Q5Computer Aided Design of Electrical Machines

Question

Q.3. (a) Derive an equation for voltage per turn in terms of phase output of a transformer. [6]

(b) Determine the main dimensions of the core, number of turns and the area of conductors for a 5kVA, 50 Hz, 11000/400 V, single phase, core type distribution transformer. The net conductor area in the window is 60% of the net cross section (square) of the iron case. Assume a flux density of 1 Wb/m^2, a current density of 1.4 A/mm^2 and a window space factor of 0.2. The window height is 3 times its width. [10]

Answer 1

(a) Voltage per Turn in Terms of Phase Output

The transformer's phase output (per-phase kVA rating, or simply kVA for a single-phase unit), Q, can be related to the transformer's voltage per turn, Et, through the fundamental transformer design (output) equation, providing a convenient design starting point that avoids needing to separately determine flux density, core area, and turns individually before establishing this key relationship.

Derivation: the EMF induced per turn, from the standard transformer EMF equation, is Et=4.44fBmAi, where f is frequency, Bm is the maximum core flux density, and Ai is the net iron core cross-sectional area. The transformer's rated phase output can be expressed as Q=Et(rated current per turn's worth of ampere-turns, summed appropriately) — more directly, using the standard relation that total window ampere-turns (AT=I1*N1, assuming equal primary/secondary MMF) relate to the window's copper-carrying capacity, and combining this with the EMF equation and the assumed window space factor/current density relationship, the output can be shown to reduce to the compact form:

Q = 2.22 f B_{m} δ K_{w} A_{i} A_{w} \times 1 0^{- 3} (kVA, single-phase)

where delta is the current density and Kw is the window space factor, Aw is the window area. Since Et=4.44fBm*Ai, this output equation can be rearranged and combined with typical design ratios (relating window area to core area) to show that, for transformers of similar proportions and design ratios (window-to-core-area ratio, aspect ratio, etc.) using similar values of Bm, delta, and Kw, the voltage per turn scales approximately with the square root of the kVA rating:

E_{t} \approx K Q

where K is an empirical constant (determined by the transformer's specific type, construction, and typical design ratios) — this square-root relationship is a well-known and practically useful rule of thumb in transformer design, typically with K in the range of approximately 0.6-0.7 for single-phase core-type distribution transformers, somewhat different values for shell-type construction, and higher values for three-phase transformers, allowing a designer to quickly estimate a reasonable starting voltage-per-turn value directly from the transformer's kVA rating alone, before proceeding to the more detailed core and window dimension calculations illustrated in part (b) of this question.

Physical reason the square-root relationship holds: the underlying reason Et scales with the square root of kVA, rather than linearly or by some other power, follows directly from the output equation Q=2.22fBmdeltaKwAiAw. If the designer keeps the window-to-core-area ratio (Aw/Ai) fixed across a family of transformers of similar type and proportions (a common simplifying design assumption, as also used in part (b) of this question), then Q becomes proportional to Ai^2 alone (since Aw is itself proportional to Ai), so Ai is proportional to sqrt(Q). Since the voltage per turn Et=4.44fBm*Ai is directly proportional to Ai (for fixed f and Bm), it follows immediately that Et is also proportional to sqrt(Q). Physically, this reflects the fact that both the core's flux-carrying capacity (through Ai) and the window's current-carrying capacity (through Aw) must grow together to accommodate a larger kVA rating while preserving similar flux density and current density throughout the design, and since kVA output depends on the product of these two areas (each scaling with the linear core dimension in a self-similar design), the voltage per turn — tied to only one of the two areas — necessarily grows more slowly than the rating itself, specifically as its square root.

Core stacking factor: in practice, the net iron cross-sectional area Ai used throughout this calculation is not the same as the gross (physical) core cross-section, since the core is built up from thin, varnish- or oxide-insulated laminations rather than a single solid block of steel, both to reduce eddy-current loss and because the stamped circular (or stepped) core cross-section does not perfectly fill a square or circular outline. The stacking factor (also called the iron factor), typically in the range of 0.90-0.97 depending on lamination thickness and insulation coating, is the ratio of net iron area to gross core area, so that the physical (gross) core cross-section actually specified to the core-cutting/stamping process must be correspondingly larger than the net iron area Ai calculated from the electromagnetic design equations, i.e., gross area=Ai/stacking factor. Neglecting this stacking factor correction would result in an under-sized physical core that could not actually accommodate the required net iron area, leading to higher-than-intended flux density, increased magnetizing current, and excessive core loss and temperature rise in the finished transformer — making the stacking-factor correction an essential, though often only briefly mentioned, practical step following the basic net-area calculation illustrated in this question.

