# Examining the Design of Different Types of DM EMI Filters and Their Effect on EMI Noise and Control Characteristics for Cuk DC-DC Converter

Authors: Erdal Şehirli
Journal: International Transactions on Electrical Energy Systems
 (2023)

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## Abstract

In this paper, DM EMI filters including LC, LC with damping, and Pi and LCL filters are designed for DC-DC Cuk converter operated in BCM. Although, DM EMI is so important for any power converter, the effect of DM filter is not investigated in detail in literature with respect to control characteristics and DM noise reduction. The main contribution of this paper is comparisons of DM EMI filter topologies regarding control characteristics and EMI noise reduction which is not presented in the literature. Another contribution is to give a practical design methodology for EMI filter design avoiding complex mathematical procedures. Besides, the paper designs an LCL filter, popular for inverter applications, for Cuk converter that is not introduced in the literature. Examining the filter effect on the converter from the control point of view is a very complicated issue because a linearized model of the converter which is naturally nonlinear should be obtained. To do so, the converter’s small signal analysis must be derived. Therefore, for examining the effect of filters, a small signal analysis of the Cuk converter is made through an average state-space model. After that, transfer functions of each filter topology are obtained. Lastly, the entire transfer functions of the system using both the Cuk converter and each filter are acquired, separately. By using the transfer functions of overall systems, the effect of each filter topology is investigated by root locus and bode plots. Moreover, experimental verification of all filters is realized over a 15 W Cuk converter with a 25 kHz switching frequency. It is validated by applications that the LCL filter provides better control characteristics on the Cuk converter, the total inductor value is reduced by 30% and the total capacitor value is reduced by 25% compared to other filters. However, with respect to DM noise Pi and LC filters give better results.


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## Body

## 1. Introduction

One of the most critical issues of power electronics is the designing of electromagnetic interference (EMI) filters. All power electronic converters because of the higher switching frequency present some high-frequency noises to their power supplies. Such noises are classified as common mode (CM), and differential mode (DM) noises. To cancel or reduce the noise level, filter structures are used and called EMI filters. Unless these high-frequency noises cancel, it causes voltage drops on the source voltages and disturbs the operation of other equipment connected to the same source. Also, with respect to renewable energy, high-frequency noises with higher magnitudes can even damage the solar panel. Therefore, the EMI filter, especially the DM EMI filter of the converter should be designed carefully. Moreover, the most preferable converter topology is the Cuk DC-DC converter providing lower input and output ripple both at the input and output side making it possible to use lower-valued filters. However, the Cuk converter output voltage is reverse polarity with the input voltage.In literature, passive DM filter types used in power converters are introduced in [1]. The use of an LC filter is given in [2] and an LC filter with series damping in [3] for the DC-DC converter, without examining the filter effect on control parameters. Also, the resonant converter uses a similar design equation to the LC filter given in [4]. The design methodology of the LCL filter is included in [5] for inverter applications and used for grid-connected inverters [6]. Although using the LCL filter in inverter applications is so popular, its usage in the Cuk DC-DC converter is not analyzed in detail in the literature, it is analyzed for boost converter [7]. Reference [8] employs a Pi filter for the boost converter without examining the filter effect from the control point of view. Cuk converter small signal model is given in [9] with reduced state variable as an inverter, for deriving a small signal model of PV [10], an exact model without experiments [11], with stability analysis [12], controller design [13], DCM model [14] [15] circuit average model of Cuk converter is presented. Reference [16] presents the comparison of the LC and LCL filter topologies used in power factor correction converters without considering the model of the converter and control parameters. Reference [17] examines the LCL filter effect for a three-phase PWM converter with the control point of view and [18] examines it for a single-phase inverter. DM LC filter effect to control parameters is examined for the PFC boost converter in [19], for the DC-DC buck converter for the single stage [20] for multistage [21] without taking into account of input DM noises. Also, [22] for the PFC boost converter, transfer functions of active filter topologies without analyzing with linear control methods are presented and the effects to DM noise are compared. Reference [23] derives transfer functions of DM filter topologies for buck DC-DC converter without integrating the converter model, in [24] transfer functions of LC with series damping DM filter and the converter are integrated just for analysis of DM noise reduction. The effect of LC and LC with series damping DM filter to transfer function of the boost converter is examined via bode diagrams without comparing DM noises in [25].In this paper, DM EMI filters cover LCL and LC with parallel damping, and Pi and LC filters are designed for the Cuk DC-DC converter operated in boundary conduction mode (BCM). Although, DM EMI is so important for any DC-DC power converter, the effect of DM filter is not investigated in detail in literature with respect to their effect to control characteristics and DM noise reduction. The main novelty of the paper to the literature is the comparisons of the DM EMI filter topologies regarding control characteristics and EMI noise reduction for the Cuk converter. Besides, the paper also designs an LCL filter, which is so popular for inverter applications, for Cuk converter that is not introduced in literature in previous studies. For examining the effect of the filters, at first, a small signal analysis of the Cuk converter is made through an average state-space model. After that, the transfer functions of each filter topology are obtained, and their characteristics are shown. Lastly, the entire transfer functions of the system using both the Cuk converter and each filter are acquired. By using the transfer functions of the overall system, the effect of each filter is investigated by root locus and bode plots.The main contributions of the paper which are not introduced in the literature are listed as follows:(i)
Deriving transfer function of the Cuk converter with an average state-space approach by cascading each filter and comparison regarding control characteristics(ii)
Comparison of the DM EMI filter structures for the Cuk converter in terms of noise reduction(iii)
Application of LCL EMI filter to the DC-DC Cuk converterMoreover, experimental verifications of all filters are realized over a 15 W Cuk converter with a 25 kHz switching frequency. It is validated by applications that the LCL filter provides better control characteristics on the Cuk converter and the total inductor value is reduced by 30% and the total capacitor value is reduced by 25% compared to other filter topologies. However, with respect to DM noise reduction Pi and LC filters give better results.


