A new active LED driver with high power factor (PF) and low total harmonic distortion (THD) compatible with a rapid-start ballast is proposed. An LC input filter is attached to the ballast to increase PF and reduce THD. A boost converter is then installed to regulate the LED current, where an unstable operating region has been newly identified. The unstable region is successfully stabilized by feedback control with two zeroes. The extremely high overall system of the 10th order is completely analyzed by the newly introduced phasor transformed circuits in static and dynamic analyses. Although a small DC capacitor is utilized, the flicker percentage of the LED is drastically mitigated to 1% by the fast controller. The proposed LED driver that employs a simple controller with a start-up circuit is verified by extensive experiments whose results are in good agreement with the design. Conventional fluorescent lamps are being replaced with light emitting diode (LED) lamps to save energy and preserve the environment [1] - [13] . An LED lamp has higher efficiency than fluorescent lamps; its lifetime is longer than 50,000 hours, which is approximately 10 times that of a fluorescent lamp. Many countries thus promote the use of LED lamps even though these lamps are expensive. The replacement of fluorescent lamps with LED lamps in some countries, such as Japan, is highly impeded by this issue. Rapid-start ballasts are already firmly installed inside ceilings and are difficult to replace with new switching converter-type LED drivers. A practical solution is to attach an LED driver to a rapid-start ballast instead of removing the ballast from the ceiling [14] - [16] . A few practical design issues must be considered in dealing with LED drivers. First, the use of a large DC input capacitor for the LED driver should be avoided because it produces a large peak inrush current and may cause occasional fires inside the rapid-start ballast [16] . Second, the power factor (PF) and total harmonic distortion (THD) of the LED lamp should not be reduced although the power level of an LED lamp is typically half that of a fluorescent lamp . This problem can be resolved by attaching an appropriate LC filter to the rapid-start ballast [16] . Third, accurate static and dynamic circuit models, including a switching converter and a highly nonlinear rapid-start ballast, are required for the design of an LED lamp. This cumbersome problem can be addressed by the recently proposed unified general phasor transformation [18] . An active LED driver that is compatible with the rapid-start ballast and meets the three design considerations mentioned above is proposed in this study. A small DC capacitor with an LC filter is attached to the ballast, and a boost converter regulates the LED array ( Fig. 1 ). Unlike a conventional boost converter, the power source of the converter is not an ideal voltage source but the small DC capacitor linked to an output of the ballast whose characteristic is highly nonlinear. Furthermore, an unstable operating region is identified in this study; such region can be avoided by selecting an appropriate duty cycle of the boost converter. The complete DC and AC phasor transformed circuit models for the 10th order LED lamp system developed in this study allow for the drastic reduction in LED ripple current even with a small DC capacitor. This design is extensively verified by experiments. The LED lamp ( Fig. 1 ) operates with a rapid-start ballast and is composed of an LED array and the proposed LED driver, which includes an input filter, a rectifier, a boost converter, and a controller. Regulated current is provided by controlling the duty cycle of the boost converter for the LED array regardless of the large rectified ripple voltage of 120 Hz caused by the small DC input capacitor. This capacitor is where the LED current obtains the feedback from the source voltage variation of the rapid-start ballast and other variations, such as temperature. A DC–DC converter is necessary to make the LED current constant regardless of the ripple voltage without any power loss. Galvanic isolation is not required for tubular-type LEDs; hence, an isolation transformer need not be used. Instead, a single-stage converter should be utilized considering the complexity of the gate driver and voltage rating of the LED array. Among various DC–DC converters, the boost converter was selected in this study because of the simple driver structure of its main MOSFET ( Fig. 1 ). Although soft switching provides high efficiency in general, it is not considered because the use of high switching frequency to reduce the converter filter size does not contribute to the overall filter size, where the input filter is already extremely large for the operating frequency of 60 Hz. For a buck converter, the n-MOSFET utilized for a power switch requires a bootstrap gate driver, such as IR2101, which does not work properly when the load current is small. The LED employed in this study (i.e., DG-82A83C-001-5S-3) has forward voltage drop V_LED = 3.0 V and equivalent series resistance r_LED = 6.3 Ω at 60 mA. The LED array can then