A Novel Buck Converter with Constant Frequency Controlled Technique

: This paper presents a buck converter with a novel constant frequency controlled technique, which employs the proposed frequency detector and adaptive on-time control (AOT) logic to lock the switching frequency. The control scheme, design concept, and circuit realization are presented. In contrast to a complex phase lock loop (PLL), the proposed scheme is easy to implement. With this novel technique, a buck converter is designed to produce an output voltage of 1.0–2.5 V at the input voltage of 3.0–3.6 V and the maximum load current of 500 mA. The proposed scheme was veriﬁed using SIMPLIS and MathCAD. The simulation results show that the switching frequency variation is less than 1% at an output voltage of 1.0–2.5 V. Furthermore, the recovery time is less than 2 µ s for a step-up and step-down load transient. The circuit will be fabricated using UMC 0.18 µ m 1P6M CMOS processes. The control scheme, design concept and circuit realization are presented in this paper.


Introduction
Power converter brings convenience to human life [1][2][3] and plays an indispensable role in various portable devices. Their applications include smartphones, tablet computers, the internet of things (IoT) and other electronic products. The power source of these products is the battery. In order to extend the battery life, the products will enter a standby mode when they are not operating. Therefore, high power efficiency is a critical issue in a wide load range.
For switching converters, the main modulation methods can be divided into two categories: pulse width modulation (PWM) and pulse frequency modulation (PFM) [4,5]. PWM can obtain high power efficiency at heavy load conditions; however, a high switching frequency at light load conditions will induce more considerable switching loss and reduce the power efficiency. Therefore, in contrast to PWM, PFM control has a lower switching frequency at light loads, to reduce the switching loss and improve the power efficiency.
Recently, the control methods of DC-DC converter have mainly been divided into two types: voltage mode control (VMC) [6] and current mode control (CMC) [1][2][3]7]. Figure 1 shows the methods of the CMC and the VMC. From Figure 1, it is obvious that CMC has one more feedback path of inductor current than VMC; consequently, the CMC has a faster transient response and better voltage regulation than VMC. Therefore, the CMC technique is widely used in today's power management ICs.
control, constant on time (COT) control, adaptive on-time (AOT) control, constant off time (CFT) control and hysteretic control, which are summarized in [2,[8][9][10][11][12][13][14][15][16][17][18][19]. The constant on-time current-mode (COTCM) control scheme has the inherent advantage of light load efficiency [20,21]. However, the variable switching frequency may cause electromagnetic interference (EMI) problems, which will impact the performance of the other integrated circuits. Moreover, the transient response of the step-load would be poor. This issue is due to the on-time remaining constant in the step-load transient, regardless of input/output conditions. For example, when the load suddenly drops after the switch is turned on, this might induce an overshot output voltage by the constant on-time.
In order to improve the issues of variable frequency and poor load transient in COTCM, a phase lock loop (PLL) based AOT control scheme is proposed [2]. The control scheme is shown in Figure 2. Similarly, the PLL technique can also be used in hysteretic control schemes [7,[22][23]. However, these topologies have a high circuit complexity.  The constant on-time current-mode (COTCM) control scheme has the inherent advantage of light load efficiency [20,21]. However, the variable switching frequency may cause electromagnetic interference (EMI) problems, which will impact the performance of the other integrated circuits. Moreover, the transient response of the step-load would be poor. This issue is due to the on-time remaining constant in the step-load transient, regardless of input/output conditions. For example, when the load suddenly drops after the switch is turned on, this might induce an overshot output voltage by the constant on-time.
In order to improve the issues of variable frequency and poor load transient in COTCM, a phase lock loop (PLL) based AOT control scheme is proposed [2]. The control scheme is shown in Figure 2. Similarly, the PLL technique can also be used in hysteretic control schemes [7,22,23]. However, these topologies have a high circuit complexity.  The constant on-time current-mode (COTCM) control scheme has the inherent advantage of light load efficiency [20,21]. However, the variable switching frequency may cause electromagnetic interference (EMI) problems, which will impact the performance of the other integrated circuits. Moreover, the transient response of the step-load would be poor. This issue is due to the on-time remaining constant in the step-load transient, regardless of input/output conditions. For example, when the load suddenly drops after the switch is turned on, this might induce an overshot output voltage by the constant on-time.
In order to improve the issues of variable frequency and poor load transient in COTCM, a phase lock loop (PLL) based AOT control scheme is proposed [2]. The control scheme is shown in Figure 2. Similarly, the PLL technique can also be used in hysteretic control schemes [7,[22][23]. However, these topologies have a high circuit complexity.  nique detailed in this paper is not necessarily superior to a PLL architecture in frequency constancy, but is better for realizing switching frequency constancy in the simplest way, which is suitable for commercialization and compatible with existing architectures.
The paper is organized as follows. Section 2 describes conventional COT/AOT control schemes. Section 3 presents the proposed control scheme and frequency detector. Section 4 introduces the design procedure and modeling analysis with MathCAD. Section 5 gives the SIMPLIS simulation results. Finally, the conclusion is given in Section 6.

