A Simple Approach to a Linear Control of Switched Capacitor DC-DC Converters in System-on-Chip Thomas Souvignet *† , Bruno Allard * , Severin Trochut † , Frederic Hasbani † * Universit´ e de Lyon, Ampere CNRS UMR 5005, INSA Lyon, Villeurbanne, France † STMicroelectronics, Crolles Email: thomas.souvignet@st.com Abstract—Switched-capacitor DC-DC converters are easily in- tegrated in CMOS technologies on the contrary to inductive DC- DC converter. They appear as an interesting solution to provide fine grain power management in system-on-chip. Previous works have demonstrated both high efficiency and high power density for several power stage topologies. Many types of controller have been analyzed as those using frequency modulation. This paper proposes a simple method to implement a linear frequency modulation based controller. The power stage is modeled using sampled-data modeling which is an uniform approach. Then the frequency response is used to size a compensation network according to stability and transient requirements. Theoretical results are supported by simulation. I. I NTRODUCTION Aggressive power management techniques such as Dy- namic Voltage Frequency Scaling (DVFS) and body biasing are required in modern Systems-on-Chip (SoC) to reduce the power consumption both in running and idle operations. Fine grain power management leads to increase the number of power domains and consequently, the number of dedicated power supplies. Increasing the number of external voltage reg- ulators is not suitable since the board area, IOs and power grid resources are limited. Moreover fast dynamic voltage scaling can not be achieved with an external supply due to the power grid, package and board parasitics’limitations. Integrated volt- age regulation can address this issue. Linear voltage regulators are only used for small output power because their efficiency depends on the output-to-input voltage ratio. Inductive DC-DC converters are widely used as external supplies because they provide high efficiency and high output power. However, their integration suffer from limitations due to magnetic devices and must be used at least with an interposer [1]. Unfortunately, the cost and fabrication difficulty are increased. Fully integrated SC converters have demonstrated high power density with high efficiency making them suitable for fine grain power management architecture [2]–[6]. Voltage Regulator Module (VRM) based on SC converter required not only a high efficiency but also a fast control. The VRM has to present a low impedance profile to achieve good load regulation and high rejection from the input voltage to achieve good line regulation. During DVFS operation, voltage operating points are changed on the fly what means accurate and fast voltage reference tracking. Consequently, the control architecture is an important part in the design of integrated VRMs. The control of the output voltage in SC converter results from its static model. SC converter static models are represented using an ideal transformer and an output impedance as shown Fig. 1. Static model of a SC converter in Fig. 1. The transformer turning ratio m:n corresponds to the topology ideal ratio. The output impedance is expressed as two asymptotic limits [7]. The Slow Switching Limit (SSL) is related to the switching frequency and the flying capacitor (1) while the Fast Switching Limit (FSL) depends on the resistive elements and the switching frequency duty cycle (2). SSL ∝ 1 C fly · f sw (1) FSL ∝ R D (2) Under a given load current I load , the output voltage V out can be expressed as V out = n m · V in - R out (C fly ,f sw ,R on ,D) · I load (3) where V in is the input voltage, n m is the converter ratio, C fly the flying capacitor, f sw the switching frequency, R on the on-state switches’resistance and D, the duty cycle. Thus from (3), the output voltage is regulated by the modulation of the impedance or by the converter ratio. The impedance modulation introduces extra losses as for a linear regulator but the efficiency can be maximized over a wide output voltage range when the topology ratio is changed. Many controls have been proposed in literature using the control variable C fly , R on , D and f sw to perform a voltage regulation. Capacitance modulation is used in [8]. The drawback is that the capacitance can only vary with discrete values. Switches’conductance can be used [9] or pulse width modulation (PWM) [10] but their values can vary only in a small range. The frequency modulation (FM) is widely used because the switching fre- quency can vary in a wide range. Moreover, the impedance is inversely proportional to the switching frequency in the SSL region. Various implementations of the FM control exist. Pulse Skipping which behaves like an “ON-OFF” control is used due to its simple implementation made with a comparator and an oscillator [11]. Single bound hysteric control exhibit 3DSHU3 :RUNVKRSRQ&RQWURODQG0RGHOLQJIRU3RZHU(OHFWURQLFV&203(/ ,(((