be characterized by r_T and V_T as follows [10] : where m is the number of series-connected LEDs. The output voltage applied to the LED array ( Fig. 1 ) becomes the following: The voltage drop of an LED at 60 mA is determined from Eq. (2) as 3.38 V, and its power becomes 0.20 W. This condition means that approximately 100 LEDs are required for an output power of 20 W. The static behavior of the proposed lighting system ( Fig. 1 ) was fully analyzed in this study through phasor transformations [17] , [18] , which enable us to determine the amplitude and phase of AC circuit variables such as in a DC circuit. By applying the transformation, an averaged DC phasor circuit was obtained ( Fig. 2 (a)). The imaginary resistors represent the reactance of inductors and capacitors in the steady state. The rapid-start ballast is highly nonlinear, but its parameters are assumed to be fixed at an operating point. The input filter composed of L_c and C_c is attached to this ballast to satisfy PF and THD regulations. The boost converter and rectifier can be transformed to equivalent auto-transformers whose turn ratios are the complementary values of their corresponding switch duty cycles [19] . The boost converter and rectifier are assumed to operate in the continuous conduction mode (CCM) so that they can be replaced with the averaged switched transformers. All variables, including V_s and I_s , are represented as phasors and the Zener diode voltage V_z1 . The LED array is modeled as DC voltage source V_T and dynamic resistance r_T . Removing the complex transformer for the equivalent circuit of the rectifier ( Fig. 2 (a)) from the left side is possible with a recently proposed method [22] . By referring all parameters into the primary side of the rapid-start ballast while removing all transformers ( Fig. 2 (b)), the source current is as follows: From Eq. (3), PF can be derived by identifying its real and imaginary parts as follows: From Eq. (6), the complex transformer is assumed to have a zero phase for simplicity so that the real part operator of Fig. 2 (a) can be neglected; only the real value of equivalent voltage source V’_tz is provided. Given the nonlinearity of the ballast, the exact values of L_c and C_c were determined by experiments, which will be described in the succeeding section. As described in the previous section, a detailed analysis of the lighting system ( Fig. 2 (a)) may yield inaccurate results because of the nonlinearity of the rapid-start ballast and rectifier. The left part of the rectifier circuit must be simplified to understand the behavior of the boost converter with the LED array. The output V-I characteristic of the rectifier in the steady state was measured as shown in Fig. 3 . It was found to be quite linear for the operating range as shown in Fig. 4 ; hence, the system at the output of the rectifier can be characterized as equivalent resistance R_i = 270 Ω and voltage source V_eq = 168 V as follows: The load current of LED array I_L versus the duty cycle of boost converter D in the steady state is derived from Fig. 4 as follows: where The critical duty cycle D_c that maximizes I_L is derived from Eq. (8) as follows: For the LED array of m = 100, the calculated I_L from Eq. (8) is compared with the experimental result as shown in Fig. 5 . I_L increases for duty cycle D up to the critical value D_c = 0.75 similar to the conventional boost converter; however, it decreases when D is larger than D_c . Load current I_L can be either controlled for the lower region ( D < D_c ) or the higher region ( D > D_c ). However, the lower region is preferred because rectifier current Ii for the higher region increases for the same load current I_L as shown in Eq. (8). The maximum duty cycle should thus be restricted to D < D_c . In this study, the Zener diode D_z2 was employed to limit the maximum duty cycle by clamping control voltage v_c . Controlling the proposed active LED driver is challenging because of the highly nonlinear characteristics and the 10th order complexity of the overall system. As shown in Fig. 1 , the proposed controller includes the differential current sensing part and frequency compensation circuit components C_T, R_z , and C_z as well as a conventional PWM control circuit. The dynamic models for the overall LED driver should be derived to design the proposed controller. As shown in Fig. 6 (a), the models can be regarded as operating in a quasi-steady state; the same static model used in the steady state ( Fig. 4 ) was thus adopted. This condition can be justified by the very slow dynamic response of the rapid-start ballast and input filter. Its operating frequency of approximately 60 Hz is more than 20 times lower than the bandwidth of the dynamics of the controller. One difference is rectified current ripple i_r , which is a surge current of 120 Hz that acts as a major disturbance to the control system. The proposed controller should mitigate these ripple harmonics by a fairly fast control loop. Considering that ripple voltage v_r that appears in capacitor C_i is easy to measure and analyze, an approximated equivalent circuit can be obtained