# Design Optimization and the Form Factor Product

Power-electronics design involves a multiplicity of tradeoffs that can be guided by criteria that are useful in assessing how close a design is to optimum. This article surveys some common guiding parameters and introduces a somewhat new optimizing parameter for converter switching efficiency, the form factor product .

Various methods have been advanced to achieve an optimal design. Some involve optimization of magnetic components. For instance, maximum magnetic power density is achieved by the optimal choice of current ripple factor and switching frequency. Maximum power transfer across windings involves an optimal power-loss ratio of winding to core loss of where Pw and Pc are winding and core losses, Rw is winding resistance, and Rc is the equivalent core resistance, referred to the same winding as Rw . (See the book, Power Magnetics Design Optimization (PMDO) at www.innovatia.com for much more on magnetics design optimization.)

Perhaps the most important converter design parameter is efficiency, which is a measure of the power transfer through the converter: or for magnetic components, it is the power transfer, Another can be regarded as structural efficiency or power-component utilization . No part should be oversized more than necessary to provide adequate reliability margins. This can be quantified as the ratio of power handled by the component over the product of its maximum voltage and current ratings, which constitute its design power : A more detailed coverage of what the power rating of a component needs to be for a given category of converter shows that Pd max = Vmax x Imax is often a worst-case maximum, and that using it for design power ratings can cause U to be less than its maximum allowed value. The utilization of magnetic components for a range of input voltages can be determined from derived formulas that are seemingly simple. (See PMDO for derivations of the formulas.)

The operating-point of the converter duty-ratio for minimal switching loss depends on the form factors of active and passive switch currents. These parameters are now examined more closely.

Waveshapes are an Important Criterion

A careful study of converter design reveals that waveform shapes are central to the optimization quest. The ideal waveshape for static-voltage power conversion is a constant (static or dc) value. A constant waveform has the same values for peak, average, and rms. These three waveform characteristics are important in design for the following reasons: Given the significance of these waveform measures, design optimization generally seeks to minimize peak and rms values and maximize average (or in inverters, rms output) values. To quantify these design criteria more directly, optimization parameters which have historically been called figures of merit – or in some cases, figures of demerit – have been defined. The first is the The form factor of a waveform is its rms value (power lost), normalized to its average value (desired or rated power). We seek to minimize κ . Its minimum value is one and typically it ranges from 1.05 to 5 in converters. As converter power increases, it becomes increasingly important that κ decrease to minimize power loss.

Another waveform figure of demerit is the This quantity is important in inverter design, where the desired output is the rms value of a bipolar (ac) waveform. It affects the sizing of power parts, normalized to the output rms value. The crest-factor-squared of waveform voltage or current relates to its power. For a sinusoid, χ 2 = 2 while for a bipolar square wave, χ 2 = 1/D . For D = 0.5, then χ is the same as for sine-waves. In low-cost inverters, to reduce χ 2 , D is made greater than 1/2 so that typically, χ 2 ≈ 1.5. The lower value reduces peak voltage and component size while delivering the same average output power, though at a lower voltage.

As important as κ is for converter design, so also is the average ripple factor,   Ripple factor is applied in optimizing the magnetic design of magnetic components (core selection) and for converter circuit waveform optimization, to reduce power loss. Ideally, γ = 1 in both magnetic and circuit waveforms, though a constant waveform conflicts with transformer or inductor operation – hence a basic conflict in converter design.

The Form-Factor Product as a Figure of Demerit

A newer figure of demerit is the form factor product . It is the product of the form factors of currents in the active (on-time) and passive (off-time) switches of the converter. Let the active-switch (transistor) form factor be κQ and the passive-switch (diode) form factor be κD . Then the The significance of κQD becomes clearer when the tradeoff in choice of duty ratio, D , is considered. In the three basic PWM-switch converter configurations (CA or boost, CP or buck, and CL), the switch form factors are As D increases, κQ decreases along with the active-switch rms current relative to average current. The tradeoff is that κD increases and the diode becomes larger. A minimum for both taken together is the optimizing switch quantity and is the total (SPDT) PWM-switch form-factor product, κQD is minimum at D = 0.5 for PWM-switch (including Cuk-derived) converters but not for every converter topology, such as the forward converter. Form-factor product, however, still applies as a design measure.