## 2. Cuk Converter

Figure1 gives the structure of the Cuk converter having two inductors with coupled structure, two capacitors, a power switch, a diode, and a load. Cuk converter is buck-boost derived topology providing negative polarity output, lower or higher magnitude regarding input voltage. There are two main advantages of using the Cuk converter regarding other DC-DC converter topologies, the first one is providing lower or higher output voltage in terms of input voltage, and the second one is having inductors at both input and output sides ensuring less current ripples reducing filter values. However, the Cuk converter provides output voltage with negative polarity regarding input voltage as a disadvantage of the Cuk converter.Figure 1
Cuk dc-dc converter.The explanation of the Cuk converter operation is as follows: at the switch is on position,L1 stores energy, and its current increases linearly, C1 discharges to the load and L2. At the switch is off, L1 transfers its energy to C1 and L2 feeds the load. Cuk converter components are chosen with (1)–(3), as in [9]. In equations, L1min and L2min are the minimum inductor values for CCM operation, d is the duty cycle, Vin is the input voltage, ∆IL is the inductor ripple current, fs is the switching frequency, ∆VC1,∆VC2 are the ripple voltages, and Io is the output current.(1)L1min,L2min≥dVin∆ILfs,(2)C1≥dIo∆VC1fs,(3)C2≥dIo∆VC2fs.The Cuk converter mathematical model is obtained via the state-space average technique with respect to switch on and off position depicted in Figure2 for continuous conduction mode (CCM) and BCM.Figure 2
Switch on and off circuits of the Cuk converter.
(a)(b)After applying Kirchhoff’s voltage law to each circuit separately, the model is achieved in (6), in the form of (4) with the assumption of (5) as in [13]. In the equations, state variables are chosen as input side inductor current iL1, output side inductor current iL2, intermediate capacitor voltage VC1, and output capacitor voltage Vo. Also, A is the system matrix, B is the input matrix, C is the output matrix, x is the state vector, u is the input vector, and R is the load resistance.(4)x˙=Ax+Bu,(5)A=dA1+1−dA2,B=dB1+1−dB2,(6)iL1VC1iL2Vo˙=0−1+dL1001−dC10−dC1000dL200−1L21Co−1RCoiL1VC1iL2Vo+1L1000Vin.The small signal approach given in (7) must be applied to the model in (6), to acquire the Cuk converter’s small signal model in (8), as in [9]. In equations, ¯ refers to the steady-state values, and refers to the small signal values.(7)x˜˙=d¯A1+1−d¯A2x¯+A1−A2x¯+B1−B2ud,(8)iL1VC1iL2Vo˜˙=0−1+dL1001−dC10−dC1000−dL200−1L21Co−1RCoiL1VC1iL2Vo+VC1L1−iL1−iL2C1VC1L20d˜.After obtaining a small signal state-space model of the converter, the transfer function is found in (10) by using the following formula:(9)Ts=Vo˜d˜=CsI−A−1B+D,(10)Ts=Vo˜d˜=s2Vc1/L2Co−sd−iL1−iL2/L2C1Co+Vc11−d2/L1L2C1Co+1−ddVc1/L1L2C1Cos4+s3/CoR+s21−d2/L1C1+1/L2C1+d2/L1C1+s1−d2/L1C1CoR+d2/L2C1CoR+1−d2/L1L2C1Co.By using (10), the root locus and bode of the converter can be depicted in Figure 3 with the parameters L1 = L2 = 45 μH, C1 = 20 μF, C2 = 940 μF, R = 13 Ω, d = 0.5. By root locus plot, it is determined that the system has 4 poles and 2 zeros, located at −0.88 ± 2.38 × 104 j, −40 ± 3.4 × 103 j, and 6.63 × 103 ± 2.26 × 104 j. The zeros are located at the right half plane making the Cuk converter nonminimum phase.Figure 3
Cuk converter: (a) root locus diagram; (b) bode diagram.
(a)(b)It is so critical for designing a controller to obtain the transfer function of the converter and make an analysis, especially by using linear methods. However, the transfer function given in (10) is not complete because it does not contain an EMI filter. Also, most of the paper presented in the literature does not include an EMI filter for the control characteristics in the transfer function. Therefore, in the next chapter EMI filter’s transfer function is added to converter transfer functions to determine the filter effect.