as shown in Fig. 6 (b) by applying the Thevenin theorem to the left part of the dotted line of the circuit of Fig. 6 (a). The detailed analysis of v_r for a given i_r is complicated and avoided in this study. Instead, it will be measured for an open loop control condition and regarded as an independent source from here on. The large signal model of Fig. 6 (b) is then perturbed, and an AC model is obtained in the frequency domain by removing all DC sources but including the ripple voltage source [19] . The harmonic ripples and perturbed small signals appear in the same equivalent circuit; this occurrence is uncommon. Finally, the most simplified model (where the auto transformer of the boost converter is removed) is obtained as shown in Fig. 6 (d). A filtered circuit with three independent voltage sources appears. For the AC analysis, DC operating point I_i is calculated from Fig. 4 as follows: The input side impedance is derived from Fig. 6 (c) as follows: The small signal transfer function of boost converter output in CCM as a function of duty cycle of the switch can be derived from Fig. 6 (d) for the two small signal-independent sources From Fig. 6 (d), the transfer function from to when is zero is as follows: The transfer function from to when is zero is derived with Eq. (11) as follows: From Eqs. (13) and (14), transfer function G_d (s) from to can be calculated as The zero frequency response is obtained from Eq. (15) as follows: where R_c is critical resistance that determines the proposed system stability as follows: When the numerator of Eq. (16) is positive (i.e., R_c > 0), the gain is positive and thus corresponds to the positive slope of I_L in Fig. 5 . This condition is justified considering that I_L is proportional to V_L as follows: Critical duty cycle D_c that maximizes I_L of Eq. (10) is also obtained from Eq. (17) when R_c = 0 as shown in the lower part of Fig. 5 . Notably, the R_c of Eq. (17) should be positive to keep the proposed system stable. The first term of Eq. (17) is always negative as can be easily identified from Eq. (18). As discussed in the previous section, ripple voltage V_i(s) is adisturbance, and its transfer function to load voltage should be obtained to mitigate the effect of this disturbance on the ripple current of the LED. This transfer function is easily derived from Fig. 6 (d) in a very similar manner to Eq. (13) as follows: The overall configuration of the proposed controller is redrawn in Fig. 7 (a). All the circuit parameters at the LED load side are highlighted. Fig. 7 (a) is modeled for the complete modeling of the proposed LED driver for use in subsequent sections. Transfer function G_v (s) is defined for perturbed load current against perturbed load voltage and obtained from Fig. 7 (a) as follows: The transfer function of the control circuit ( Fig. 1 ) from to can be determined as follows: where g_m and r_m are the gain and output resistances of the operational transconductance amplifier (OTA), respectively. The amplitude of the saw tooth of the PWM is denoted as V_saw . The overall control block diagram of the proposed LED driver is drawn in Fig. 7 (b) by combining the transfer functions of Eqs. (13) to (22). The overall transfer function for the reference current input is derived from Fig. 7 (b) with Eq. (15) as follows: The overall transfer function for the ripple voltage input is derived from Fig. 7 (b) with Eq. (15) again as follows: From Eqs. (23) and (24), the proposed LED driver is determined to have loop gain as follows: To make the system stable, converter transfer function G_d (s) was examined with Eq. (15) and plotted in Fig. 8 for the circuit parameters of Table I used in the experiment. For clarity, these calculation results were verified by the circuit simulator MMSIM72 (Cadence) using the equivalent circuit in Fig. 6 , which is found to be very accurate. As shown in Eq. (15), three poles and two zeroes exist in G_d (s) , which is rewritten as follows: From Eq. (26), the first pole and zero (i.e., p₁ and z₁ , respectively) can be approximately obtained by neglecting the high-order terms at low frequency as follows: As calculated from Eq. (28), the first pole and zero are p₁ /(2π) = ˗47 Hz and z₁ /(2π) = –13 Hz, respectively, which are mainly formed by large input capacitance C_i . The low-frequency pole and zero do not reduce the system stability because their values are negative and only a slight change occurs in the phase of much less than 90 degrees ( Fig. 8 ). However, the high-frequency zero, z₂ , should be positive as far as R_c > 0. This condition can be proved from Eq. (26), considering z₁ < 0, as follows: Complex poles p₂ and p₃ and positive zero z₂ are determined from Eq. (28) and Table I as | p₂ | /(2π) = | p₃ | /(2π) = 951 Hz and z₂ /(2π) = 27 kHz, respectively. They may affect the system stability significantly because the phase of G_d (s) approaches 180 degrees because of the poles and even crosses over it because of the positive zero. The complex poles are mainly formed by the L_i and C_L of the boost converter as identified from the highest-order coefficient of Eq. (27). However, the dominant