Conventional COT/AOT Control Scheme
The conventional COT/AOT control scheme is as shown in Figure 3. For a COT scheme, the on-time is constant and fixed by the constant voltage V R_COT . For an AOT scheme, the on-time is variable and controlled by V R_AOT . As the scheme can easily get a fast transient response, the AOT scheme is becoming more and more popular.
Based on the above reasons, this paper proposes a novel constant frequency control scheme to maintain the switching frequency constant, instead of PLL. The proposed technique detailed in this paper is not necessarily superior to a PLL architecture in frequency constancy, but is better for realizing switching frequency constancy in the simplest way, which is suitable for commercialization and compatible with existing architectures.
The paper is organized as follows. Section 2 describes conventional COT/AOT control schemes. Section 3 presents the proposed control scheme and frequency detector. Section 4 introduces the design procedure and modeling analysis with MathCAD. Section 5 gives the SIMPLIS simulation results. Finally, the conclusion is given in Section 6.

Conventional COT/AOT Control Scheme
The conventional COT/AOT control scheme is as shown in Figure 3. For a COT scheme, the on-time is constant and fixed by the constant voltage VR_COT. For an AOT scheme, the on-time is variable and controlled by VR_AOT. As the scheme can easily get a fast transient response, the AOT scheme is becoming more and more popular.
To illustrate why the AOT can easily get a fast transient response, we can use the conventional COT/AOT schemes in Figure 3. The key waveforms for the AOT/COT scheme are shown in Figure 4. The AOT transient response is better than the COT scheme when the load current is step-up due to the adaptive on-time.  To illustrate why the AOT can easily get a fast transient response, we can use the conventional COT/AOT schemes in Figure 3. The key waveforms for the AOT/COT scheme are shown in Figure 4. The AOT transient response is better than the COT scheme when the load current is step-up due to the adaptive on-time.

Conventional AOT Scheme with PLL
Various control schemes based on AOT have been proposed to remedy the drawbacks. The most obvious example is the AOT scheme with PLL, which controls the on time, as shown in Figure 5.
Under a verified AOT control scheme, this paper proposes a novel and simple circuit to replace PLL.

Conventional AOT Scheme with PLL
Various control schemes based on AOT have been proposed to remedy the drawbacks. The most obvious example is the AOT scheme with PLL, which controls the on time, as shown in Figure 5.
Under a verified AOT control scheme, this paper proposes a novel and simple circuit to replace PLL.