## 3. EMI Filter Design

Input differential EMI filter must be used in the connection of the converter with the input source to have continuous input current without high-frequency noises, especially in BCM and DCM operation of the converter. In this paper, different kinds of filter topologies, LC, LC with series damping, and LCL and Pi filters are investigated, designed, and implemented with the converter. Furthermore, the effect of filters on to converter is analyzed by root locus and bode plots.


### 3.1. LC Filter

The structure of the LC filter havingLf at the input and parallel Cf at the output side is shown in Figure 4. It is the most preferable filter topology for power converters [26].Figure 4
LC filter.Lf and Cf values are calculated by using (11) and (12) as in [26]. In the equations, Lf is the input filter, fc is the filter corner frequency, and Cf is the filter capacitor.(11)Lf=14π2fc2Cf,(12)fs≫fc.Also, the filter transfer function is derived in (13). It is seen that the filter is second order so in the transfer function it has two roots.(13)Ts=VoVin=1s2LfCf+1.


### 3.2. LC Filter with Series Damping

In Figure5. LC filter with series damping is shown. In addition, a parallel RC circuit is added to the LC filter. The RC structure provides damping of the filter.Figure 5
LC filter with series damping.Passive components of the filter can be determined by using (14) and (15) as in [27]. In the equations, Rb is the damping resistor, Cb is the damping capacitor, Rin is the input resistance, and n is the capacitance ratio.(14)LfCfRb≪Rin,(15)ζ=n+1nLf2RbLfCf,n=CbCf.Also, the filter transfer function is obtained in (16). The filter includes two capacitors and an inductor, so it is a third-order filter. Therefore, in its transfer function, it has three poles and a zero.(16)Ts=VoVin=sRbCb+1s3RbCbCfLf+s2RbCfLf+RbCbLf+sRbCb+1.


### 3.3. PI Filter

The Pi filter is depicted in Figure6. There are two capacitors and an inductor in the filter.Figure 6
PI filter.Passive components of the Pi filter can be designed by using (11) and (12). Also, both filter capacitors have the same value as the LC filter. Although the Pi filter is a third-order filter having two capacitors and an inductor, the Pi filter transfer function is derived in (17), the same as of LC filter by neglecting source impedance.(17)Ts=VoVin=1s2LfCf+1.


### 3.4. LCL Filter

Figure7 gives the LCL filter topology. It has two inductors and a series RC circuit middle of them. Although the LCL filter is very common for inverter circuits, the usage for DC-DC converter is rare and not analyzed in detail in the literature.Figure 7
LCL filter.Passive components of the LCL filter can be designed by using (18)–(21) as in [5]. In equations, ωres is the filter angular resonance frequency, fres is the filter resonance frequency, fg is the grid frequency equals to “0” because of DC supply, r is the inductance ratio, Lf1 is the input side filter inductance, and Lf2 is the output side filter inductance.(18)Rb=13ωresCf,(19)ωres=Lf1+Lf2Lf1Lf2Cf,(20)10fg<fres<0.5fs,(21)Lf2=rLf1.Besides, the filter transfer function is derived in (22), by assuming, the output side inductor is added to the input side inductor L1 of the converter. Therefore, the filter has second-order characteristics even though it has two inductors and a capacitor.(22)Ts=VoVin=Rbs2CfLf+sCfRb+1.


## 3.1. LC Filter

The structure of the LC filter havingLf at the input and parallel Cf at the output side is shown in Figure 4. It is the most preferable filter topology for power converters [26].Figure 4
LC filter.Lf and Cf values are calculated by using (11) and (12) as in [26]. In the equations, Lf is the input filter, fc is the filter corner frequency, and Cf is the filter capacitor.(11)Lf=14π2fc2Cf,(12)fs≫fc.Also, the filter transfer function is derived in (13). It is seen that the filter is second order so in the transfer function it has two roots.(13)Ts=VoVin=1s2LfCf+1.


## 3.2. LC Filter with Series Damping

In Figure5. LC filter with series damping is shown. In addition, a parallel RC circuit is added to the LC filter. The RC structure provides damping of the filter.Figure 5
LC filter with series damping.Passive components of the filter can be determined by using (14) and (15) as in [27]. In the equations, Rb is the damping resistor, Cb is the damping capacitor, Rin is the input resistance, and n is the capacitance ratio.(14)LfCfRb≪Rin,(15)ζ=n+1nLf2RbLfCf,n=CbCf.Also, the filter transfer function is obtained in (16). The filter includes two capacitors and an inductor, so it is a third-order filter. Therefore, in its transfer function, it has three poles and a zero.(16)Ts=VoVin=sRbCb+1s3RbCbCfLf+s2RbCfLf+RbCbLf+sRbCb+1.


## 3.3. PI Filter

The Pi filter is depicted in Figure6. There are two capacitors and an inductor in the filter.Figure 6
PI filter.Passive components of the Pi filter can be designed by using (11) and (12). Also, both filter capacitors have the same value as the LC filter. Although the Pi filter is a third-order filter having two capacitors and an inductor, the Pi filter transfer function is derived in (17), the same as of LC filter by neglecting source impedance.(17)Ts=VoVin=1s2LfCf+1.