pole compensation that allows the high-order dynamics to be buried cannot be used for the proposed design because fast response is crucial to eliminate the 120 Hz ripple caused by the feedback loop control. Two additional zeroes are thus inserted to the frequency at the two poles so that they cancel each other out. The first zero is inserted to the OTA output, and the second one is added at the current sensing circuit as follows: From Eq. (30b), C_T is determined as 270 nF for r_T = 620 Ω; however, the value of R_zC_z only can be determined from Eq. (30a) for now. Considering the increased concern over the flicker in LED lamps and its effect on human health [20] , the ripple current of LEDs is mitigated in this study. The small voltage change across LEDs causes a large current variation because of the small dynamic resistance. The flicker percentage (percent flicker) is defined as follows [21] : where A_max and A_min are the maximum and minimum brightness, I_max and I_min are the maximum and minimum currents, and I_ripple and I_avg are the ripple and average currents of the LED lamp, respectively. In Eq. (31), the brightness of the LED is assumed to be linearly proportional to its current. The targeted flicker percentage was set to approximately 1% to be nearly flicker-free. Given that the average current of the LED array in this study is 60 mA, I_ripple should be less than 1.2 mA to meet the specification in Eq. (31). From Eq. (24), the ripple current can be obtained for the ripple frequency of ω_r (= 2 π x 120 rad/s ) in the steady state as follows: where it has been approximated considering that the loop gain of Eq. (25) is much larger than 1 at the ripple frequency. Applying Eqs. (13), (15), and (26) to (32) and neglecting all poles and zeroes higher than 120 Hz result in the following: For the operating conditions in Table I , peak-to-peak ripple voltage V_r is measured to be approximately 13 V when C_i is 47 μF. C_z can thus be calculated from Eq. (33b) as 330 nF for the given value of | H₂(jω_r) |=1.2 mA/13 V = –80.7 dB. C_z is thus selected as 250 nF considering a 25% margin. From Eqs. (30a) and (33b), R_z is finally deter mined to be 670 Ω as follows: As shown in Fig. 9 , the compensated loop gain and phase margin at 120 Hz are 34.2 dB and 44º, respectively. This finding confirms the design of fairly high gain and appropriate margin. This calculation was verified by simulation using the equivalent circuit of Fig. 6 (b); good agreement was observed. For the operation of the proposed LED driver without an external power supply, a cheap and simple on-board power supply is required for the controller. A Zener regulator in series with the LED array was used as shown in Fig. 1 . At start-up, this regulator cannot generate any voltage because the boost converter does not operate until the regulator activates the control circuit. A start-up circuit is thus implemented by simply adopting the diode D_s connected between v_i and an appropriated intermediate point of the LED array. During start-up, D_s is turned on to provide the regulator with LED current directly from v_i . D_s is automatically turned off after start-up because load voltage v_L is increased as the boost converter operates. The point near the center of the array was carefully selected to meet this condition. Several design requirements, such as load power of 20 W, PF higher than 90%, low THD characteristics, and high efficiency, should be considered for the source voltage variable range to determine the suitable values of L_c and C_c of the input filter (i.e., from 94 V_rms to 106 V_rms ). The proposed boost converter-type LED driver is very similar to the passive LED driver in terms of output power and resonant frequency [16] . Therefore, the inductor of the input filter is considered to be approximately 0.1 H to 0.3 H. The suitable value of C_c that increases PF to above 0.9 is approximately 2 μF to 4 μF from Eq. (4). However, determining the exact values of L_c and C_c is theoretically impossible because of the non-linearity of the rapid-start ballast. Eq. (4) is inaccurate because the rectifier does not always operate in CCM and the elimination of the complex transformer is approximated and imperfect. The equations can thus be used as a guideline to select L_c and C_c , which were actually determined by experiments. The load power and PF for different source voltages were measured as shown in Fig. 10 . The results are shown in Table II . The combination of 0.1 H, 3 μF or 0.1 H, 4 μF cannot satisfy the operating point of 20 W. The value of L_c was selected as 0.2 H in consideration of the THD of the source current and the cost. Two additional experiments were conducted to determine the exact value of C_c considering the power loss and PF by changing C_c from 2 μF to 4 μF as shown in Fig. 11 . As a result, 0.2 H and 2.4 μF were selected as the optimal values of L_c and C_c , respectively. The proposed boost converter type LED driver was built based on the proposed design procedure and verified in the laboratory. The value of input capacitor C_i was set to as small as 47 μF to decrease the maximum peak