(b) Numerical: Design of 5kVA Single-Phase Core-Type Distribution Transformer

Given: Q=5 kVA, f=50 Hz, V1=11000V, V2=400V, Bm=1.0 Wb/m^2, current density delta=1.4 A/mm^2=1.4x10^6 A/m^2, window space factor Kw=0.2, net conductor area in window=60% of net iron core cross-section (Ai), window height=3 times window width.

Step 1: Relate Window Area to Core Area

Given that the net copper (conductor) area in the window equals 60% of the net iron core area, Acu=0.6*Ai, and the window space factor relates window area to copper area as Aw=Acu/Kw:

A_{w} = \frac{0.6 A _{i}}{0.2} = 3 A_{i}

Step 2: Apply the Output Equation to Find Core Area

Using the single-phase transformer output equation Q=2.22fBmdeltaKwAiAw (in VA), and substituting Aw=3*Ai:

5000 = 2.22 \times 50 \times 1.0 \times (1.4 \times 1 0^{6}) \times 0.2 \times A_{i} \times (3 A_{i})

A_{i}^{2} = \frac{5000}{2.22 \times 50 \times 1.4 \times 1 0 ^{6} \times 0.2 \times 3} \Rightarrow A_{i} \approx 0.007323 m^{2} = 73.23 cm^{2}

Step 3: Core and Window Dimensions

Since the net iron cross-section is stated to be square, the side of the square core is:

a = A_{i} = 0.007323 \approx 0.0856 m = 85.6 mm

The window area is Aw=3Ai=0.02197 m^2=219.7 cm^2, and with window height Hw=3 times window width Ww (HwWw=Aw, Hw=3Ww, so 3Ww^2=Aw):

W_{w} = \frac{A _{w}}{3} \approx 0.0856 m = 85.6 mm, H_{w} = 3 W_{w} \approx 256.7 mm

Step 4: Voltage per Turn and Number of Turns

E_{t} = 4.44 f B_{m} A_{i} = 4.44 \times 50 \times 1.0 \times 0.007323 \approx 1.626 V/turn

N_{1} = \frac{V _{1}}{E _{t}} = \frac{11000}{1.626} \approx 6766 turns, N_{2} = \frac{V _{2}}{E _{t}} = \frac{400}{1.626} \approx 246 turns

Step 5: Area of Conductors

The primary and secondary rated currents are I1=Q/V1=5000/11000=0.4545A and I2=Q/V2=5000/400=12.5A, giving the required conductor cross-sectional areas at the assumed current density of 1.4 A/mm^2:

a_{1} = \frac{I _{1}}{δ} = \frac{0.4545}{1.4} \approx 0.3247 mm^{2}, a_{2} = \frac{I _{2}}{δ} = \frac{12.5}{1.4} \approx 8.929 mm^{2}

Summary of results: net core area Ai approximately 73.2 cm^2 (square core, side approximately 85.6mm); window dimensions approximately 85.6mm (width) by 256.7mm (height); voltage per turn approximately 1.626V; primary turns N1 approximately 6766; secondary turns N2 approximately 246; primary conductor area approximately 0.325 mm^2; secondary conductor area approximately 8.93 mm^2 — these results represent a complete, internally-consistent first design pass for the specified 5kVA distribution transformer, from which further detailed design steps (actual conductor gauge/shape selection, insulation thickness allowance, and loss/efficiency verification) would proceed in a complete transformer design exercise.

The voltage-per-turn design constant derived above serves as the essential starting point for the entire main-dimension design procedure carried out in this problem, since every subsequent quantity, including the core cross-sectional area, the number of primary and secondary turns, and the conductor cross-sectional areas, is derived directly or indirectly from this single output-equation-based constant, making its correct derivation the foundation upon which the rest of the transformer design calculation depends.