## 3.4. LCL Filter

Figure7 gives the LCL filter topology. It has two inductors and a series RC circuit middle of them. Although the LCL filter is very common for inverter circuits, the usage for DC-DC converter is rare and not analyzed in detail in the literature.Figure 7
LCL filter.Passive components of the LCL filter can be designed by using (18)–(21) as in [5]. In equations, ωres is the filter angular resonance frequency, fres is the filter resonance frequency, fg is the grid frequency equals to “0” because of DC supply, r is the inductance ratio, Lf1 is the input side filter inductance, and Lf2 is the output side filter inductance.(18)Rb=13ωresCf,(19)ωres=Lf1+Lf2Lf1Lf2Cf,(20)10fg<fres<0.5fs,(21)Lf2=rLf1.Besides, the filter transfer function is derived in (22), by assuming, the output side inductor is added to the input side inductor L1 of the converter. Therefore, the filter has second-order characteristics even though it has two inductors and a capacitor.(22)Ts=VoVin=Rbs2CfLf+sCfRb+1.


## 4. Applications

Application of the Cuk converter is carried out by using each DM EMI filter topology including LC, LC with series damping, and Pi and LCL filters separately. The application environment is shown in Figure8. Applications and measurements are conducted by TPS2024B oscilloscope, A621 current probe, and TPP0201 voltage probes. DM noises for higher frequencies are measured by using a line impedance stabilizer network (LISN) with an oscilloscope. In the study self-made LISN which can provide satisfactory results [28] is included to avoid the higher cost of commercial LISN. Also [29] presents self-made LISN devices which also give proper results. Moreover, in the presented study, EMI filter structures are compared by themselves with self-made LISN resulting in the same possible measurement error which does not change the comparisons. So commercial LISN devices just increase the cost of the study even if they may provide better measurement.Figure 8
Application environment.Passive components used in the applications after calculation by (11), (12), (14), (15), and (18)–(21) are presented in Table 1.Table 1
Passive components used in applications.
FilterLf (Lf1) (μH)Lf2Cf (μF)RdCdLC100—40——LC with damping100—402.8 Ω160μFPi100—40——LCL3535μH100.44 Ω—


### 4.1. LC Filter

By using the transfer function given in (13), the LC filter bode, and root locus diagram are depicted in Figure 9. From the root locus graph, it is seen that the filter has two poles on the imaginary axis at ±1.58 × 104 j. Also, by the bode graph, the filter has 40 dB attenuation. It is because the filter does not have damping so at 1.58 × 104 rad/s corner frequency, it has a higher peak of 143 dB.Figure 9
For LC filter: (a) root locus; (b) bode.
(a)(b)Besides, to show the filter effect for control characteristics, the transfer function of the overall system is rearranged in (23) by integrating the filter transfer function in (13) and the Cuk converter in (10). It is seen that connecting the LC filter causes two poles and a zero to transfer function. The system is a nonminimum phase having a zero at the right half plane.(23)Ts=Vo˜d˜=−32104s3+56.74s2−8,026x1012s+3,152x10174x10−9s6+3,23x10−7s5+3,317s4+263,7s3+6,055x108s2+4,546x1010s+6.657x1015.The root locus and bode diagram of (23) is sketched in Figure 10. Root locus gives that there are 6 poles and 3 zeros. It has a right half plane zero causing nonminimum phase characteristics. Poles and zeros are located at −0.889 ± 2.38 × 104 j, ±1.58 × 104 j, −40 ± 3.4 × 103 j, −3.31 × 103 ± 1.99 × 104 j, and 2.42 × 104, respectively. When the gain increased up to 0.0055 the overall systems become unstable so it should be considered while choosing control parameters.Figure 10
Cuk converter with LC filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure11 the frequency spectrum of the converter using an LC filter is shown. Figure 11(a) shows the frequency spectrum of L1 current, and Figures 11(b) and 11(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 11(b) and 11(c) that maximum noises are 24 dB at 25 kHz, 16 dB at 123 kHz, 12 dB at 174 kHz, 18 dB at 10.6 MHz, and 18 dB at 21.2 MHz.Figure 11
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LC filter.
(a)(b)(c)Furthermore, in Figure12 PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 12
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with LC filter.
(a)(b)(c)


