“Modern signal processing system designs that utilize ADCs, PLLs, and RF transceivers often require lower power consumption and higher system performance. Choosing the right power supply for these noise-sensitive devices is always a challenge for the system designer. These designs always require a trade-off between high efficiency and high performance.

“

**Introduction**

Modern signal processing system designs that utilize ADCs, PLLs, and RF transceivers often require lower power consumption and higher system performance. Choosing the right power supply for these noise-sensitive devices is always a challenge for the system designer. These designs always require a trade-off between high efficiency and high performance.

Traditionally, LDO regulators have been used to power noise-sensitive devices. LDO regulators suppress low-frequency noise often found in system power supplies and provide clean power supplies for ADCs, PLLs, or RF transceivers. But LDO regulators are generally less efficient, especially in those systems where the LDO regulator must step down a supply rail several volts above the output voltage. In this case, LDO regulators typically provide 30% to 50% efficiency, while switching regulators are used to achieve efficiencies of 90% or more.

Switching regulators, while more efficient than LDO regulators, are too noisy to directly power ADCs or PLLs without significantly degrading their performance. One of the noise sources of switching regulators is output ripple, which can appear as distinct tones or spurs in the ADC’s output spectrum. To avoid degrading the signal-to-noise ratio (SNR) and spurious-free dynamic range (SFDR), it is important to minimize the output ripple and output noise of the switching regulator.

To maintain high efficiency and high system performance at the same time, it is usually necessary to add a secondary LC filter (L2 and C2) at the output of the switching regulator to reduce ripple and suppress noise (as shown in Figure 1). However, the two-stage LC output filter also has corresponding disadvantages. Ideally, the power stage transfer function is modeled as a fourth-order system, which is very unstable. If the sampled data effect of current loop 1 is considered again, the complete control-to-output transfer function is a fifth-order system. Another alternative solution is to sense the output voltage at the primary LC filter (L1 and C1) points to stabilize the system. However, when the load current is large, applying this method results in poor output voltage regulation due to the large voltage drop across the secondary LC filter, which is unacceptable in some applications.

This paper presents a new hybrid feedback method that enables the application of switching regulators with secondary LC filters to provide high-efficiency, high-performance power supplies to ADCs, PLLs, or RF transceivers, while operating under all load conditions provides sufficient stability margin and maintains output accuracy.

There have been published research articles 2-5 on DC-DC converters with secondary LC output filters, specifically, Control of Secondary DC-DC Converters with Low Voltage/High Current Outputs Loop Design” and “Comparative Evaluation of Multi-Loop Control Schemes for High-Bandwidth AC Power Supplies with Two-Stage LC Output Filters” discuss the modeling and control of a two-stage voltage-mode converter (the converter). not directly applicable to current mode converters). The articles “Secondary LC Filter Analysis and Design Techniques for Current-Mode Controlled Converters” and “Three-Loop Control for Multi-Module Converter Systems” discuss the advantages of current-mode converters with secondary LC filters. Analysis and Modeling. However, both articles assume that the inductance value of the secondary inductance is much smaller than the primary inductance, which is not always appropriate in practical applications.

Figure 1. Circuit diagram of a current-mode buck converter with a secondary LC filter.

This article analyzes the small-signal modeling of a buck converter with a secondary LC filter. A new fifth-order control-to-output transfer function is proposed that is very accurate regardless of the peripheral inductance and capacitance parameters. A new hybrid feedback method is proposed to maintain good DC accuracy of the output voltage while providing sufficient stability margin. The limit value of the feedback parameter is analyzed for the first time, which provides the basic basis for the actual design. Based on the power stage small signal model and the new hybrid feedback method, a compensation network is designed. The stability of the closed-loop transfer function was evaluated using Nyquist plots. A simple design example based on the power management product ADP5014 is provided. With the secondary LC filter, the output noise performance of the ADP5014 is even better than that of an LDO regulator in the high frequency range.

Appendices I and II list the required small-signal transfer functions for the power stage and feedback network, respectively.

**Power Stage Small Signal Modeling**

Figure 2 shows a small signal block diagram corresponding to Figure 1. The control loop consists of an inner current loop and an outer voltage loop. The sampled data coefficient He(s) in the current loop refers to the model proposed by Raymond B. Ridley in “A Novel Continuous Time Model for Current Mode Control”. Note that in the simplified small-signal block diagram shown in Figure 2, the input voltage disturbance and load current disturbance are assumed to be zero, as the input voltage and load current dependent transfer functions are not discussed in this paper.

Figure 2. Small-signal block diagram of a current-mode buck converter with a secondary LC filter.

