Analysis of Multiphase DC-DC Converter

Dear All,

I am designing a multiphase DC-DC boost converter working in DCM mode of operation. However, I am not sure how to go about the small-signal analysis of a multiphase converter. I know that all the phases have to be mapped to an equivalent single-phase circuit and then the standard state-space averaging technique can be applied. The problem is what should be the values of the components and design parameters (viz. Inductance [L], Switching Frequency [Fs]) for that equivalent single-phase circuit. I am unable to find a suitable paper which may enlighten me. Please help me regarding this.

Regards
Samiran.

State space averaging can be applied directly to multiphase converters, but it doesn't really change the results. An N phase converter can be modeled as a single phase converter with the impedance scaled down by 1/N. In the case of a voltage mode converter this means your effective inductance is reduced, for a current mode converter it means the effective output current is increased, etc.

Design of multiphase converters is usually guided by ripple requirements, so you should start there.

mtwieg said:

State space averaging can be applied directly to multiphase converters, but it doesn't really change the results. An N phase converter can be modeled as a single phase converter with the impedance scaled down by 1/N. In the case of a voltage mode converter this means your effective inductance is reduced, for a current mode converter it means the effective output current is increased, etc.

Design of multiphase converters is usually guided by ripple requirements, so you should start there.

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Hi mtwieg,

In my case it is a voltage-mode control. So, according to your suggestion I should decrease the inductance (L) by 1/N times. What about the switching frequency (F_s)? Will it go up N times? And the load current, in case of voltage-mode control, would not be changed, right?

Actually, I have seen one paper where, for multiphase buck converter, it was shown that equivalent L = (actual L in each phase)/N and equivalent F_s=actual F_s*N. In case of buck converter, this explains well because from the average sense, the input port can be modeled as DV_g and rest is just a L-C filter. So, each phase is modeled by its averaged input voltage and the L's are coming parallel. So equivalent L is L/N. However for the boost converter, I am not getting any paper which explains how to derive the equivalent single-phase circuit.

So, if you have any knowledge about any good paper which addresses this issue, please point me.

Regards
Samiran.

samiran_dam said:

In my case it is a voltage-mode control. So, according to your suggestion I should decrease the inductance (L) by 1/N times.

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No, that's not a design recommendation, it's just what happens to the effective inductance of a multiphase converter.

Actually, I have seen one paper where, for multiphase buck converter, it was shown that equivalent L = (actual L in each phase)/N

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Correct, that's what I meant.

and equivalent F_s=actual F_s*N.

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No, this isn't really correct. Keep in mind that from a state space averaging perspective, the switching frequency is completely irrelevant (except in determining what frequency range the SSA model is valid for). In terms of input and output ripple, a multiphase converter will also be different from an equivalent single phase converter operating at N times the frequency. The ripple vs duty cycle dependence for multiphase converters is very different to single phase converters.

However for the boost converter, I am not getting any paper which explains how to derive the equivalent single-phase circuit.

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It's the same result. It's a somewhat trivial thing, so I'm not aware of any papers that actually derive it, but it shouldn't be difficult to do so by hand.

Here is the datasheet for an interleaved boost PFC (with average-CMC control) controller I've used in the past. It doesn't show any fancy derivations, but it does demonstrate some of the benefits of multiphase converters, and how to design for them. Though since it's a CMC operating in CCM, it's not exactly what you want. For deriving the input and output ripple for a multiphase DCM boost, I doubt you'll find any references giving the formula, but it is not very difficult to derive by hand. I can assist if you need help.

mtwieg said:

No, this isn't really correct. Keep in mind that from a state space averaging perspective, the switching frequency is completely irrelevant (except in determining what frequency range the SSA model is valid for).

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I think for DCM, I need to know the switching frequency. So from DCM point of view, I need to ensure what would be the equivalent switching frequency. Am I correct?

mtwieg said:

In terms of input and output ripple, a multiphase converter will also be different from an equivalent single phase converter operating at N times the frequency. The ripple vs duty cycle dependence for multiphase converters is very different to single phase converters.

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Let me do some analysis and simulations to convince myself with this point.

mtwieg said:

For deriving the input and output ripple for a multiphase DCM boost, I doubt you'll find any references giving the formula, but it is not very difficult to derive by hand.