### 4.2. LC Filter with Series Damping

By using the transfer function given in (16), LC with series damping filter’s bode and root locus diagram is depicted in Figure 13. It is the most preferable filter topology for DC-DC power converters [27] because of its damping capability [1]. However, the damping resistor efficiency may be relatively lower than the LC filter. By the root locus graph, it is seen that the filter has 3 poles and a zero located at −2.03 × 104, −7.09 × 103, −3.88 × 103, and −2.23 × 103. It is seen by the bode diagram that the filter has 40 dB attenuation.Figure 13
LC filter with series damping: (a) root locus; (b) bode.
(a)(b)Besides, to show the effect of LC with a series damping filter on the Cuk converter for control characteristics, the transfer function of the overall system is rearranged in (24) by integrating the transfer functions of the LC series damping filter in (16) and Cuk converter in (10). It is seen that LC with a series damping filter adds 3 poles and a zero to the transfer function.(24)Ts=Vo˜d˜=−14,38s4−195276s3−3,028x109s2+1,332x1014s+3,152x10171,792x10−12s7+5,615x10−8s6+0,001491s5+33,55s4+2,739x105s3+9,673x108s2+2,987x1012s+6,567x1015.The root locus and bode diagram of (24) is sketched in Figure 14. By the root locus plot, the overall system has 7 poles and 4 zeros. Also, it has right half plane zeros making nonminimum phase characteristics. The location of the poles and zeros are −0.889 ± 2.38 × 104 j, −40 ± 3.4 × 103 j, −3.88 × 103, −7.09 × 103, −2.03 × 104, −1.36 × 104 ± 2.07 × 104 j, and −2.26 × 103, 1.58 × 104, respectively. Increasing the gain up to 9.7119 × 10−4 the system becomes unstable so it should be considered while choosing control parameters.Figure 14
Cuk converter plus LC series damping filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure15, the frequency spectrum of the converter using an LC filter with series damping is shown. Figure 15(a) shows the frequency spectrum of L1 current, and Figures 15(b) and 15(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 15(b) and 15(c) that maximum noises are 24 dB at 25 kHz, 16 dB at 123 kHz, 10 dB at 174 kHz, 19 dB at 10.6 MHz, and 20 dB at 21.2 MHz.Figure 15
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LC filter with series damping.
(a)(b)(c)Furthermore, in Figure16, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 16
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo LC filter with series damping.
(a)(b)(c)


### 4.3. PI Filter

The Pi filter transfer function is the same as with the LC filter, so the same cascaded transfer function in (23) can be used for the Pi filter. Therefore, root locus and bode graphs are depicted in Figures 9 and 10 and are valid for the Pi filter as well.On the other hand, the frequency response of the input side inductor of the Cuk converter with the Pi filter is given in Figure17. Figure 17(a) shows the frequency spectrum of L1 current, and Figures 17(b) and 17(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 17(b) and 17(c) that maximum noises are 10 dB at 25 kHz, 20 dB at 123 kHz, 10 dB at 174 kHz, 17 dB at 10.6 MHz, and 19 dB at 21.2 MHz.Figure 17
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with Pi filter.
(a)(b)(c)Furthermore, in Figure18, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 18
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with Pi filter.
(a)(b)(c)


### 4.4. LCL Filter

By using the transfer function given in (22), LCL filter bode, and root locus diagram are depicted in Figure 19. By root locus plot, the filter has 2 poles and a zero, located at −6.29 × 103 ± 5.31 × 104 j, and −2.27 × 105. Also, by bode plot, it is derived that the filter has 20 dB attenuation with damping. Although the LCL filter is the most preferable filter topology, especially for inverter applications as in [6] and PWM rectifiers as in [30], the use of an LCL filter is not included for Cuk DC-DC converters in the literature. Besides, because of having a small valued damping resistor, the LCL filter has a 34.4 dB peak magnitude at 5.27 × 104 rad/s.Figure 19
LCL filter: (a) root locus; (b) bode.
(a)(b)Furthermore, to show the effect of the LCL filter on the Cuk converter for control characteristics, the transfer function of the overall system is rearranged in (25) by integrating the transfer functions of the LCL filter in (22) and Cuk converter in (10). It is seen that the LCL filter adds 2 poles and a zero to the transfer function.(25)Ts=Vo˜d˜=11913s3+3,971x108s2−5,95x1012s+9,782x10161,6x10−11s6+2,122x10−5s5+1,008s4+8308s3+3,883x108s2+7,304x1010s+2,038x1015.The root locus and bode diagram of (25) is sketched in Figure 20. It is seen by the root locus plot that the system has 6 poles and 3 zeros, located at −1.28 × 106, −1.99 ± 1.96 × 104 j, −4.89 × 104, −39 ± 2.3 × 103 j, −4.75 × 104, and 7 × 103 ± 1.1 × 104 j, respectively. Increasing the controller gain by more than 34.108 the overall system becomes unstable so it should be considered while choosing control parameters.Figure 20
Cuk converter with LCL filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure21, the frequency spectrum of the converter using an LCL filter is shown. Figure 21(a) shows the frequency spectrum of L1 current, and Figures 21(b) and 21(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 21(b) and 21(c) that maximum noises are 23 dB at 25 kHz, 17 dB at 123 kHz, 12 dB at 174 kHz, 18 dB at 10.6 MHz, and 19 dB at 21.2 MHz.Figure 21
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LCL filter.
(a)(b)(c)Furthermore, in Figure22, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 22
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with LCL filter.
(a)(b)(c)Comparison of filters with respect to controller gain by using cascaded transfer functions, attenuation of filters, and maximum gain at corner frequencies are given in Table2. It is concluded by Table 2 that for controller gain LCL filter gives the best result, however for filter attenuation by decade, the LCL filter has lower than others. Also, as a maximum gain at corner frequency, the LCL filter is better than LC and Pi filters but LC with a series filter having no peak magnitude is the best one because of the damping.Table 2
Comparison of the filter with respect to controller gain, filter attenuation, and maximum gain.
LCLC with dampPiLCLControl gain0.00559.72 × 10−40.005534.1Attenuation by decade (dB)40404020Max. gain at corner freq.143 dB at 1.58 × 104 rad/s—143 dB at 1.58 × 104 rad/s34.4 dB at 5.27 × 104 rad/sDM noise magnitudes by frequency comparisons are given in Table3. It can be concluded that 25 kHz Pi filter, 174 kHz Pi and LC with series damping filter, 23 kHz LC and LC with series damping filter, 10.6 MHz LC and LCL filter, and 21.2 MHz LC filter give better results with respect to DM noise.Table 3
Comparison of the DM noises for each filter.
Magnitude (dB)Freq.LC2425 kHz16123 kHz12174 kHz1810.6 MHz1821.2 MHzLC with damp2425 kHz16123 kHz10174 kHz1910.6 MHz2021.2 MHzPi1025 kHz20123 kHz10174 kHz1710.6 MHz1921.2 MHzLCL2325 kHz17123 kHz12174 kHz1810.6 MHz1921.2 MHzAlso, efficiencies are measured as 77.3% for LC, 76.6% for LC with damping, 78.1% for Pi, and 78.3% for LCL filter by Figures12, 16, 18, and 22.


