Q2Computer Aided Design of Electrical Machines

Numerical: Specific Magnetic and Electric Loading

Given data: Output P=350 kW, terminal voltage V=500 V, speed N=450 rpm, poles=6, armature diameter D=0.87 m, core length L=0.32 m, lap winding, total conductors Z=660.

Step 1: Armature Current

Step 2: Current per Conductor

For a lap winding, the number of parallel paths a equals the number of poles (a=P=6), so the current in each conductor is:

Step 3: Specific Electric Loading

Step 4: Flux per Pole (from EMF Equation)

Using the DC generator EMF equation E=(phiZNP)/(60a), and approximating the generated EMF E by the terminal voltage V (a standard simplifying assumption in this class of design problem, since armature resistance drop is not given):

Justification for the E-approximately-equal-to-V approximation: the exact relationship between the generated armature EMF and the terminal voltage of a DC generator is E=V+Ia*Ra (generator convention, where Ia is the armature current and Ra is the armature resistance, neglecting brush contact drop for simplicity). In a preliminary design calculation of this type, where the objective is only to establish approximate main dimensions and specific loadings before the detailed winding and resistance calculations have been carried out, the armature resistance Ra is not yet known (it depends on the conductor size and length, which are themselves outputs of a later design stage), so it is standard design practice to approximate E by V at this stage, since the armature resistance drop in a well-designed machine is typically only a small percentage (commonly 2-5%) of the rated terminal voltage. This approximation introduces only a correspondingly small error into the calculated flux per pole and specific magnetic loading, an error that is subsequently corrected, if required, once the actual winding resistance is computed in the later, detailed design stage; using the exact E=V+IaRa relation at this preliminary stage would in any case be circular, since Ra itself cannot be found until the conductor dimensions (which depend on the specific loadings being calculated here) are known. Once the detailed winding design is complete and the actual armature resistance is known, the designer normally revisits this calculation, recomputing E from E=V+IaRa and correspondingly refining the flux per pole and specific magnetic loading values, so that the preliminary approximate result obtained here is properly understood as the first iteration of what is, in general, an iterative design process rather than a single-pass exact calculation, converging quickly since the resistance-drop correction itself is small relative to the rated terminal voltage.

Step 5: Specific Magnetic Loading

Result: the specific electric loading is approximately 28,172 A/m and the specific magnetic loading is approximately 0.693 Wb/m^2 (Tesla), both physically reasonable values falling within the typical design range used for medium-sized DC machines (specific magnetic loading commonly in the range 0.4-0.9 Wb/m^2, specific electric loading commonly in the range 15,000-50,000 A/m depending on cooling method and machine size), confirming this is a well-proportioned design consistent with standard DC machine design practice.

Real and Apparent Flux Density

Apparent (nominal) flux density refers to the flux density calculated using the simplified, idealized assumption that the total flux per pole is uniformly distributed across the full geometric pole-pitch/core-length area, i.e., Bapp=phi/(tau*L), without accounting for any non-uniformity in the actual flux distribution caused by ventilating ducts, slot openings, or fringing effects at the pole edges.

Real (actual/true) flux density refers to the actual, physically-realized flux density within the iron teeth and core, which differs from the apparent value because the total flux per pole must actually pass through only the net (reduced) iron cross-sectional area available after accounting for the space occupied by ventilating (radial cooling) ducts and the insulation between laminations (represented by an iron/stacking factor, typically around 0.90-0.95, since the laminated core is not 100% solid iron), and because slot openings in the teeth locally concentrate flux into a smaller effective tooth width, increasing the real flux density at the tooth compared to the apparent smoothed-out value. The real flux density in the teeth, in particular, is calculated as: Breal(tooth)=phi/(number_of_teeth_per_pole tooth_width net_iron_length), and is always higher than the apparent gap flux density due to this cross-sectional area reduction caused by slotting — this distinction is of critical practical importance in machine design, since the real (actual) flux density in the teeth (which can be substantially higher than the apparent, smoothed-out average value) is what actually determines whether local magnetic saturation occurs in the teeth, directly affecting the machine's magnetizing current requirement and core loss, making it essential for the designer to calculate and check the real, rather than merely the apparent, flux density at each critical section of the magnetic circuit.

Practical Significance of the Calculated Loadings

The specific electric loading obtained here, approximately 28,172 A/m, and the specific magnetic loading, approximately 0.693 Wb/m^2, both fall comfortably within the ranges customarily adopted for medium-rating DC machines (specific magnetic loading commonly 0.4-0.9 Wb/m^2, specific electric loading commonly 15,000-50,000 A/m, as noted in the companion answer on specific loadings and design limitations elsewhere in this paper). A specific magnetic loading of 0.693 Wb/m^2 indicates the machine's core and teeth are operating comfortably below the saturation knee of typical silicon-steel lamination material (saturation generally becoming pronounced above roughly 1.5-1.8 Wb/m^2 in the core, and somewhat lower, around 1.6-2.0 Wb/m^2, tolerable in the more heavily-utilized teeth on account of their smaller net cross-section), leaving adequate margin for the additional local flux concentration caused by slotting (the real, as opposed to apparent, tooth flux density discussed above) without risking excessive magnetizing current or core loss. Similarly, a specific electric loading of approximately 28,172 A/m corresponds to a moderate current density in the armature conductors, consistent with a machine using conventional (rather than forced or liquid) ventilation, and indicates the winding design should achieve an acceptable temperature rise within standard insulation class limits. Both calculated values therefore confirm that the assumed design (D=0.87m, L=0.32m, Z=660, at the specified rating and speed) represents a realistic, well-proportioned DC generator design consistent with standard industrial practice, rather than an over- or under-utilized machine that would require revisiting the assumed dimensions.

Real flux density and apparent flux density both play essential roles in accurately estimating the iron losses and magnetizing current of an electrical machine at the design stage, and the gap between the two, driven by the reluctance of interlamination air gaps and any residual stacking imperfections, is one of the practical reasons why designers apply an empirically determined stacking factor when translating an idealized magnetic circuit calculation into a physically realizable core design.