κQD minimization not only optimizes D for the power switches; it also optimizes D for the inductor of a Cuk-derived converter. For inductor form factor, κL , κL = κQ for the CA (boost) and κL = κD for the CP (buck) configurations κQD also optimizes the inductor for the CL, such as for flyback or Cuk-derived converters.

One method (such as that used by Maniktala in Switching Power Supply Design and Optimization , McGraw Hill, 2014) optimizes converter design for inductor and input and output capacitors. The form-factor product optimizes for switches and also for inductor for the CL configuration. For constant input and output voltages, the capacitor currents are the switch currents and can be included as part of the optimization for CL-derived converters – those with D/D’ in their transfer functions.

Form-Factor Product Example

To demonstrate use of the form-factor product, the following graphs show the form-factor products for three converter topologies for a low-Rin (low voltage, high current) converter design, such as those found in localized microcomputer supplies: boost push-pull (BPP), Cuk-derived, and push-pull (PP) (which is a secondary peak-charging “chopper” converter without an intermediate energy-storage inductance). These are plotted as a function of D below. While the PWM-switch and Cuk-derived converters have a minimum (and therefore optimum) value at D = 0.5, the other two topologies are optimal at the extremes of D . The PP chopper has minimum κQD at D = 1 while the BPP is optimum at D = 0. The form-factor product also shows the superiority of the BPP over both the PP and Cuk converters for high-power applications in that it has a lower κQD over a wider range than either of the other topologies.

In low-cost inverters that operate from 12 V battery supplies, a 1.5 kW inverter will require more than 125 A of input current. A low form-factor input-current waveform is thus a high priority in efficient design. These curves show that PP switch current minimization favors the addition of a series input inductance to the PP scheme, making it a CA-derived (boost) converter. The inductor, however, must conduct the full input current, but its inductance is not large with a 12 V input. The PP circuit is found in low-cost inverters as a chopper, without the buck inductor on the secondary. They are run at a high value of D – typically in the mid to high 90 % range to minimize the form factor product. Both primary-circuit transistors and secondary-circuit diodes conduct during on-time, leading to a much-decreased κQD with D .

The form-factor product also shows that the Cuk-derived topologies are not optimal for low-Rin converters, with a minimum κQD = 2. Neither is the push-pull chopper. In 12 V lead-acid battery applications, the voltage range can vary by 33 %, placing the nominal 12 V value of D no higher than about 0.8 and the low-end value of D for high battery voltage at 0.67, corresponding to a κQD = 2.12, about the same as the Cuk converter at that duty-ratio. The BPP has κQD = φ (the Golden Ratio), which is about 1.618 for D = 1 − 1/ φ ≈ 0.382. At this value of D , κQ = κD ≈ 1.27. At worst-case, which is the high end of the battery-voltage range, D is a maximum of 0.45 at which κQD = 1.67, or 21 %, and significantly less than that of the PP. The BPP is an unexploited topology for low–Rin conversion. For this category of converters, κQ is more important than κD because of the high primary and relatively low secondary currents.

Form-factor product is a useful power-electronics design criterion and complementary to other optimization schemes. Figures of merit are guides and are not rigorous means of optimization because the optimization of various power components also depends on other factors such as size, price, availability, and control. Yet they provide insight, and I hope that if you use the form-factor product for optimizing the switch (and components in series with the active or passive switches) that you too will find it a useful addition to your design optimization repertoire.

## 3 comments on “Design Optimization and the Form Factor Product”

1. ross008
April 22, 2017

Great news on design

2. ross008
April 22, 2017

Great news on design

3. healthinfo88
May 14, 2017