## 4.1. LC Filter

By using the transfer function given in (13), the LC filter bode, and root locus diagram are depicted in Figure 9. From the root locus graph, it is seen that the filter has two poles on the imaginary axis at ±1.58 × 104 j. Also, by the bode graph, the filter has 40 dB attenuation. It is because the filter does not have damping so at 1.58 × 104 rad/s corner frequency, it has a higher peak of 143 dB.Figure 9
For LC filter: (a) root locus; (b) bode.
(a)(b)Besides, to show the filter effect for control characteristics, the transfer function of the overall system is rearranged in (23) by integrating the filter transfer function in (13) and the Cuk converter in (10). It is seen that connecting the LC filter causes two poles and a zero to transfer function. The system is a nonminimum phase having a zero at the right half plane.(23)Ts=Vo˜d˜=−32104s3+56.74s2−8,026x1012s+3,152x10174x10−9s6+3,23x10−7s5+3,317s4+263,7s3+6,055x108s2+4,546x1010s+6.657x1015.The root locus and bode diagram of (23) is sketched in Figure 10. Root locus gives that there are 6 poles and 3 zeros. It has a right half plane zero causing nonminimum phase characteristics. Poles and zeros are located at −0.889 ± 2.38 × 104 j, ±1.58 × 104 j, −40 ± 3.4 × 103 j, −3.31 × 103 ± 1.99 × 104 j, and 2.42 × 104, respectively. When the gain increased up to 0.0055 the overall systems become unstable so it should be considered while choosing control parameters.Figure 10
Cuk converter with LC filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure11 the frequency spectrum of the converter using an LC filter is shown. Figure 11(a) shows the frequency spectrum of L1 current, and Figures 11(b) and 11(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 11(b) and 11(c) that maximum noises are 24 dB at 25 kHz, 16 dB at 123 kHz, 12 dB at 174 kHz, 18 dB at 10.6 MHz, and 18 dB at 21.2 MHz.Figure 11
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LC filter.
(a)(b)(c)Furthermore, in Figure12 PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 12
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with LC filter.
(a)(b)(c)


## 4.2. LC Filter with Series Damping

By using the transfer function given in (16), LC with series damping filter’s bode and root locus diagram is depicted in Figure 13. It is the most preferable filter topology for DC-DC power converters [27] because of its damping capability [1]. However, the damping resistor efficiency may be relatively lower than the LC filter. By the root locus graph, it is seen that the filter has 3 poles and a zero located at −2.03 × 104, −7.09 × 103, −3.88 × 103, and −2.23 × 103. It is seen by the bode diagram that the filter has 40 dB attenuation.Figure 13
LC filter with series damping: (a) root locus; (b) bode.
(a)(b)Besides, to show the effect of LC with a series damping filter on the Cuk converter for control characteristics, the transfer function of the overall system is rearranged in (24) by integrating the transfer functions of the LC series damping filter in (16) and Cuk converter in (10). It is seen that LC with a series damping filter adds 3 poles and a zero to the transfer function.(24)Ts=Vo˜d˜=−14,38s4−195276s3−3,028x109s2+1,332x1014s+3,152x10171,792x10−12s7+5,615x10−8s6+0,001491s5+33,55s4+2,739x105s3+9,673x108s2+2,987x1012s+6,567x1015.The root locus and bode diagram of (24) is sketched in Figure 14. By the root locus plot, the overall system has 7 poles and 4 zeros. Also, it has right half plane zeros making nonminimum phase characteristics. The location of the poles and zeros are −0.889 ± 2.38 × 104 j, −40 ± 3.4 × 103 j, −3.88 × 103, −7.09 × 103, −2.03 × 104, −1.36 × 104 ± 2.07 × 104 j, and −2.26 × 103, 1.58 × 104, respectively. Increasing the gain up to 9.7119 × 10−4 the system becomes unstable so it should be considered while choosing control parameters.Figure 14
Cuk converter plus LC series damping filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure15, the frequency spectrum of the converter using an LC filter with series damping is shown. Figure 15(a) shows the frequency spectrum of L1 current, and Figures 15(b) and 15(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 15(b) and 15(c) that maximum noises are 24 dB at 25 kHz, 16 dB at 123 kHz, 10 dB at 174 kHz, 19 dB at 10.6 MHz, and 20 dB at 21.2 MHz.Figure 15
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LC filter with series damping.
(a)(b)(c)Furthermore, in Figure16, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 16
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo LC filter with series damping.
(a)(b)(c)


