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Dynamic Analysis of Switching Converters Using Fast Analysis Techniques (Part 1)

If the transfer function of a circuit is well solved using mesh-node analysis, it is often not possible to immediately obtain a meaningful symbolic formula and additional work is required to arrive at it. Applying classical analytical techniques to obtain so-called low-entropy expressions – i.e. fractional forms from which you can identify gains, poles, and zeros – often leads to what Dr. Middlebrook has called an algebraic paralysis in his papers [1], [2].

This is where Fast Analysis Circuit Techniques (FACTs) can help you expand on what you learned in college to greatly simplify your analysis. By using FACTs, not only will your execution speed be faster, but the final result will appear in the form of an ordered polynomial, often eliminating the need for further factorization work [3], [4].

This article begins by introducing the FACTs that will be used later in this article to determine the control-to-output transfer function of a switching converter. This is a large topic and we will only scratch the surface here in the hope of inspiring you to explore further. We have chosen a voltage-mode coupled-inductor single-ended primary-inductor converter (SEPIC) operating in discontinuous conduction mode (DCM). The PWM switch [5] will be used to form the small signal model.

01

The basic principle behind FACTs is the determination of the circuit time constant – t = RC or t = L/R – where the circuit under investigation is observed under two different conditions: when the excitation signal is reduced to zero and when the response is zeroed. By using this technique, you will appreciate how fast and intuitive it is to determine a particular transfer function. Analysis techniques based on this approach began decades ago, as documented in [6] and [7].

A transfer function is a mathematical relationship that relates an excitation signal, the stimulus, to the response signal generated by that stimulus. If we consider a linear time-invariant (LTI) system with no delay and a static gain H0 – such as a linear ideal power stage of a switching converter – its transfer function H connecting the control signal Verr (excitation) and the output Vout (response) can be expressed as:

(1)

The first term, H0, is the gain or attenuation exhibited by the system evaluated at s=0. This term will have the units (or dimensions) of the transfer function, if any. If both the response and the stimulus are expressed in volts, as we have here as Verr and Vout, H is unitless. The numerator, N(s), controls the zeros of the transfer function. Mathematically, a zero is a root where the magnitude of the function is zero. With FACTs, we use mathematical abstraction to easily uncover these zeros. Rather than considering only the vertical axis in the s plane as is usually done in harmonic analysis (s=jw), we cover the entire plane taking into account negative roots.

Therefore, if the circuit has a zero point, it will appear as a signalless output response when the input signal is tuned to zero angular frequency sz. In this case, some impedance in the deformed circuit blocks the signal propagation and the response is zero, despite the presence of an excitation source: when the deformed circuit is excited at s=sz, the series impedance in the signal path tends to infinity or branches, shunting the excitation to ground.

Note that this convenient mathematical abstraction provides a tremendous help in finding the zeros by observation, often without having to write a line of passive circuit algebra. Figure 1 provides a simple flow chart that details the process. More details on this approach can be found in [8].

Figure 1

The denominator D(s) consists of the natural time constants of the circuit. These time constants are found by setting the excitation signal to zero and determining the impedance "seen" by the capacitor or inductor in question if it were temporarily removed from the circuit.

By "observation", you can imagine placing an ohmmeter across the temporarily removed energy storage element (C or L) and reading the resistance it displays. This is actually a fairly simple application, as detailed in the second flow chart in Figure 2.

Figure II

Looking at Figure 3, there is a first-order passive circuit involving an injection source—the excitation source—biasing the network on the left. The input signal Vin propagates through the grid and nodes, forming the response Vout you see on resistor R3. We are interested in deriving the transfer function G connecting Vout and Vin.

Figure 3

To determine the time constant of this example circuit, we set the excitation source to zero (a short circuit replaces the 0V voltage source and an open circuit replaces the 0A current source) and remove the capacitor. We then connect an ohmmeter to determine the resistance presented by the capacitor terminals. Figure 4 guides you through these steps.