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I am not really bothered about the derivation of input current ripple or output voltage ripple by the equivalent single-phase circuit. My motivation is to derive the control-to-output transfer function of the converter in DCM and then verify the same by simulating the equivalent circuit in Cadence (through PSS/PAC analysis).

mtwieg said:

I can assist if you need help.

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Surely, that would be helpful. Let me start with the knowledge I have gathered from you. I will surely ping you whenever I stumble . Meanwhile, it would be very helpful if you find any (I mean "any") document regarding this.

samiran_dam said:

I think for DCM, I need to know the switching frequency. So from DCM point of view, I need to ensure what would be the equivalent switching frequency. Am I correct?

Click to expand...

I suppose that for DCM converters, frequency can be thought of as affecting the state space model, in that the relationship between average currents/voltages and duty cycle will depend on frequency (which is not the case with CCM operation). However, when actually deriving the SSA model, what you ultimately want is your bias conditions, partial state matrices, and interval weights, all of which are affected by frequency.

I am not really bothered about the derivation of input current ripple or output voltage ripple by the equivalent single-phase circuit.

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That's strange, because usually the main objective of using multiphase converters is to lower ripple, sometimes almost eliminating it completely. It's also quite uncommon for interleaving to be applied to DCM converters, or to voltage mode converters. In fact, you may end up being concerned about imbalances in duty cycle between phases which lead to strange behavior. Such is a problem with voltage mode interleaved converters, but I'm not sure about voltage mode ones.

mtwieg said:

I suppose that for DCM converters, frequency can be thought of as affecting the state space model, in that the relationship between average currents/voltages and duty cycle will depend on frequency (which is not the case with CCM operation). However, when actually deriving the SSA model, what you ultimately want is your bias conditions, partial state matrices, and interval weights, all of which are affected by frequency.

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Yes, reduction of ripple and improvement of efficiency are the primary motivations to go for the interleaved converter and same is true in my case. However, for the time being, I want to derive the small-signal transfer function of the converter. Hence, right now I am concerned about that.

As you agreed that frequency is of concern for DCM, then what about the equivalent Switching Frequency - what will be its value?

Another thing is the equivalent converter will not give exactly same transient performance (switching ripple, settling and overshoot) as that by the actual multiphase converter, right? If that is true, then the equivalence only lies from the averaged perspective - is it so? - please explain.

Going by a simplistic approach, if I derive the G_vd(s) considering only one phase (not equivalent, just one of the N phases) considering the load current divided by N (as the load current is equally shared by all the phases), and then design the compensator based on that, will it be a problem when actual multiphase converter is working is closed loop? Please have a look at the attached plots (from simulation). Here I have shown the inductor current ripple and output voltage ripple for a single-phase and a 5-phased DCM boost converter. L=1 uH (for each phase), F_s=500 KHz, C=10 uF, V_g=5 V. For the 5-phased converter, load current is 1 A while for the single-phase, the load current is scaled down to 1/5 i.e. 0.2 A. Duty ratio (D) is same (=24.4%) for both the cases. Inductor current ripple is same for both the cases, however, average output voltage is 12 V in case of 5-phase whereas it is 11.77 V in case of single-phase. So, what is your suggestions?

It's also quite uncommon for interleaving to be applied to DCM converters, or to voltage mode converters. In fact, you may end up being concerned about imbalances in duty cycle between phases which lead to strange behavior. Such is a problem with voltage mode interleaved converters, but I'm not sure about voltage mode ones.

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If I am controlling all the phases by a single compensator then what could cause the imbalance of duty cycle? I am not clear about this point.

Many many thanks for your time for addressing my concerns!!! I really appreciate that :grin:

samiran_dam said:

As you agreed that frequency is of concern for DCM, then what about the equivalent Switching Frequency - what will be its value?

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I'm not sure that trying to go directly to a single phase equivalent circuit with an "equivalent switching frequency" is the right approach for a DCM converter... at least if you take that approach then you should find biasing conditions before converting to a single phase model (since for biasing in very nonlinearly dependent on frequency and duty cycle). Then once biasing for the phases is established you could find the SSA model for each one (with 1/N of the load impedance) or combine them into a single phase model (with 1/N of the inductance).

Another thing is the equivalent converter will not give exactly same transient performance (switching ripple, settling and overshoot) as that by the actual multiphase converter, right? If that is true, then the equivalence only lies from the averaged perspective - is it so? - please explain.