## 4.3. PI Filter

The Pi filter transfer function is the same as with the LC filter, so the same cascaded transfer function in (23) can be used for the Pi filter. Therefore, root locus and bode graphs are depicted in Figures 9 and 10 and are valid for the Pi filter as well.On the other hand, the frequency response of the input side inductor of the Cuk converter with the Pi filter is given in Figure17. Figure 17(a) shows the frequency spectrum of L1 current, and Figures 17(b) and 17(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 17(b) and 17(c) that maximum noises are 10 dB at 25 kHz, 20 dB at 123 kHz, 10 dB at 174 kHz, 17 dB at 10.6 MHz, and 19 dB at 21.2 MHz.Figure 17
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with Pi filter.
(a)(b)(c)Furthermore, in Figure18, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 18
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with Pi filter.
(a)(b)(c)


## 4.4. LCL Filter

By using the transfer function given in (22), LCL filter bode, and root locus diagram are depicted in Figure 19. By root locus plot, the filter has 2 poles and a zero, located at −6.29 × 103 ± 5.31 × 104 j, and −2.27 × 105. Also, by bode plot, it is derived that the filter has 20 dB attenuation with damping. Although the LCL filter is the most preferable filter topology, especially for inverter applications as in [6] and PWM rectifiers as in [30], the use of an LCL filter is not included for Cuk DC-DC converters in the literature. Besides, because of having a small valued damping resistor, the LCL filter has a 34.4 dB peak magnitude at 5.27 × 104 rad/s.Figure 19
LCL filter: (a) root locus; (b) bode.
(a)(b)Furthermore, to show the effect of the LCL filter on the Cuk converter for control characteristics, the transfer function of the overall system is rearranged in (25) by integrating the transfer functions of the LCL filter in (22) and Cuk converter in (10). It is seen that the LCL filter adds 2 poles and a zero to the transfer function.(25)Ts=Vo˜d˜=11913s3+3,971x108s2−5,95x1012s+9,782x10161,6x10−11s6+2,122x10−5s5+1,008s4+8308s3+3,883x108s2+7,304x1010s+2,038x1015.The root locus and bode diagram of (25) is sketched in Figure 20. It is seen by the root locus plot that the system has 6 poles and 3 zeros, located at −1.28 × 106, −1.99 ± 1.96 × 104 j, −4.89 × 104, −39 ± 2.3 × 103 j, −4.75 × 104, and 7 × 103 ± 1.1 × 104 j, respectively. Increasing the controller gain by more than 34.108 the overall system becomes unstable so it should be considered while choosing control parameters.Figure 20
Cuk converter with LCL filter: (a) root locus; (b) bode.
(a)(b)Besides, in Figure21, the frequency spectrum of the converter using an LCL filter is shown. Figure 21(a) shows the frequency spectrum of L1 current, and Figures 21(b) and 21(c) show input current with 25 kHz and 2.5 MHz scale, respectively. It is seen by Figures 21(b) and 21(c) that maximum noises are 23 dB at 25 kHz, 17 dB at 123 kHz, 12 dB at 174 kHz, 18 dB at 10.6 MHz, and 19 dB at 21.2 MHz.Figure 21
(a)IL1 at 25 kHz, (b) Iin at 25 kHz, and (c) Iin at 2.5 MHz with LCL filter.
(a)(b)(c)Furthermore, in Figure22, the PWM signal, L1, input-output currents, and input-output voltages are shown. It is concluded that the continuous input current having low ripple is obtained by the filter while the converter is operated at BCM.Figure 22
(a) PWM,Vin, IL1, and Vo; (b) PWM, Vin, Iin, and Vo (c) PWM, Vin, Io, and Vo Cuk converter with LCL filter.
(a)(b)(c)Comparison of filters with respect to controller gain by using cascaded transfer functions, attenuation of filters, and maximum gain at corner frequencies are given in Table2. It is concluded by Table 2 that for controller gain LCL filter gives the best result, however for filter attenuation by decade, the LCL filter has lower than others. Also, as a maximum gain at corner frequency, the LCL filter is better than LC and Pi filters but LC with a series filter having no peak magnitude is the best one because of the damping.Table 2
Comparison of the filter with respect to controller gain, filter attenuation, and maximum gain.
LCLC with dampPiLCLControl gain0.00559.72 × 10−40.005534.1Attenuation by decade (dB)40404020Max. gain at corner freq.143 dB at 1.58 × 104 rad/s—143 dB at 1.58 × 104 rad/s34.4 dB at 5.27 × 104 rad/sDM noise magnitudes by frequency comparisons are given in Table3. It can be concluded that 25 kHz Pi filter, 174 kHz Pi and LC with series damping filter, 23 kHz LC and LC with series damping filter, 10.6 MHz LC and LCL filter, and 21.2 MHz LC filter give better results with respect to DM noise.Table 3
Comparison of the DM noises for each filter.
Magnitude (dB)Freq.LC2425 kHz16123 kHz12174 kHz1810.6 MHz1821.2 MHzLC with damp2425 kHz16123 kHz10174 kHz1910.6 MHz2021.2 MHzPi1025 kHz20123 kHz10174 kHz1710.6 MHz1921.2 MHzLCL2325 kHz17123 kHz12174 kHz1810.6 MHz1921.2 MHzAlso, efficiencies are measured as 77.3% for LC, 76.6% for LC with damping, 78.1% for Pi, and 78.3% for LCL filter by Figures12, 16, 18, and 22.


