In the last chat, Magnetics Design Chat: Part 1, Optimal Design of Magnetics Components, our goal was to maximize transfer-power density in a magnetic component so that for a given maximum power, the component is as small as possible. The basic formula for transfer of power through a magnetic core was expressed by the simple relationship:
The two basic limitations on cores are:
This basic conclusion is so important, it bears repeating in another way:
The magnetic (core) and electric (winding) designs are connected through the choice of number of turns, N of winding, a central magnetic design parameter. Maximum core transfer-power density, or core utilization, determines turns limits. The turns minimum is Nλ:
Δφ(p̄c) = ΔB(p̄c)A, where A is the core magnetic path cross-sectional area. Both A and magnetic path length, l, are given in core catalogs along with core magnetic volume, V. The maximum turns is limited by saturation:
The current referred to or “seen” by the core field is Nī = N⋅ī , where ī is the average current; Nī = H̄⋅l. It sets the magnetic op-pt and the extent of saturation through ksat.
- Maximum turns that fit winding window, Nw, is another turns limitation → allowable current density
The range of N is bounded by power-loss and saturation limits, and maximum window turns:
With adequate window area for turns, the design range of N is bracketed by core power-loss (Nλ) and saturation (Ni). The core is fully utilized when Nλ = Ni:
where circuit flux, Δλ = Vp⋅ton, Ip = average (primary) winding on-time current amplitude and Vp = average (primary) winding on-time voltage amplitude. The condition for Nopt is
Solve for the primary-winding on-time power amplitude for the circuit,
where Vp and Ip are the values during on-time, when the primary winding is driven. The on-time power relates to the average primary power, P̄p, by the duty-ratio, D, the fraction of the switching cycle that is the on-time: ton = D⋅Ts = D/fs and 1/ton = fs/D. Substitute
and core volume,
Then relating to on-time circuit power,
Average power transferred = on-time power, Pp, times the fraction of its duration, D, or
We have maximized P̄p by equating the turns limits: Nλ = Ni, where Nλ is the power-loss minimum N and Ni is the saturation-limited maximum N.
In the next chat, we will look at the circuit-design consequences of these new-found criteria for magnetics design.