Optimize Compensation with ZCS Gain/Phase Prediction

Apr 1, 2002 12:00 PM
By Akram M. Sultan, Vicor Corp., Andover, Mass.

Analytical approach demonstrates how the open-loop gain and phase prediction can determine the optimal compensation network for a zero current switched (ZCS) dc-dc, variable frequency forward converter.

By minimizing switching losses, the ZCS variable frequency forward converter provides high power density and a relatively high operating frequency. As seen in Fig. 1, an LC tank circuit using the transformer leakage inductance and a capacitor in the secondary provides the energy transfer medium between the input and output. By turning on the switch, energy transfers from the input source to the tank circuit capacitor, during which time an approximately half-sinusoidal current flows through the switch. Resonant bidirectional energy flow can't occur because the forward rectifier (D_F) only permits unidirectional energy transfer. Switch turn-on occurs at zero-current, and switch turn-off occurs when the current returns to zero. In this way, transfer of energy from the input to the output of the converter is virtually loss-free.

You can divide each switching cycle into four intervals, T1 through T4, as shown in Fig. 2, on page 48. During T1 interval, Q1, D_F, and D_S conduct, the transformer leakage inductance stores energy, and the resonant capacitor voltage is zero. In T2 interval, the shunt diode becomes reverse biased and main switch, Q1, and D_F continue conducting. Leakage inductance resonates with the resonant capacitor and the T2 interval ends when the leakage inductance current returns to zero. During T3 interval, the semiconductors don't conduct, and the resonant capacitor discharges linearly, releasing its energy to the load. T4 interval starts when the resonant capacitor voltage reaches zero, which causes the shunt rectifier to become forward biased. The total energy transfer per pulse is:

Where:
= Normalized output current
L_Lkg = Leakage inductance
N = Transformer turns ratio
ω_r =
V_in = Input voltage
I_out = Output load current

The gain of the ZCS converter is a function of the operating conditions and varies with the input voltage and load current. The average voltage produced during the converter's resonant mode is the result of three variables: Vin, Iout, and F ^[1]. The incremental control gain is proportional to the inverse of operating frequency, and the input voltage and output load current set the operating frequency.

The ZCS dc-dc power converter block diagram in Fig. 3, on page 48, indicates how the operating switching frequency changes as the operating conditions (input voltage and output current) change. When the input voltage or the output load current changes, the total energy transferred per pulse changes. The operating frequency is directly proportional to the output current and the average voltage at the resonant capacitor, and is inversely proportional to the total energy transfer per pulse.

Analytical prediction of the uncompensated open-loop transfer function of the system requires an understanding of the controller used with this converter. The controller (Fig. 4) consists of two amplifiers, one is a low-frequency, narrow bandwidth, very high gain amplifier — which, in the closed-loop system, delivers a signal whose average value is just sufficient to drive the average value of the loop error to zero. The second amplifier is a wide bandwidth, variable gain type whose gain is proportional to the output of the low-frequency amplifier. Fig. 5 shows the gain vs. frequency characteristic of this controller. The controller's transfer function is:

The low-frequency high-gain amplifier dominates below 1 kHz, and the high-frequency low-gain amplifier dominates above 1 kHz. The high frequency, low gain amplifier provides a constant mid-band loop gain and a controlled roll-off in gain above 1 MHz, which insures that the gain and phase margins at the crossover frequency are consistent with stable closed-loop operation.

Output Filter

The output filter transfer function is:

Where:
w_of = Output filter break frequency

The output filter poles are complex conjugates that lie in the left-half s-plane at ω_of. An effective non-dissipative static resistance, R_R, in the resonant mode controls the output filter's Q ^[3]:

Where:
fc = Conversion frequency
Llk_s = Secondary leakage inductance of the transformer

The output filter corner frequency is a function of the output filter inductance (L_out) and the output filter capacitance (C_out). Any variation in the output filter inductance (output filter inductance decreases as the load current increases) and the output filter capacitance causes the output filter corner frequency to change.

Capacitance of some output filter ceramic capacitors varies with the applied voltage (capacitance decreases as the applied voltage increases), which results in a variation of the output filter corner frequency. Capacitance of some tantalum output capacitors vary with frequency, and their ESR is a function of frequency. Fig. 6, on page 50, is a Bode plot of the output filter.