Conventional AOT Scheme with PLL
Various control schemes based on AOT have been proposed to remedy the drawbacks. The most obvious example is the AOT scheme with PLL, which controls the on time, as shown in Figure 5.
Under a verified AOT control scheme, this paper proposes a novel and simple circuit to replace PLL.   Figure 6 shows the proposed constant frequency AOT control scheme. In Figure 6, the proposed frequency detector produces a control voltage proportional to the switching frequency. The proposed AOT module generates an appropriate on-time to control the switches, S 1 , S 2 . The on-time labeled T on is defined as the turn-on time of switch S 1 . The state of S 1 and S 2 in Figure 6 are complementary and non-overlapping. The S 1 operates as follows: S 1 turns off at the end of the on-time, until V SEN drops to V CMP , and then turns on S 1 again. The R i block in Figure 6 is the current sensor, which senses the inductor current I L and converts the voltage V SEN . The current sensor is another research topic. Relevant studies are fully introduced in [24]. The g m in Figure 6 is the trans-conductance of the V-I converter. The path through R i to AOT is called the current sensing path, which is fast to improve the transient response. In addition, the path through the error amplifier (EA) to AOT is called the voltage sensing path. This is a slow path, mainly to regulate the output to the defined value. In addition, the EA includes a compensation network, not only an error amplifier. the proposed frequency detector produces a control voltage proportional to the switching frequency. The proposed AOT module generates an appropriate on-time to control the switches, S1, S2. The on-time labeled is defined as the turn-on time of switch S1. The state of S1 and S2 in Figure 6 are complementary and non-overlapping. The S1 operates as follows: S1 turns off at the end of the on-time, until VSEN drops to VCMP, and then turns on S1 again. The Ri block in Figure 6 is the current sensor, which senses the inductor current IL and converts the voltage VSEN. The current sensor is another research topic. Relevant studies are fully introduced in [24]. The gm in Figure 6 is the trans-conductance of the V-I converter. The path through Ri to AOT is called the current sensing path, which is fast to improve the transient response. In addition, the path through the error amplifier (EA) to AOT is called the voltage sensing path. This is a slow path, mainly to regulate the output to the defined value. In addition, the EA includes a compensation network, not only an error amplifier. In this paper, the proposed control scheme is based on a AOT scheme, which has been verified and fabricated by 0.18 um/0.35 um CMOS processes in many literatures [17,[25][26]. The proposed frequency detector only replaces PLL, and does not change the control topology. In other words, the purpose of this paper is not to propose a new control architecture. Instead, a frequency detector is added under the verified control architecture to replace the PLL. This technique can easily be realized with only D Flip-Flop, SR Flip-Flop, passive RC components, and about 60 dB of an operational amplifier. Figure 7 shows the proposed frequency detector. This block mainly converts the switching frequency information into an analog voltage. The operation of the frequency detector can be understood from the key waveforms in Figure 8. The VSET_PUL, Vsaw, Vfc, and Vgate are all defined in Figure 6 or Figure 7. The shape converter converts the VSET_PUL into a sawtooth wave. The frequency information is hidden in the sawtooth wave. There- In this paper, the proposed control scheme is based on a AOT scheme, which has been verified and fabricated by 0.18 um/0.35 um CMOS processes in many literatures [17,25,26]. The proposed frequency detector only replaces PLL, and does not change the control topology. In other words, the purpose of this paper is not to propose a new control architecture. Instead, a frequency detector is added under the verified control architecture to replace the PLL. This technique can easily be realized with only D Flip-Flop, SR Flip-Flop, passive RC components, and about 60 dB of an operational amplifier. Figure 7 shows the proposed frequency detector. This block mainly converts the switching frequency information into an analog voltage. The operation of the frequency detector can be understood from the key waveforms in Figure 8. The V SET_PUL , V saw , V fc , and V gate are all defined in Figure 6 or Figure 7. The shape converter converts the V SET_PUL into a sawtooth wave. The frequency information is hidden in the sawtooth wave. Therefore, it can be extracted by following a low-pass filter (LPF). Figure 7b shows the implantation of the frequency detector. In contrast to the realization of PLL in Figure 5, the proposed method is simple and easy.

Proposed Frequency Detector
Overall, the proposed frequency detector's design concept involves how to find out the frequency information in the switching signal. Based on this purpose, this frequency detector can basically be divided into two parts: (a) a shape converter and (b) a low pass filter. The shape converter generates the sawtooth wave with switching period information. The low pass filter converts the sawtooth wave into a steady voltage and feeds it to the following stage. The switching frequency can keep constant, as long as the error amplifier locks the V fc through the feedback path. Therefore, the proposed detector does not need special foundry processes to support the circuit implementation and layout issue. the frequency information in the switching signal. Based on this purpose, this frequency detector can basically be divided into two parts: (a) a shape converter and (b) a low pass filter. The shape converter generates the sawtooth wave with switching period information. The low pass filter converts the sawtooth wave into a steady voltage and feeds it to the following stage. The switching frequency can keep constant, as long as the error amplifier locks the Vfc through the feedback path. Therefore, the proposed detector does not need special foundry processes to support the circuit implementation and layout issue.  Moreover, the switching frequency accuracy depends on the dc gain of the amplifier. In this paper, the dc gain of the amplifier is 60 dB, which means that the accuracy can reach 0.1%.