Click to expand...

I would only expect equivalent performance for averaged small signal disturbances.

Going by a simplistic approach, if I derive the G_vd(s) considering only one phase (not equivalent, just one of the N phases) considering the load current divided by N (as the load current is equally shared by all the phases), and then design the compensator based on that, will it be a problem when actual multiphase converter is working is closed loop? Please have a look at the attached plots (from simulation). Here I have shown the inductor current ripple and output voltage ripple for a single-phase and a 5-phased DCM boost converter. L=1 uH (for each phase), F_s=500 KHz, C=10 uF, V_g=5 V. For the 5-phased converter, load current is 1 A while for the single-phase, the load current is scaled down to 1/5 i.e. 0.2 A. Duty ratio (D) is same (=24.4%) for both the cases. Inductor current ripple is same for both the cases, however, average output voltage is 12 V in case of 5-phase whereas it is 11.77 V in case of single-phase. So, what is your suggestions?

If I am controlling all the phases by a single compensator then what could cause the imbalance of duty cycle? I am not clear about this point.

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For a multiphase VMC you'll need interleaved ramp signals which are ideally identical except they're out of phase. But in reality there will be offsets and slope differences, as well as skew in the PWM signal path, so you should expect some imbalance in the phase duty cycles. This will lead to current imbalances, which can potentially defeat your attempts at analysis. On the other hand, if you are in DCM then such errors probably won't have a huge effect since their effects die off every cycle.

If you want to start analyzing, I recommend just doing it in the following approach:
1. Remember you're in DCM so the dynamic analysis is pretty simple. You might not even have to use SSA.
2. Figure out the amount charge delivered by each inductor per cycle based on the bias conditions.
3. Figure out the derivative of that charge vs d.
4. That turns boost converters into controllable current sources (in the averaged small signal sense). They feed into the output capacitor and load, so you'll end up with a single pole response (with a zero if you include capacitor ESR).
5. The effect of multiple phases is just to multiple that current source be N, which is the same thing you'd get if you took a single phase and analyzed it with L/N (but with the same switching frequency!!).

mtwieg said:

The effect of multiple phases is just to multiple that current source be N, which is the same thing you'd get if you took a single phase and analyzed it with L/N (but with the same switching frequency!!).

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Hi mtweig,

I have attached some simulation results (dynamic and steady-state responses) -

1. CCM boost converters - with 2 phases and equivalent single phase (Leq=L/2, Fsw=2*Fsw)
2. DCM boost converters - with 2 phases and equivalent single phase (Leq=L/2, Fsw=Fsw)

I am eagerly waiting for your comments. Is the equivalent circuit's dynamics acceptable in case of DCM???

Why are you trying to use large-signal transient response to verify the small-signal behavior of a very nonlinear system? The results are believable, but they don't seem to prove anything either.

Also please don't attach BMP images...

mtwieg said:

Why are you trying to use large-signal transient response to verify the small-signal behavior of a very nonlinear system? The results are believable, but they don't seem to prove anything either.

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Hi mtweig,

I have done a PSS/PAC analysis of the 2-phased and equivalent single phase (Leq=L/2) circuits. Attached is the small-signal transfer function plot. There is a mismatch of 6 dB in dc Gain. The equivalent circuit gives 30 dB, however the actual circuit provides about 24 dB of DC gain.

Average output voltage (Vo) of the equivalent circuit is 11.7V and duty(D)=0.6. So, theoretically, DC Gain= Vo/(1-D) which is equal to 29.32 dB. So, that means the DC gain of the equivalent circuit measured from the PSS/PAC analysis is correct. Then why there is a 6 dB mismatch in the actual circuit? When I operated both the channels of the actual circuit with same PWM clocks that is, both the switches work in-phase, then the DC gain is 30 dB which is matching with the equivalent circuit's gain. Hence this suggests that the phase-shifted operation of two switches actually causing the fall of 6 dB gain in the actual circuit. Am I right? If this is so, then is there any way to incorporate this characteristic of the actual circuit in the equivalent one?

Sam.

First of all that DC gain equation is a large-signal equation, not a small signal gain. Second, it's only valid for CCM operation. In DCM, the DC gain will depend on the load, inductance, frequency, and duty cycle. You should try and derive it yourself, then you will probably see where things went wrong.