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:

Optimal Turns

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 N_λ :

Δφ (p&#772_c ) = ΔB(p&#772_c )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 Ni&#772 = N⋅i&#772 , where ī is the average current; Ni&#772 = H&#772⋅l . It sets the magnetic op-pt and the extent of saturation through k_sat .

• Maximum turns that fit winding window, N_w , 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 (N_λ ) and saturation (N_i ). The core is fully utilized when N_λ = N_i :

where circuit flux, Δλ = V_p ⋅t_on , I_p = average (primary) winding on-time current amplitude and V_p = average (primary) winding on-time voltage amplitude. The condition for N_opt is

Solve for the primary-winding on-time power amplitude for the circuit,

where V_p and I_p are the values during on-time, when the primary winding is driven. The on-time power relates to the average primary power, P&#772_p , by the duty-ratio, D , the fraction of the switching cycle that is the on-time: t_on = D⋅T_s = D /f_s and 1/t_on = f_s /D . Substitute

and core volume,

Then

Then relating to on-time circuit power,

Average power transferred = on-time power, P_p , times the fraction of its duration, D , or

We have maximized P&#772_p by equating the turns limits: N_λ = N_i , where N_λ is the power-loss minimum N and N_i 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.