Figure 4

If you use the approach of Figure 4, you "see" R1 in parallel with R2 in series with R4, all of which are in parallel with R3 in series with rC. The time constant of this circuit can be calculated using only R and C1:

(2)

We can show that the poles of a first-order system are the inverse of its time constant. Therefore:

(3)

Now, what is the quasi-static gain of this circuit when s = 0? Under DC conditions, the inductor is shorted and the capacitor is open. Applying this concept to the circuit of Figure 3, it is drawn as shown in Figure 5. Imagine disconnecting the connection before R4, and you see a resistor divider consisting of R1 and R2. The Thévenin voltage across R2 is:

(4)

The output resistance Rth is the value of R1 in parallel with R2. Therefore the complete transfer function involves a resistor divider (formed by R4 in series with Rth and R3 as loading). rC is disconnected since capacitor C1 is removed in this DC analysis. Therefore:

(5)

Figure 5

That’s basically it, we are missing the zero. As we mentioned earlier, the zero manifests itself in the circuit by blocking the propagation of the excitation signal, resulting in a signalless output response (see Figure 1).

If we consider a variant of the circuit – where C1 is replaced by – as shown in Figure 6, what specific conditions mean no signal response when the excitation source biases the circuit? No signal response simply means that the current flowing through R3 is zero. This is not a short circuit, but is equivalent to a virtual ground.

Figure 6

If there is no current in R3, then the series connection of rC is transformed into a short circuit:

(6)

The root sz is the desired zero position:

(7)

Thus:

(8)

We can now combine all of these results to form the final transfer function that characterizes the circuit of Figure 3:

(9)

This is the so-called low entropy expression, from which you can immediately identify the static gain G0, the poles wp and the zeros wz. A high entropy expression is obtained by applying a large-scale external force to the original circuit when considering an impedance divider, such as:

(10)

Not only can you make mistakes when deriving the expression—but formatting the result like (9) requires more effort. Also, note that in this particular example, we did not write a single line of algebra when writing (9). If we later discover a mistake, it is easy to go back to a single drawing and fix it individually. The correction of (9) is simple. Now try to make the same correction to (10), and you will probably start from scratch.

02

FACTs are equally applicable to n-order passive or active circuits. The order of the circuit is determined by counting the number of storage elements whose state variables are independent. If we consider a second-order system with a finite static gain H0, its transfer function can be expressed as follows:

When H0 has the units of the transfer function, then the ratio N:D has no units. This means that the units of a1 and b1 are time [s]. When a1 has no signal response and the stimulus for b1 is zero, you add the determined time constants.

For the second-order coefficients, a2 or b2, the dimension is time squared [s²], and you combine the time constants into a product. However, in this time constant product, you reuse a time constant that has already been determined as a1 or b1, while the determination of the second-order time constant requires a different sign:

(12)

In this definition, you set the energy storage element whose label appears in the "power" to a high frequency state (capacitor is short-circuited, inductor is open-circuited), and when we temporarily remove the second-order element terminal from the circuit (see subscript), you can determine the resistance from it. You can use this method when a2 must be the output without a signal and the excitation source of b2 is reduced to 0.

Of course, when observation is useful, it is always the fastest and most efficient way to find N. It may seem a bit hard to understand at first glance, but it is nothing insurmountable and we will explain it in a few words.

Figure 7

Figure 7 shows a classic second-order filter used to determine the output impedance of a voltage-mode buck converter operating in continuous conduction mode (CCM). Impedance is a transfer function connecting an excitation signal, Iout, to a response signal, Vout.

Here, Iout is our installed test generator and Vout is the voltage generated across it. To determine the various coefficients from (11), we follow the flow chart of Figure 2, starting with s = 0: As shown, the inductor is short-circuited and the capacitor is open-circuited. The circuit is simple, and the resistance R0 of the current source is just a simple parallel combination of rL and Rload:

(13)

Is there a zero point in this circuit? Let's look at the variant circuit shown in Figure 8. Let's see what combination of components will make the response Vout zero when the excitation source current Iout is adjusted to zero angular frequency sz. We can find that the two variant short circuits involve rL–L1 and rc–C2.