**Buck Converter Example**

The new small-signal model demonstrated using a current-mode buck converter has the following parameters:

► Vg = 5V

► Vo = 2 V

► L1 = 0.8 μH

► L2 = 0.22 μH

► C1 = 47 μF

► C2 = 3 × 47 μF

► RESR1 = 2 mΩ

► RESR2 = 2 mΩ

► RL = 1Ω

► Ri = 0.1 Ω

► Ts = 0.833 μs

**current loop gain**

The first transfer function we care about is the current loop gain measured at the output of the duty cycle modulator. The resulting current loop transfer function (see Equation 16 in Appendix I) behaves as a fourth-order system with two pairs of complex conjugate poles, which produces two system resonant frequencies (ω1 and ω2). Both of these resonant frequencies are determined by L1, L2, C1 and C2. The load resistance RL and C1 and C2 create the main zero. A pair of complex conjugate zeros (ω3) is determined by L2, C1 and C2. Additionally, the sampled data coefficients He(s) in the current loop will introduce a pair of complex right half-plane (RHP) zeros at 1/2 the switching frequency.

Compared to a conventional current-mode buck converter without a secondary LC filter, the new current loop gain adds a pair of complex conjugate poles and a pair of complex conjugate zeros that are placed very close to each other.

Figure 3. Buck converter current loop gain.

Figure 3 shows the current loop gain plot with different external ramp values. For the case without external slope compensation (Mc = 1), it can be seen that the phase margin in the current loop is very small, which can lead to sub-harmonic oscillations. By adding external slope compensation, the shape of the gain and phase curves will not change, but the magnitude of the gain will decrease and the phase margin will increase.

**Control to output gain**

When the current loop is closed, a new control-to-output transfer function is created. The resulting control-to-output transfer function (see Equation 19 in Appendix I) behaves as a fifth-order system with one dominant pole (ωp) and two pairs of complex conjugate poles (ωl and ωh). The dominant pole is mainly determined by the load resistances RL, C1, and C2. The lower frequency pair of conjugate poles is determined by L2, C1, and C2, while the higher frequency pair is located at 1/2 the switching frequency. In addition, the ESR of C1 and the ESR of C2 affect the two zeros respectively.

Figure 4 shows the control-to-output loop gain plot with different external ramp values. Compared to a conventional current-mode buck converter, a pair of complex conjugate poles (ωl) is added to the control-to-output gain of a current-mode buck converter with a secondary LC filter. Additional resonant poles can provide additional phase delay up to 180°. The phase margin will drop sharply, and even with a Type III compensation system, it will be very unstable. Furthermore, Figure 4 clearly shows the transition from current mode control to voltage mode control with the addition of slope compensation.

Figure 4. Control-to-Output Transfer Function for a Buck Converter

**Hybrid feedback method**

This paper will introduce a new hybrid feedback structure, as shown in Fig. 5(a). The idea of hybrid feedback is to stabilize the control loop by utilizing additional capacitive feedback from the primary LC filter. The external voltage feedback from the output through the resistor divider is defined as the remote voltage feedback, while the internal voltage feedback through the capacitor CF will be defined as the local voltage feedback in the following. Remote feedback and local feedback carry different information in the frequency domain. Specifically, the remote feedback detects low frequency signals to provide good DC output regulation, while the local feedback detects high frequency signals to provide good AC stability for the system. Figure 5(b) shows a simplified small-signal block diagram corresponding to Figure 5(a).

Figure 5. A current-mode buck converter using the proposed hybrid feedback approach, with the circuit diagram in (a) and the small-signal model in (b).

**The transfer function of the feedback network**

The equivalent transfer function of the resulting hybrid feedback structure (see Equation 31 and Equation 32 in Appendix II) is significantly different from that of conventional resistive divider feedback. The transfer function of the new hybrid feedback has more zeros than poles, and the extra zeros will result in a 180° phase advance at the resonant frequency determined by L2 and C2. Therefore, with the hybrid feedback approach, the additional phase delay in the control-to-output transfer function will be compensated for by additional zeros in the feedback transfer function, which enables a compensation design based on the entire control-to-feedback transfer function.

**Limits for feedback parameters**

In addition to those in the power stage, two parameters are included in the feedback transfer function. As is well known, the parameter β (see Equation 30 in Appendix II) is the output voltage amplification. The parameter α is a completely new concept.

The feedback parameter α (see Equation 29 in Appendix II) can be adjusted to understand the behavior of the feedback transfer function. Figure 6 shows the trend of the zero point in the feedback transfer as α decreases. The figure clearly shows that as α gradually decreases, a pair of conjugate zeros will advance from the left half-plane (LHP) to the RHP.

Figure 6. The effect of the feedback parameter α on the zero point of the feedback network.