Proposed AOT
The proposed AOT is shown in Figure 9. Differently from a conventional AOT, the on-time of the AOT is decided to regulate the switching frequency. In other words, the proposed control scheme contains two regulation loops: a voltage regulation loop and a frequency regulation loop. The voltage regulation loop keeps the output voltage constant, and the frequency regulation loop maintains the switching frequency constant. Figure 10 shows the key waveforms and allows the reader to easily and quickly to understand the operation. the frequency information in the switching signal. Based on this purpose, this frequency detector can basically be divided into two parts: (a) a shape converter and (b) a low pass filter. The shape converter generates the sawtooth wave with switching period information. The low pass filter converts the sawtooth wave into a steady voltage and feeds it to the following stage. The switching frequency can keep constant, as long as the error amplifier locks the Vfc through the feedback path. Therefore, the proposed detector does not need special foundry processes to support the circuit implementation and layout issue.  Moreover, the switching frequency accuracy depends on the dc gain of the amplifier. In this paper, the dc gain of the amplifier is 60 dB, which means that the accuracy can reach 0.1%.

Proposed AOT
The proposed AOT is shown in Figure 9. Differently from a conventional AOT, the on-time of the AOT is decided to regulate the switching frequency. In other words, the proposed control scheme contains two regulation loops: a voltage regulation loop and a frequency regulation loop. The voltage regulation loop keeps the output voltage constant, and the frequency regulation loop maintains the switching frequency constant. Figure 10 shows the key waveforms and allows the reader to easily and quickly to understand the operation. Moreover, the switching frequency accuracy depends on the dc gain of the amplifier. In this paper, the dc gain of the amplifier is 60 dB, which means that the accuracy can reach 0.1%.

Proposed AOT
The proposed AOT is shown in Figure 9. Differently from a conventional AOT, the on-time of the AOT is decided to regulate the switching frequency. In other words, the proposed control scheme contains two regulation loops: a voltage regulation loop and a frequency regulation loop. The voltage regulation loop keeps the output voltage constant, and the frequency regulation loop maintains the switching frequency constant. Figure 10 shows the key waveforms and allows the reader to easily and quickly to understand the operation.

Mathematical Modeling for Main Body Converter
Before entering the circuit design phase, we needed to derive the closed-loop transfer function of the control scheme, establish the mathematical model, and determine the pole/zero positions of the system. However, it can be seen from Figure 6 that the whole system includes a comparator, switch, amplifier, and compensation network. Therefore, it is not easy to establish an accurate mathematical model for the CMC scheme. Relevant modeling studies are presented in [10,[27][28][29][30]. Among these prior works, the small-signal model proposed in [30] is most commonly used. Furthermore, Jian Li et al. [10] also proposed a more effective and intuitive current mode control circuit model for system analysis. In this section, we use MathCAD to obtain relevant design parameters and draw a Bode plot for stability analysis.
In order to derive the system transfer function, we divided Figure 6 into three parts: (a) a main body converter Gp(s), (b) divider, and (c) error amplifier including compensation network A(s), as shown in Figure 11. In [30], we can express Gp(s) as Equation (1).

Mathematical Modeling for Main Body Converter
Before entering the circuit design phase, we needed to derive the closed-loop transfer function of the control scheme, establish the mathematical model, and determine the pole/zero positions of the system. However, it can be seen from Figure 6 that the whole system includes a comparator, switch, amplifier, and compensation network. Therefore, it is not easy to establish an accurate mathematical model for the CMC scheme. Relevant modeling studies are presented in [10,[27][28][29][30]. Among these prior works, the small-signal model proposed in [30] is most commonly used. Furthermore, Jian Li et al. [10] also proposed a more effective and intuitive current mode control circuit model for system analysis. In this section, we use MathCAD to obtain relevant design parameters and draw a Bode plot for stability analysis.
In order to derive the system transfer function, we divided Figure 6 into three parts: (a) a main body converter Gp(s), (b) divider, and (c) error amplifier including compensation network A(s), as shown in Figure 11. In [30], we can express Gp(s) as Equation (1).