The operating frequency of a ZCS forward converter is a function of the variation of the input voltage and the load current. An antilog error amplifier provides predictable closed-loop system performance over a wide range of the operating conditions. The power converter's gain is known when the output voltage and the switching frequency are known; it is inversely proportional with the converter's operating frequency. Fig. 7, on page 50, shows the system's open-loop uncompensated Bode plot. Thus, the uncompensated open-loop transfer function of the system is:

Where:
G_OL=Uncompensated open-loop transfer function of the system
G_ERROR=Controller transfer function
G_OF=Output filter transfer function
G_ZCS=Forward converter transfer function

The antilog wideband controller compensates for the gain variation of the ZCS converter. The controller has a mid-band gain, above 1 kHz, of approximately 34 dB.

You can use a lead-lag compensation network to obtain the desired phase and gain margins that improve the stability of the ZCS dc-dc forward converter. Also, the lead-lag compensation can help restore the lost phase margin when the output filter capacitors have a low ESR that produces a zero at higher frequencies. The open-loop compensated system transfer function is:

Here, G_C is the transfer function of the lead-lag network which, in the frequency domain, consists of a zero followed by a pole. Whether you should add a lead-lag compensation network will depend on the predicted phase and gain margins of the open-loop system, G_OL. Fig. 8 is the compensated open-loop plot of G_{OL_comp} (s)

If lead-lag compensation does not produce an acceptable gain margin, it may be necessary to add a low frequency RC network. The low-frequency RC network will have a significant effect on the lead-lag pole frequency, but it will have no effect on the lead-lag zero frequency. Therefore, use of an RC low frequency network will require a redesign of the lead-lag network.

Fig. 8 shows the error amplifier circuit with output voltage divider and single pole-single zero lead-lag compensation network. The lower resistor of the output voltage divider (R2) can affect the dc gain of the loop and the dc gain of the closed-loop transfer function. Effects of R2 on stability can be significant due to the mid-band limited gain characteristics of the antilog error amplifier (these effects aren't encountered in the use of conventional error amplifiers).

To determine the effect of the output voltage divider's lower resistor on the loop-gain and the lead-lag compensation network, we first have to look at the closed-loop transfer function:

Where:
G_CL=Closed-loop transfer function

(Output voltage divider ratio)

When using a single zero-pole lead-lag compensation network, the pole frequency is a function of R2 while the zero frequency is independent of R2, as can be seen from the pole and zero equations:

R_C and C_C are the lead-lag compensation network components shown in Fig. 9, on page 52. You can see the benefits gained from predicting the open-loop phase and gain margins in optimizing the design of the lead-lag compensation network in an example, where:

Vin_min=180V
Vin_max=375V
Vo=18V
Io=8.33A
Po=150W

Bode plots of the predicted crossover frequency, phase margin, and gain margin of the uncompensated open-loop system at low line, full-load conditions are in Fig. 10, on page 52, and high line, full load in Fig. 11, on page 52. Bode plots of the compensated open-loop system at low line, full load are in Fig. 12 and high line, full-load conditions in Fig. 13. Looking at the Table, above, you can see this data listed.

At light load, the conversion frequency decreases and causes stability problems (open-loop gain margin attenuation), which requires open — loop gain reduction for stabilization. You can achieve this with a low-frequency lag-lead compensation network in parallel with the lower resistor of the output voltage divider.

We measured the open-loop phase and gain margins at low line, full load and high line, full load of the above design. The predicted open-loop phase and gain at LLFL and HLFL conditions agrees with the measured values as you can see in Figs. 14, 15, 16, and 17.

This algorithm has been coded and implemented in an automated bill of materials design generator program that generates feasible ZCS dc-dc forward converter designs for given specifications.

References

1. Patrizio Vinciarelli and Louis Bufano, U.S. Patent No. 5,490,057, “Feedback Control System Having Predictable Open-Loop Gain.”

2. Montminy et al, U.S. Patent No. 5,946,210, “Configuring Power Converters.”

3. Louis A. Bufano, PCIM June 1998, “Gain and Frequency Compensation Optimize Performance of ZCS DC-DC Forward Converter.”

4. IEEE Transactions on Power Electronics Vol. 4, No. 2, April 1989. Pp.205-214.

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