Figure 7 is a graph of feedback transfer functions with different values of alpha. It shows that when α is reduced to 10-6 (eg: RA = 10k, CF = 1 nF), the transfer function of the feedback network will exhibit a phase delay of 180°, which means that the complex zero has become the RHP zero. The feedback transfer function has been simplified to a new form (see Equation 33 in Appendix II). To keep the zero point in the LHP, the parameter α should always satisfy the following conditions:

(Formula 1)

Equation 1 gives the minimum limit reference for the feedback parameter α. As long as this condition is met, the control system can easily remain stable. However, since RA and CF will work as RC filters for output voltage changes during load transient jumps, the load transient performance will be degraded by large values of α. So the alpha value should not be too large. In practical designs, it is recommended that the parameter α be about 20% to 30% larger than the minimum limit.

Figure 7. Transfer functions of hybrid feedback networks with different parameters α.

**Loop Compensation Design Design Compensation**

The control-to-feedback transfer function GP(s) can be derived by the product of the control-to-output transfer function Gvc(s) and the feedback transfer function GFB(s). The compensation transfer function GC(s) is designed to have one zero and one pole. The asymptotic Bode plots of the control-to-feedback transfer function and the compensation transfer function and the closed-loop transfer function TV(s) are shown in Figure 8. The following steps illustrate how to design a compensation transfer function.

Determine the crossover frequency (fc). Since the bandwidth is limited by fz1, it is recommended to choose fc smaller than fz1

Calculate the gain of GP(s) at fc and the mid-band gain of GC(s) should be the inverse of GP(s)

Place the compensation zero at the dominant pole (fp1) of the power stage

Place the compensation pole at the zero (fz2) created by the ESR of the output capacitor C1.

Figure 8. Loop gain design based on the proposed control-to-output and hybrid feedback transfer functions.

**Stability Analysis Using Nyquist Plots**

According to Figure 8, the closed-loop transfer function TV(s) has passed the 0 dB point three times. The Nyquist plot is used to analyze the stability of the closed-loop transfer function, as shown in Figure 9. Since the graph is far away from (C1, j0), the closed loop is stable and has sufficient phase margin. Note that points A, B, and C in the Nyquist plot correspond to points A, B, and C in the Bode plot.

Figure 9. Nyquist plot of closed-loop transfer function.

Figure 10. RF transceiver powered by the ADP5014 with a secondary LC filter.

**Design example**

The ADP5014 is optimized for many analog blocks to achieve lower output noise in the low frequency range. The unity-gain voltage reference structure also makes the output noise independent of the output voltage setting when VOUT is set to be less than the VREF voltage. A secondary LC filter was added to the design, which attenuates output noise in the high frequency range, especially for the switching ripple and its harmonics at the fundamental. Figure 10 shows the design details.

Figure 11 shows the noise spectral density measurements from 10 Hz to 10 MHz and the integrated rms noise from 10 Hz to 1 MHz for the ADP5014 compared to the ADP1740, another conventional 2A Low noise LDO regulator. The output noise performance of the ADP5014 is even better than the ADP1740 in the high frequency range.

Figure 11. Comparison of the output noise performance of the ADP5014 and ADP1740. Figure (a) shows the noise spectral density and (b) shows the integrated rms noise.

in conclusion

This paper presents a general analytical framework for modeling and controlling a current-mode buck converter with a secondary LC output filter, discusses the precise control-to-output transfer function, and proposes a new hybrid feedback structure, And the feedback parameter limits are deduced.

Design examples show that a switching regulator with a secondary LC filter and a hybrid feedback approach can provide a clean, stable power supply with performance comparable to or better than an LDO regulator.

The modeling and control in this article focuses on current-mode buck converters, but the methods described here are also applicable to voltage-mode buck converters.

Appendix I

The power stage transfer function in Figure 2 is as follows.

in:

Among them: L1 is the primary inductance.

C1 is the primary capacitor.

RESR1 is the equivalent series resistance of the primary capacitor.

L2 is the secondary inductance.

C2 is the secondary capacitor.

RESR2 is the equivalent series resistance of the secondary capacitor. RL is the load resistance.

The gain block in the current loop is as follows.

in:

Among them: Ri is the equivalent current detection resistance

Se is the sawtooth slope with slope compensation

Sn is the on-time slope of the current detection waveform

Ts is the switching period

The current loop gain is

in:

in:

D is the duty cycle

According to Figure 2, the gain block kr is calculated as follows

The transfer function from control to output is

in:

Appendix II

In Figure 5, the local feedback and remote feedback transfer functions are

From Equation 1 to Equation 27, the control-to-feedback transfer function is calculated as follows

in

Where: RA is the upper resistor of the feedback resistor divider

RB is the lower resistor of the feedback resistor divider

CF is the local feedback capacitor

The equivalent feedback network transfer function is

The approximate feedback transfer function is

in:

In typical low noise applications, a unity gain voltage reference structure is usually applied, so the parameter β will be equal to 1.Then, the feedback transfer function is

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