Mathematical Modeling for Main Body Converter
Before entering the circuit design phase, we needed to derive the closed-loop transfer function of the control scheme, establish the mathematical model, and determine the pole/zero positions of the system. However, it can be seen from Figure 6 that the whole system includes a comparator, switch, amplifier, and compensation network. Therefore, it is not easy to establish an accurate mathematical model for the CMC scheme. Relevant modeling studies are presented in [10,[27][28][29][30]. Among these prior works, the small-signal model proposed in [30] is most commonly used. Furthermore, Jian Li et al. [10] also proposed a more effective and intuitive current mode control circuit model for system analysis. In this section, we use MathCAD to obtain relevant design parameters and draw a Bode plot for stability analysis. In order to derive the system transfer function, we divided Figure 6 into three parts: (a) a main body converter G p (s), (b) divider, and (c) error amplifier including compensation network A(s), as shown in Figure 11. In [30], we can express G p (s) as Equation (1).
where R i is the gain for the current sensor, ω = π T on and Q = 2 π .
Energies 2021, 14, x FOR PEER REVIEW 8 of 18 where is the gain for the current sensor, = = 2 . Equation (1) is an approximate result. There is a dominant pole at the output, i.e., ≈ • . In addition, the ESR of the capacitor will form a zero. Generally, this zero frequency is very high. For example, if Co=10 μF and RESR = 5 mΩ, the zero frequency is about 20 MHz.

Compensation Network Design for A(s)
For the buck converter system, the A(s) design is critical and directly affects the crossover frequency, DC gain, and gain/phase margin of the closed-loop. As shown in Figure  12, the A(s) consists of an error amplifier and compensation network. The compensation network is composed of a resistor R1 and capacitor C1. This is a type II compensator. It is an off-chip compensation. The resistor Ro in Figure 12 signifies the output resistance of the error amplifier. The gm is the trans-conductance of the error amplifier. Thus, the A(s) can be expressed by Equation (2). There is one pole and one zero in A(s). The static error is less than 0.1%, due to the error amplifier gain of 60 dB. From Figure 12, the converter loop gain T(s) can be expressed by Equation (3). In this paper, we set the zero of A(s) at the output pole of the buck converter, as shown in Equation (4). The pole of A(s) is extrapolated to make the feedback system stable. Equation (1) is an approximate result. There is a dominant pole at the output, i.e., w pout ≈ 1 R LOAD ·C o . In addition, the ESR of the capacitor will form a zero. Generally, this zero frequency is very high. For example, if C o =10 µF and R ESR = 5 mΩ, the zero frequency is about 20 MHz.

Compensation Network Design for A(s)
For the buck converter system, the A(s) design is critical and directly affects the crossover frequency, DC gain, and gain/phase margin of the closed-loop. As shown in Figure 12, the A(s) consists of an error amplifier and compensation network. The compensation network is composed of a resistor R 1 and capacitor C 1 . This is a type II compensator. It is an off-chip compensation. The resistor R o in Figure 12 signifies the output resistance of the error amplifier. The g m is the trans-conductance of the error amplifier. Thus, the A(s) can be expressed by Equation (2). There is one pole and one zero in A(s). The static error is less than 0.1%, due to the error amplifier gain of 60 dB. From Figure 12, the converter loop gain T(s) can be expressed by Equation (3). In this paper, we set the zero w z of A(s) at the output pole of the buck converter, as shown in Equation (4). The pole w p of A(s) is extrapolated to make the feedback system stable. where w z = 1 where R LOAD , C out is the output resistor and output capacitor in Figure 6, respectively.
Energies 2021, 14, x FOR PEER REVIEW . Figure 12. Error amplifier and compensation network.

Stability Analysis of Mathematical Model
Step 1 Substitute the Table 1 value into Equation (1), and set Ri = 490 m, then ca the poles and zeros with MathCAD, and draw a Bode plot of the ( ). Th space of the MathCAD is shown in Figure 13. The Bode plot of the shown in Figure 14, where ≈ • i. e. , ≈ 4.4 .

