Continuing the chat from last time, we turn now to a circuit-related topic, that of current waveforms. The typical converter input waveform is shown with a static (dc) component and a ripple (ac) component.

This is analogous to BJTs, which are operated at a static current that determines their operating-point (op-pt) around which small variations occur. Magnetic cores are also operated with a static current op-pt around which small variations of ripple current occur;

Magnetics operates almost linearly when the ripple is small relative to the static (average) component of current. This is the *small-ripple approximation*. For it, Δ*W* ≈ Δ*W*_{L} = linear energy transferred through core. The op-pt is shown on the graphs of *B*(*H*) below in circuit-referred variables, *λ* and *i*, circuit flux and circuit current, and field-referred in *B* and *H*.

*Waveforms* are defined as electrical functions of time. They express the behavior of circuits and they are often too messy to work with directly, so we abstract various *waveform parameters* from them. For instance, the ideal of a constant waveform is approximated by the *average* of an actual waveform.

The *ripple factor* is defined as the ripple amplitude relative to the average;

DCM applies to a waveform that has a zero value for a finite time. Zero-crossings and extrema points at zero do not make a waveform DCM. A DCM waveform is zero in its cycle for a finite time. Thus, for *γ* = 1, the waveform is CCM and at the boundary. Ripple factor, *γ*, is important in magnetics design because

and a maximum transfer-power density, for

Choice of core material determines

Maximum power transfer through the core occurs whenever *γ* is

where *B̂*_{∼} is the maximum design power-loss value and *N*_{î} corresponds to the minimum allowable design *k*_{sat} value. As *k*_{sat} decreases, the sawtooth current waveform becomes superlinear, curving increasingly upward until comparators cannot respond quickly enough to it and

Also determined by γ is:

**Circuit Limitations of Full Core Utilization**

Maximum core utilization (max transfer power) is achieved when the field is matched to the circuit. Field and circuit are related through circuit ↔ field referrals. Circuit quantities are related to size-dependent core field quantities through turns, *N*, making *N* a central design parameter. The core field quantities are related to the material size-independent field quantities through *core geometry*: *A*, *l*, and *V*. The following table summarizes these relationships.

The resistance that the circuit presents to the primary winding is the average on-time circuit resistance,

The average on-time quantities, *V*_{p} and *I*_{p}, are their average values while not zero during the switching cycle.
*N*_{λ} is expressed in *γ:* using “Magnetic Ohm’s Law”,

where *I*_{p} is the average on-time current; under the small-ripple assumption, on-time current is approximated as constant. Solving for *N*^{2 },

*R*_{ckt} in the numerator is from the winding terminal voltage and current during on-time. The denominator can be interpreted as the steady-state field resistance, *R*_{fld}. Substituting *γ*_{opt} = *γ* in *R*_{fld},

Δ*φ* and *N*_{î} are at their design maximums as ratioed by *γ*_{opt}. *R*_{fld} can be derived alternatively from *R*_{ckt}:

Conclusions?

So when you pick a core material, you have picked an optimal ripple factor for the current waveform if you want maximum transfer power through the core. If *γ* is determined by the circuit design, then a matching core material is sought for an optimum design. *γ*_{opt} for iron-powder cores is typically 0.1 while for ungapped ferrite cores, it is typically 0.6. Inductor applications with small ripple current are thus optimized by selecting iron-powder (or other lower-*γ*_{opt}) cores. Transformers, with bipolar waveforms having unipolar (half-cycle) *γ* = 1, are optimized with high-*γ*_{opt} materials such as ferrites.

**Optimal Turns from ***N*_{λ} and *N*_{i}

We can derive an equation for the optimal turns, *N*_{opt} from *N*_{λ} and *N*_{i} as follows.

Solve the previous equation relating circuit and field resistance for *N*:

*N* is a function of

*γ*. When solved,

*N*(

*γ*) can be expressed more simply by substituting

into N(*γ*):

Then *N* simplifies to

The tradeoff for maximum transfer-power density (or put another way, minimum core volume for a given transfer power) is that circuit input voltage and current (thus, *R*_{ckt}) is determined. This might be too constraining for the circuit requirements, but by knowing it, you can choose a core material that most closely matches circuit to core field and gives design freedom for minimizing core volume for a given circuit input specification. For ferrites, “core material” is varied by introducing an air gap. Thus, there is an optimal air gap corresponding to an optimal L that results in maximum power transfer.