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, ΔW ≈ ΔWL = 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 Nî corresponds to the minimum allowable design ksat value. As ksat 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, Vp and Ip , are their average values while not zero during the switching cycle. Nλ is expressed in γ: using “Magnetic Ohm’s Law”,
where Ip is the average on-time current; under the small-ripple assumption, on-time current is approximated as constant. Solving for N2 ,
Rckt 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, Rfld . Substituting γopt = γ in Rfld ,
Δφ and Nî are at their design maximums as ratioed by γopt . Rfld can be derived alternatively from Rckt :
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. γopt 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-γopt ) cores. Transformers, with bipolar waveforms having unipolar (half-cycle) γ = 1, are optimized with high-γopt materials such as ferrites.
Optimal Turns from Nλ and Ni
We can derive an equation for the optimal turns, Nopt from Nλ and Ni 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, Rckt ) 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.