Stability Analysis of Mathematical Model
Step 1 Substitute the Table 1 value into Equation (1), and set R i = 490 m, then calculate the poles and zeros with MathCAD, and draw a Bode plot of the G p (s). The workspace of the MathCAD is shown in Figure 13. The Bode plot of the G p (s) is shown in Figure 14, where w pout ≈ 1 R LOAD ·C o i.e., f pout ≈ 4.4 kHz.
Step 2 As expressed in (4), the zero f z = f pout = 4.4 kHz. Using Equation (2), suppose C 1 = 100 pF, then R 1 = 250 kΩ can be obtained. In order to obtain a better output regulation, the A(s) gain is set to at least 60 dB. Thus, R o = 10 MΩ, then g m = 100 µA/V, and f p ≈160 Hz. Finally, the values of g m , R o , C 1 , and R 1 are substituted into Equation (2).

Step 3 We substitute Equation (2) into Equation (3) and draw the Bode diagram of T(s) with
MathCAD. Here, in Equation (3), the k is substituted by 0.5. The Bode diagrams of T(s), G p (s), and A(s) are drawn in Figure 15. As can be seen from Figure 15, the T(s) phase margin is about 40 degrees, DC gain is about 71 dB, and the crossover frequency f c is about 400 kHz.  Step 2 As expressed in (4), the zero = = 4.4 . Using Equation (2), suppos = 100 pF, ℎ = 250 kΩ can be obtained. In order to obtain a better outpu regulation, the A(s) gain is set to at least 60 dB. Thus, = 10 MΩ, then gm = 10 μA/V, and ≈160 Hz. Finally, the values of gm, Ro, C1, and R1 are substituted int Equation (2).
Step 3 We substitute Equation (2) into Equation (3) and draw the Bode diagram of T(s) with MathCAD. Here, in Equation (3), the k is substituted by 0.5. The Bode diagrams of T(s), Gp(s), and A(s) are drawn in Figure 15. As can be seen from Figure  15, the T(s) phase margin is about 40 degrees, DC gain is about 71 dB, and the crossover frequency is about 400 kHz.

SIMPLIS Schematic Building
In this section, the proposed control scheme is simulated by SIMPLIS. The schematic is presented in Figure 16.

Stability Analisis with SIMPLIS
For investigating stability issues, the proposed scheme was verified by SIMPLIS, and Bode diagrams are shown in Figure 17. The legend label names in Figure 17 follow the previous section definitions. As shown in Figure 17, the crossover frequency of the loop response is about 371 kHz, the phase margin is about 41 degrees, and DC gain is about 69 dB.
In Section 4.3, we used MathCAD to show the Bode diagrams of the T(s) by numerical approach. However, in this section, we build a circuit level schematic and confirm the stability by SIMPLIS. In order to compare the differences between MathCAD and SIMPLIS, we put the waveforms of Figures 14 and 17 into Figure 18. From Figure 18, we can see that the numerical approach and the circuit level have some differences. There is a possible reason for this difference: Equation (1) is an approximated result, and Equation (1) does not consider the parasitic effect of the inductor series resistance. Nevertheless, Equation (1) is very close to the real behavior of the system. Therefore, the difficulty of switching converter design is to build an accurate model to meet the actual circuit behavior. Unfortunately, we could not find a mathematical model to completely represent the circuit behavior from the prior research literature. However, this does not mean that the mathematical model is useless for the design, because the relevant design parameters can be quickly obtained through MathCAD, such as the loop gain, capacitance, and resistance of the compensator.

SIMPLIS Schematic Building
In this section, the proposed control scheme is simulated by SIMPLIS. The schematic is presented in Figure 16.