voltage across capacitor C_b in the initial transient period [16] . This LED driver can thus ensure a stable initial start-up. The experimental results for load voltage V_L , load current I_L , source power P_s , ballast power loss P_b , load power P_L , PF, and the efficiency for source voltages from 94 V _rma to 106 V _rms are summarized in Table III . Compared with a previous study [16] , the load power in the present study is perfectly regulated within 0.1% offset error without system instability. Source voltage v_s , source current i_s , rectified voltage v_i , and LED current i_L were measured in ms ( Fig. 12 ) to verify the line regulation. The measured rectified ripple voltage v_r and LED ripple current are approximately 13 V _p-p and 1 mA _p-p , respectively. With regard to the ratio of these peak-to-peak values to that of the 120 Hz ripples, themeasured loop gain at 120 Hz is 33.2 dB, whereas the simulated one ( Fig. 9 ) is 34.2 dB. The discrepancy of 1.0 dB is of no practical concern; the experiment thus verifies that the proposed design guarantees the required mitigation of an LED ripple current of 1.2 mA _p-p . LED ripple current with line frequency is substantially decreased compared with that in a previous study [16] . This experiment also shows that the selected combination of L_c and C_c meets the THD regulation for source voltages from 94 V _rms to 106 V _rms as shown in Fig. 13 . Excluding the power dissipation of the rapid-start ballast, the efficiency of the proposed boost converter-type LED driver was measured to be 88.5%, which is decent considering the hard switching. The switching waveforms of MOSFET drain voltage v_x , inductor current i_Li , load voltage v_L , and LED current i_L were measured in μs as shown in Fig. 13 ; a large reverse recovery current appeared in i_Li as expected. The prototype of the proposed LED driver is shown in Fig. 14 . The prototype is compact in size and inexpensive. The proposed boost converter-type LED driver compatible with a rapid-start ballast was proven to be very stable with fairly high PF and low THD characteristics. The very complicated 10 ^th order overall system was completely analyzed for static and dynamic characterizations through reasonably simplified models. A systematic design procedure was fully established and verified by extensive simulations and experiments as having good agreement. The flicker percentage was successfully mitigated by the proposed fast feedback controller to as low as 1% even though a small DC capacitor was used. The experiments also showed that all the design requirements and standards were completely satisfied for source voltages from 94 V _rms to 106 V _rms . Chang-Byung Park received his B.S. and M.S degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2008 and 2010, respectively He is currently pursuing his Ph.D. degree at KAIST. He has developed an LED driver IC and a column driver IC for LCDs as well as a wireless power transfer system. His current research interests include wireless power transfer systems and touch read-out circuits. Bo-Hwan Choi was born in Korea in 1988. He received his B.S. degree in electrical engineering from Sungkyunkwan University, Seoul, Korea, in 2011. He is pursuing his integrated M.S. and Ph.D. degrees at the Department of Nuclear and Quantum Engineering, KAIST, Daejeon, Korea. Jun-Pil Cheon was born in Korea in 1987. He received his B.S. degree in electrical engineering from Kwangwoon University, Seoul, Korea, in 2013. He is pursuing his M.S. degree at the Department of Nuclear and Quantum Engineering, KAIST, Daejeon, Korea. Chun-Taek Rim was born in Korea in 1963. He received his B.S. degree in electrical engineering from Kumoh Institute of Technology, Gumi, Korea, in 1985, and his M.S. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Technology (KAIST), Daejeon, Korea, in 1987 and 1990, respectively. He has been an associate professor of nuclear and quantum engineering and an adjunct to aerospace engineering in power electronics at KAIST since 2007. From 1990 to 1995, he served as a military officer at the Ministry of National Defense in Korea. He worked as a senior researcher at the Agency for Defense Development, Daejeon, Korea, from 1995 to 2003. From 1997 to 1999, he worked for Astrium in Portsmouth, U.K. He served as a senior director at the Presidential Office, Seoul, Korea, from 2003 to 2007. He was one of the developers of Korea’s first airborne and spaceborne synthetic aperture radars. He has obtained three awards from the Korean government. He is currently developing inductive power transfer systems for online electrical vehicles and is the head of the Mobile Power Electronics Lab (Tesla Lab) at KAIST. His research area includes green modes of transportation, such as electric vehicles, ships, and airplanes, and wireless power systems for robots, home appliances, and bio-medical applications. He has authored and co-authored 42 technical papers, has written three books, and holds more than 40 patents (awarded and pending). Prof. Rim is a member of the Korea Aerospace Engineering Society and the Korean Political Science Association.