The crankshaft, flywheel and attached load form a rotating system whose angular acceleration, multiplied by its moment of inertia, equals the net torque acting on it. That net torque is the sum of the gas torque from cylinder pressure, the torque from the inertia of the reciprocating masses, friction and the load. Gas torque is the product of the pressure difference across the piston, the piston area, the crank radius and a geometric function of crank angle and connecting rod ratio (Heywood, 1988). Rizzoni (1989) cast this balance as a dynamic model and showed that indicated torque can be estimated from measured speed, which opened the field. The textbook treatments of Kiencke and Nielsen (2005) and Guzzella and Onder (2010) give the model in a form suited to implementation.

Three families of method have been published. The first inverts the torque balance directly: Citron et al. (1989) recovered pressure and torque waveforms for each cylinder from speed fluctuation with a lumped model of the crankshaft, and Moro et al. (2002) used frequency response functions relating harmonics of speed to harmonics of pressure. Taraza et al. (2001) showed that the lowest harmonic orders of speed carry the information needed for diagnosis of diesel engines. The second family uses observers, in which a model of the engine is driven by the error between measured and predicted speed (Shiao and Moskwa, 1995). The third trains a non-parametric model, such as a radial basis function network, on recorded pairs of speed and pressure (Johnsson, 2006). Connolly and Yagle (1994) supplied an idea used below: describe the pressure trace by a few parameters and estimate those, instead of estimating pressure at every crank angle.

A single-cylinder engine has no overlapping firing events, so the gas torque of one cylinder is observed without interference from its neighbours. The crankshaft is short and stiff, and torsional resonance, which complicates the problem in long multi-cylinder shafts (Schagerberg and McKelvey, 2003), lies well above the frequencies of interest. The speed fluctuation is large because one firing in two revolutions must carry the whole cycle. Against these advantages, the reciprocating inertia torque is comparable in size to the gas torque at high speed, the driven machine may impose an unsteady load, and the flywheel teeth on a low-cost engine are cut to loose tolerances.

The microcontroller records the time between successive tooth edges with a hardware timer, so that the resolution is set by the timer clock and not by software latency. Angular resolution equals the tooth pitch, which on a typical ring gear is a few degrees of crank angle. The reference pulse identifies the tooth nearest top dead centre, and a cam or injection signal, or simply the asymmetry of the speed trace, distinguishes the compression stroke from the exhaust stroke. Records of at least a hundred consecutive cycles at steady load are stored for averaging.

Unequal tooth spacing appears as a speed fluctuation that repeats every revolution and has nothing to do with combustion. The usual remedy, described in work on misfire detection (Kiencke, 1999), is to learn a correction factor for each tooth while the engine coasts down with fuel cut off, when the true speed changes smoothly. The factors are stored and applied to all later records. Without this step the acceleration signal is dominated by manufacturing error.

Corrected tooth periods give the mean speed over each tooth interval. Acceleration is obtained by differentiating with respect to crank angle, and differentiation amplifies noise. The record is therefore averaged over cycles first and then smoothed with a zero-phase low-pass filter whose cut-off is expressed in harmonic orders of the engine cycle. Retaining roughly the first ten to twenty orders is a reasonable starting point, since the pressure trace of a diesel engine has little energy above that range apart from the sharp rise at ignition.

The moment of inertia of the rotating assembly and the reciprocating mass are taken from drawings or by weighing. Friction and load torque are harder to know. The proposal is to identify them from the compression stroke, where pressure follows a polytropic law fixed by intake conditions and compression ratio, so that the torque balance contains only the unknown resistance. A mean friction torque with a speed-dependent term, in the manner of Chen and Flynn (1965), is fitted to that interval. A more detailed description of instantaneous friction exists (Rezeka and Henein, 1984), but it has more parameters than this signal can support.

With inertia, reciprocating torque, friction and load accounted for, the torque balance yields the gas torque at each crank angle. Dividing by piston area, crank radius and the geometric function gives pressure. The division fails near top dead centre, where the geometric function passes through zero: the piston has no lever arm on the crank, pressure produces no torque, and any noise is amplified without limit. This is an ill-posed inverse problem in the sense treated by Hansen (1998), and it is the reason for the next step.

Instead of accepting the pointwise result, the procedure fits a thermodynamic model with a handful of parameters to the gas torque over the whole compression and expansion interval. The model is a single-zone energy balance: polytropic compression from known intake conditions, heat release described by a Wiebe function with start, duration and shape parameters (Ghojel, 2010), and wall heat loss from a standard correlation (Woschni, 1967). The parameters are adjusted by least squares so that the gas torque predicted by the model matches the torque obtained in step 5, with the angles close to top dead centre given low weight. The fitted model then supplies the pressure where direct inversion cannot.

From the reconstructed trace the usual indicators are computed: indicated mean effective pressure, peak pressure and its crank angle, and the angle at which half the heat has been released. Indicated mean effective pressure is sensitive to errors in the assumed position of top dead centre, as Brunt and Emtage (1996) showed for measured pressure, and the same holds here. Two checks are proposed. The indicated work should equal the measured brake work plus the friction identified in step 4, and the fitted start of combustion should fall after the known start of injection by a plausible ignition delay. If either check fails, the parameters of steps 4 and 6 are revisited.