Stability Analisis with SIMPLIS
For investigating stability issues, the proposed scheme was verified by SIMPLIS, and Bode diagrams are shown in Figure 17. The legend label names in Figure 17 follow the previous section definitions. As shown in Figure 17, the crossover frequency of the loop response is about 371 kHz, the phase margin is about 41 degrees, and DC gain is about 69 dB.
In Section 4.3, we used MathCAD to show the Bode diagrams of the T(s) by numerical approach. However, in this section, we build a circuit level schematic and confirm the stability by SIMPLIS. In order to compare the differences between MathCAD and SIM-PLIS, we put the waveforms of Figures 14 and 17 into Figure 18. From Figure 18, we can see that the numerical approach and the circuit level have some differences. There is a possible reason for this difference: Equation (1) is an approximated result, and Equation (1) does not consider the parasitic effect of the inductor series resistance. Nevertheless, Equation (1) is very close to the real behavior of the system.  Therefore, the difficulty of switching converter design is to build an accurate model to meet the actual circuit behavior. Unfortunately, we could not find a mathematical model to completely represent the circuit behavior from the prior research literature. MathCAD helps us quickly obtain system parameters through mathematical models, and SIMPLIS verifies these design parameters in circuits. By comparing the two results, we confirm again that although MathCAD can help us obtain the design parameters quickly, it is challenging to represent the actual circuit behavior. As mentioned earlier, an accurate mathematical model is difficult to obtain. Therefore, we suggest verifying the circuit behavior with SIMPLIS first. Then, to modify the designed parameters appropriately, to approach the actual circuit behavior. Finally, the circuits can deliver to transistor-level simulation efficiently.
It is unwise to bypass SIMPLIS verification and directly deliver the design parameters to Cadence Virtuoso SPICE simulation. Since the switching converter is a circuit with both analog and digital parts, if the circuit is simulated directly with SPICE, the simulation time will be very long, which is inefficient. We can say that the advantage of SIMPLIS is to quickly determine the stability of the system. SPICE is the final step in the design phase before delivering the layout. Therefore, MathCAD and SIMPLIS must design and verify the circuit with each other.
The design of a power converter is a tedious and time-consuming process. Especially when considering the circuit stability, we should pay attention to AC analysis and transient response. Besides MathCAD and SIMPLIS, many computer-aided design (CAD) software platforms can also help designers design the compensator for stability issues, such as LTspice and LTpowerCAD.

Transient Performance
The load transient response is shown in Figure 19. In this paper, the step transition of load current is between 0.1 A and 0.5 A. The test condition is described as below: under 3.3 V input voltage, the output voltage is 1.8 V. The recovery time is defined as the output voltage recovered to within 1% of the expected voltage of 1.8 V during load transition.

Transient Performance
The load transient response is shown in Figure 19. In this paper, the step transition of load current is between 0.1 A and 0.5 A. The test condition is described as below: under 3.3 V input voltage, the output voltage is 1.8 V. The recovery time is defined as the output voltage recovered to within 1% of the expected voltage of 1.8 V during load transition.
From the simulation results in Figure 19, the step-up and step-down load transition recovery times are 1.69 μs and 1.62 μs, respectively. In other words, the recovery time is less than 2 μs for the proposed converter. Moreover, the overshoot and undershoot voltages are measured as 24 mV and 20 mV, respectively; all within 25 mV.
At a 3.0-3.6 V input voltage, 500 mA load current, and 1.0-2.5 V output voltage, the electrical performance of the proposed converter is shown in Figure 20. The maximum ripple voltage can be measured as 2.24 mV at a 3.6 V input voltage and 2.5 V output voltage.   From the simulation results in Figure 19, the step-up and step-down load transition recovery times are 1.69 µs and 1.62 µs, respectively. In other words, the recovery time is less than 2 µs for the proposed converter. Moreover, the overshoot and undershoot voltages are measured as 24 mV and 20 mV, respectively; all within 25 mV.
At a 3.0-3.6 V input voltage, 500 mA load current, and 1.0-2.5 V output voltage, the electrical performance of the proposed converter is shown in Figure 20. The maximum ripple voltage can be measured as 2.24 mV at a 3.6 V input voltage and 2.5 V output voltage.

Load Regulation/Line Regulation
Load/Line regulation is an important specification for a power converter system. The load regulation is defined in Equation (5). In general, the load regulation should be as small as possible. In order to measure the load regulation, the test conditions of the load regulation are as follows: 3.3 V input voltage, 1.8 V output voltage, and a load current from 0.1 A to 0.5 A. As shown in Figure 21, the load regulation can be calculated as −0.03% by Equation (5).
where V o_max_load is the voltage at maximum load and V o_min_load is the voltage at minimum load. V o_normal_load is the voltage at the typical load.