Figure 8

The roots of these two impedances are immediately determined:

(14)

(15)

Therefore, the denominator N(s) is expressed as

(16)

The first-order coefficient b1 of the denominator D(s) is provided by the impedance across L1 while C2 is in the DC state (open circuit): we have t1. Then we look at the impedance when driving C2 while L1 is set to the DC state (short circuit): we get t2. As shown in Figure 9, the definition of b1 is immediately obtained from this sketch:

(17)

Figure 9

The second-order coefficient b2 is determined using the sign introduced in (12). L1 is set in its high-frequency state (open circuit), driving C2 to obtain the impedance, and C2 is in its high-frequency state (short circuit), driving L1 to obtain the impedance. Figure 10 shows two possible sorting results.

You usually choose the simplest expression, or one that avoids uncertainty, if any (such as ∞×0 or ∞/∞). The following two definitions of b2 are equivalent, and you can see that the above is the simplest:

(18)

Now we have all the ingredients to combine into the final transfer function, defined as:

(19)

We have determined this transfer function without writing a single line of algebra, just by breaking the circuit down into several simple sketches to solve individually.

Again, as expected, (19) is already a canonical expression, and you can easily see a static gain, two zeros, and a second-order denominator that can be further organized with a resonant component w0 and a quality factor Q. We would not have gotten this result if we had not quickly considered the parallel combination of Z1, Z2, and Rload.

Figure 10

With FACTs, transfer functions can be derived by observation, especially for passive circuits. Because the circuit is complex and includes voltage or current controlled sources, it is not so obvious to observe, and you need to use classic mesh and node analysis. But FACTs provides several advantages: Because you break the circuit into small individual sketches that are used to determine the coefficients of the final polynomial expression, if you find an error in the final expression, you can always go back to a specific drawing and correct it individually.

Furthermore, when you determine the terms related to ai and bi of the transfer function, you naturally obtain a polynomial expression without having to invest further effort in collecting and rearranging the terms. Finally, as shown in [4], SPICE can be of great help in verifying the calculation of individual poles and zeros in complex passive and active circuits.

1. R. D. Middlebrook, Methods of Design-Oriented Analysis: Low-Entropy Expressions, Frontiers in Education Conference, Twenty-First Annual conference, Santa-Barbara, 1992.

2. RD Middlebrook, Null Double Injection and the Extra Element Theorem, IEEE Transactions on Education, Vol. 32, NO. 3, August 1989.

3. V. Vorpérian, Fast Analytical Techniques for Electrical and Electronic Circuits, Cambridge University Press, 2002.

4. C. Basso, Linear Circuit Transfer Functions – An Introduction to Fast Analytical Techniques, Wiley, 2016.

5. V. Vorpérian, Simplified Analysis of PWM Converters Using the Model of the PWM Switch, Parts I and II, Transactions on Aerospace and Electronics Systems, vol. 26, no. 3, May 1990.

6. D. Feucht, Design-Oriented Circuit Dynamics, http://www.edn.com/electronics-blogs/outside-the-box-/4404226/Design-oriented-circuit-dynamics

7. D. Peter, We Can do Better: A Proven, Intuitive, Efficient and Practical Design-Oriented Circuit Analysis Paradigm is Available, so why aren't we using it to teach our Students?,

http://www.icee.usm.edu/ICEE/conferences/asee2007/papers/1362_WE_CAN_DO_BETTER__A_PROVEN__INTUITIVE__E.pdf

8. C. Basso, Fast Analytical Techniques at Work with Small-Signal Modeling, APEC Professional Seminar, Long Beach (CA), 2016, http://cbasso.pagesperso-orange.fr/Spice.htm

9. J. Betten, Benefits of a coupled-inductor SEPIC, slyt411, application note, Texas-Instruments.

10. C. Basso, Switch-Mode Power Supplies: SPICE Simulation and Practical Designs, McGraw-Hill, 2nd edition, 2014.

11. D. Maksimovic, R. Erickson, Advances in Averaged Switch Modeling and Simulation, Power Electronic Specialist Conference Professional Seminar, Charleston, 1999