## 5. Discussion

This paper examines the effect of LC, LC with series damping, and Pi and LCL DM EMI filters used at the input of the Cuk converter regarding the control point of view and DM noise reduction. In literature, generally, LC type filter is preferred for DC-DC converters, especially for Cuk converter implementation of LCL, Pi and LC with damping filters, and their comparisons are not presented as given in this paper.Firstly, transfer functions integrating the converter and filters are obtained to compare from the control point of view. It can be clearly derived by the root locus graph that the LCL filter has higher control gain for stability at 34.1 than other filter structures. The Pi and LC filter has the same controller gain but LC with a damping filter has a lower maximum controller gain for stability. Furthermore, from the bode graphs, filter attenuation of the LCL filter without considering the converter is lower than other types of filters. However, the maximum filter gain at the corner frequency of the LCL filter is lower than the Pi and LC filters because of having a damping filter. Also, LC with a damping filter has close to zero peak magnitude at the corner frequency. On the other hand, from the bode graph of integrated transfer functions, the LCL filter has a lower peak magnitude of 76.6 dB at 19.6 kHz and LC with series damping filter has a 78.1 dB peak magnitude at 23.8 kHz. Also, LC and Pi filter has higher peak magnitude. Moreover, for DM noise reduction as given in Table3, even if similar results are obtained, LC and Pi filters give better results.By using an LCL filter, total inductor value reduction is achieved by 30% and filter capacitor value is also reduced by 25% compared to other filters. Besides, the measured efficiency of the LCL filter is 78.3% higher than other types, even though the LCL filter has passive damping, thanks to the reduced passive components.However, as a limitation of the study, by using optimization methods as given in [31] for inverters, better filter value options ensure that better comparison may be chosen with the condition of convenience to (11), (15), and (19). Such optimization methods can be covered in the scope of another study due to the complicated procedure for each filter. In addition, a disadvantage of the LCL filter using passive damping is power loss on the filter resistor which may reduce the efficiency, which is another limitation of the study. By using an active filter structure or different LCL filter structure without resistor-based damping, such power loss may be avoided. Nevertheless, the active damping method can be achieved through closed-loop control of the converter which is not the scope of the presented paper, as in [32] which adds extra difficulty to the design process of the filter.


## 6. Conclusions

In this paper, an examination of the effect of different DM EMI filter structures covering LC, LC with series damping, and Pi and LCL filters for DC-DC Cuk converter is conducted in terms of control characteristics, DM noise, and efficiency which was not presented in the literature. All filters are tested through experiments with a 15 W, 25 kHz DC-DC Cuk converter. First, a linear mathematical model of the converter is achieved by the average state-space method. Then, the transfer function of the converter and integrated transfer functions with each filter are derived. After that, all integrated transfer functions are compared through Bode and root locus graphs included in linear theory. Such analysis for the DC-DC Cuk converter covering the filter structure was not presented in the literature. Owing to the linear theory analysis, it is concluded that for the DC-DC Cuk converter, the LCL filter gives better results from the control point of view.Besides, through the simplified filter design procedure, an LCL filter is designed and applied for the DC-DC Cuk converter, which was not presented in the literature, and 30% total inductor and 25% capacitor reduction are achieved compared to other structures. Furthermore, overall efficiency by using an LCL filter is higher than other structures although an LCL filter has passive damping. However, regarding DM noise comparison, which was not presented in the literature, even if similar results are obtained for each filter structure, LC and Pi filters ensure better results.As a future task, it is aimed to use one of the optimization methods for the selection of the filter values. Also, another future work is to use an active damping method through the controller design to avoid using a damping resistor.



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*Source: 7101065-2023-03-15.xml*