Load Regulation/Line Regulation
Load/Line regulation is an important specification for a power converter system. The load regulation is defined in Equation (5). In general, the load regulation should be as small as possible. In order to measure the load regulation, the test conditions of the load regulation are as follows: 3.3 V input voltage, 1.8 V output voltage, and a load current from 0.1 A to 0.5 A. As shown in Figure 21, the load regulation can be calculated as −0.03% by Equation (5).
where _ _ is the voltage at maximum load and _ _ is the voltage at minimum load. _ _ is the voltage at the typical load. The line regulation is defined as Equation (6). Similarly to the load regulation, the line regulation should be as small as possible. Unfortunately, the line regulation cannot be presented in this system-level simulation. The calculation result is close to 0.
where ΔVin is the change of the input voltage, ΔVo is the change of the output voltage.

Switching Frquency Regulation
By using the proposed constant frequency technique, the converter has good switching frequency regulation. For example, with a 3.3 V input voltage and 500 mA load current, Figure 22 shows the switching frequency variation for the deferent outputs. As been seen from the results in Figure 22, the frequency variation is smaller than 1%. The line regulation is defined as Equation (6). Similarly to the load regulation, the line regulation should be as small as possible. Unfortunately, the line regulation cannot be presented in this system-level simulation. The calculation result is close to 0.
where ∆V in is the change of the input voltage, ∆V o is the change of the output voltage.

Switching Frquency Regulation
By using the proposed constant frequency technique, the converter has good switching frequency regulation. For example, with a 3.3 V input voltage and 500 mA load current, Figure 22 shows the switching frequency variation for the deferent outputs. As been seen from the results in Figure 22, the frequency variation is smaller than 1%.
The line regulation is defined as Equation (6). Similarly to the load regulation, the line regulation should be as small as possible. Unfortunately, the line regulation cannot be presented in this system-level simulation. The calculation result is close to 0.
where ΔVin is the change of the input voltage, ΔVo is the change of the output voltage.

Switching Frquency Regulation
By using the proposed constant frequency technique, the converter has good switching frequency regulation. For example, with a 3.3 V input voltage and 500 mA load current, Figure 22 shows the switching frequency variation for the deferent outputs. As been seen from the results in Figure 22, the frequency variation is smaller than 1%. Therefore, the switching frequency can keep almost constant for the different outputs. The constant frequency technique can be applied to the buck converters and effectively control the switching frequency. A comparison between the constant frequency technique and non-constant frequency technique is shown in Figure 23. The constant switching frequency can resolve the EMI issue in applications. Therefore, the switching frequency can keep almost constant for the different outputs. The constant frequency technique can be applied to the buck converters and effectively control the switching frequency. A comparison between the constant frequency technique and non-constant frequency technique is shown in Figure 23. The constant switching frequency can resolve the EMI issue in applications.

Performance List
The performance of the proposed converter is summarized in Table 2. For a 3.0-3.6 V input voltage and 1.8 V output voltage, the recovery time is smaller than 2 μs for a 100 mA~500 mA load current transition. Moreover, by using the proposed constant frequency technique, the switching frequency is well controlled. Finally, performance comparisons with reported converters are listed in Table 3. From Table 3, there is no significant difference in switching frequency constancy between PLL architectures and the proposed architecture.

Performance List
The performance of the proposed converter is summarized in Table 2. For a 3.0-3.6 V input voltage and 1.8 V output voltage, the recovery time is smaller than 2 µs for a 100 mA~500 mA load current transition. Moreover, by using the proposed constant frequency technique, the switching frequency is well controlled. Finally, performance comparisons with reported converters are listed in Table 3. From Table 3, there is no significant difference in switching frequency constancy between PLL architectures and the proposed architecture.

Conclusions
This paper presents a novel constant frequency technique and illustrates the concept and implementation of the proposed control scheme in detail. Modeling was derived and verified by MathCAD and SIMPLIS. The proposed frequency detector is designed to detect and lock the switching frequency, rather than using PLL. Compared with a control scheme based on PLL, the proposed method is easy to implement and compatible with the existing schemes. The simulation results show that the switching frequency variation is less than 1% for a 1.0-2.5 V output. The recovery times for the step-up and step-down load transition are 1.69 µs and 1.62 µs, respectively. This novel constant frequency technique can resolve the EMI issue for various applications in converters.