assuming a series generator is a good general-purpose voltage source — its rising-then-drooping curve makes it unsuitable except in special applications, unlike shunt or compound generators.

Ask the tutor: want more back-EMF/speed numerical practice, or a walkthrough of why series-motor torque scales as the square of armature current before saturation?

Why does the back-EMF equation E_b = V minus I_a R_a apply to a shunt motor's speed but not directly to a series motor's speed

The equation itself holds for any DC motor — it is just Kirchhoff's voltage law around the armature loop. The difference is what happens to flux. In a shunt motor the field current, and hence flux, barely changes with load, so E_b alone tracks speed through N proportional to E_b over Phi with Phi treated as constant. In a series motor the same current that sets E_b also sets flux, so a change in load changes both E_b and Phi at once, and you cannot read off speed from E_b without separately tracking how Phi has moved.

Why is a series motor chosen for traction and cranes despite the no-load overspeed risk

Because its torque characteristic is exactly what heavy starting loads need. Since Phi grows with armature current before saturation, torque rises roughly as I_a squared at low currents — a series motor delivers very high torque right at standstill, precisely when a loaded train or crane needs the biggest push to get moving. The overspeed risk is manageable in practice because traction and crane loads are permanently, mechanically coupled to the motor shaft, so the "no load" condition that would let flux collapse never actually occurs in normal operation.

Is a compound motor just a worse version of both shunt and series motors

No — it is a deliberate compromise, not a mediocre average. A cumulative-compound motor keeps a shunt field for steadier speed regulation and adds a series field for extra starting torque beyond what a plain shunt motor can give. Its speed-versus-load curve sits between the shunt motor's near-flat line and the series motor's steep droop, and unlike a series motor it never truly unloads to zero flux, because the shunt field alone still holds up E_b's denominator, avoiding the dangerous overspeed. It is chosen exactly where both moderate speed regulation and higher-than-shunt starting torque are wanted together, such as in some rolling mills and elevators.

DC generator and motor characteristics

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads