Transient Response Counts When Choosing Phase Margin

Nov 1, 2008 12:00 PM
By Christophe Basso, Applications Manager, ON Semiconductor, Toulouse, France

Looking for Roots

A study of Eq. 2's denominator reveals the roots for which H(s) goes to infinity. Mathematically, it corresponds to:

In Eq. 7, the term under the square root can either be positive or negative, depending on the quality coefficient value. For values below 0.5, the so-called overdamped case, the term under the square root remains positive and both roots s₁ and s₂ are separated real roots.

The step response is sluggish, as shown in Fig. 2. When the quality coefficient reaches 0.5, called the critically damped case, the roots are still real but are now coincident. The step response is much faster, but still does not exhibit overshoot.

Now, if the quality coefficient grows further, this is an underdamped case and the roots welcome an imaginary portion that increases as the quality coefficient goes up. This results in a fast-step response now featuring overshoot and oscillations.

If the quality coefficient reaches infinity, the real portion of roots s₁ and s₂ fades away and the system freely oscillates. This means there is no more damping (losses) brought by the real terms. Analyzing the trajectory of these roots is called root locus analysis. Such an analysis shows how the roots are positioned in the s-plane and give an indication of how they move relative to some parameters.

Keep in mind that it is Q in this example here, but it could be the gain k of a system where, at some point when k increases, the roots migrate in the right-half plane and cause instability. Fig. 3 describes the path taken by s₁ and s₂ as the quality coefficient changes.

Approximation of an Open-Loop Response

Based on what has already been disclosed, it would be interesting to model the closed-loop dc-dc converter with an equation where a quality coefficient term would appear. That way, a designer could select the parameter that affects this quality coefficient to shape the output response he or she is looking for: a response that is slow but without any overshoot, or vice versa, a response that is faster but accepts a little overshoot. Let us start the derivation process by looking at Fig. 4.

Fig. 4 shows the complete loop gain T(s) made of the converter power-stage transfer function, H(s), further shaped by the compensator transfer function, G(s). The example here is dealing with a continuous-conduction mode (CCM) buck converter operated in voltage-mode control. In this figure, concentrate on the area around the crossover frequency, which represents one important design parameter of the dc-dc converter trying to be stabilized. Asymptotically looking at the curve within the frame reveals the effects of an origin pole (ω₀) and a high frequency pole (ω₂). Mathematically, this approximation is:

In this approximated expression, extra poles and zeros are considered far away from the crossover frequency, naturally limiting their impact on the transfer function. However, what is interesting is the response the dc-dc converter delivers once its loop is closed. In other terms, let us identify the closed-loop transfer function derived from Eq. 8. To obtain the closed-loop expression, evaluate:

Eq. 9 is similar in form to Eq. 2. Therefore, it can be put under the familiar form of a second-order system as described in Eq. 10:

The identification of the quality coefficient and the resonant frequency is straightforward:

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