Transient and Steady-State Analysis of Single Switched Capacitor Converter

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Abstract


The paper derives closed form expressions for transient and steady-state operation of a DC-DC converter with single switched capacitor. To this end, the result of each switching is considered as a point in the iterative process, and the function between the points is reconstructed. As opposed to the commonly accepted approach, when each of the topologies is approximated by a first order circuit, the proposed analysis is carried out for second order circuits. This allows obtaining the waveform of output voltage ripple and paves the way to more accurate calculation of equivalent resistance. The obtained analytical expressions were verified by simulations and an excellent agreement between the results was found.

 


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Introduction

Switched capacitor converters (SCCs) are favored in some applications due to low EMI and compatibility with IC technology. Over the past few decades, ongoing research has shifted towards sophisticated SCCs with large number of capacitors and advanced control circuits. However, only a few studies use analytical methods. The analysis presented in this paper is based on the method of difference equations [1], [2], [3], which allows finding the solution for the transient and steady-state operation. Although this method is unified, it was applied previously only to the switched inductor converters [4], [5], [6]. However, some analytical methods for analysis of switched capacitor circuits that bear resemblance of the proposed one, were presented in [7], [8], [9], [10].

Let us consider the SCC shown schematically in Fig. 1(a). It comprises four switches S1S4 with on-resistances r1r4. The corresponding pair of switches is turned on/off by two non-overlapping clocks φ1 and φ2 shown in Fig. 1(b). Thus, during t1 the capacitor C1 is charged by Vin through S1, S3 and then, during t2, is discharged to the load through S2, S4.Thus, we have two topologies, which are considered separately in Sections II. Applying to each topology the Kirchhoff’s voltage and current laws (KVL and KCL), we write a system of two first-order differential equations. These equations are then solved using the Laplace transform. The solution for the first topology defines the initial conditions for the second one. This enables us to compose a system of two first-order difference equations, which is solved using the Z-transform. Thus, for a given period, n, we know the initial values of state variables and functions according to which these variables change. This is the solution is closed form.

        

Fig. 1: Considered SCC (a) and two non-overlapping clocks φ1 and φ2 (b).

 

Differential equations

  1. First topology

The switches S1 and S3 in Fig. 1(a) are turned on during nTt<nT+t1, where n is the number of period. Thus, in the first topology (Fig. 2), R1=r1+r3.

 

Fig. 2: First topology of the considered SCC.

 

The KVL and KCL equations for this circuit are:

 Vin=i1R1+VC1+VCoi1=i2+i3

or

 Vin=C1dVC1dtR1+VC1+VCoC1dVC1dt=CodVCodt+VCoRo

(1)

Let us write (1) in the Laplace domain using

 LK=KS

and

 Ldf(x)dx=sF(s)-f(0)

(2)

such that

 (sR1C1+1)VC1+VCo=R1C1VC1(n)+Vins-sC1VC1+(sCo+1Ro)VCo=CoVCo(n)-C1VC1(n)

(3)

The solution of (3) can be written as:

 VC1=(s-r1)(s-r2)s(s-s1)(s-s2)VC1(n); VCo=(s-p)(s-s1)(s-s2)VCo(n)

(4)

where

 r1,2=-λ±λ2-ω02VinVC1(n); λ=121R1+1Ro1Co+Vin-VCo(n)R1C1VC1(n); p=-Vin-VC1(n)R1CoVCo(n)+1R1C1

(5)

and

 s1,2=-α±(α2-ω02); α=121C1+1Co1R1+1RoCo; ω02=1R1C1RoCo

(6)

The inverse Laplace transform of (4) is:

 L-1VC1=(r1-s1)(r2-s1)s1(s1-s2)es1t-(r1-s2)(r2-s2)s2(s1-s2)es2t+r1r2s1s2VC1(n)

L-1VCo=(s2-p)es2t-(s1-p)es1ts2-s1VCo(n)

(7)

Using  and

  t=t1

 r1r2=ω02VinVC1(n); s1s2=ω02; r1+r2=-2λ

(8)

we can rewrite (7) as:

 VC1(t1)=a+b(Vin-VCo(n))+cVC1(n)

 VCo(t1)=d(Vin-VC1(n))+eVCo(n)

(9)

where

 a=1+s1es2t1-s2es1t1s2-s1Vin; b=es2t1-es1t1s2-s11R1C1; 

c=s2+1R1+1Ro1Coes2t1-s1+1R1+1Ro1Coes1t1s2-s1; 

d=es2t1-es1t1s2-s11R1Co; e=s2+1R1C1es2t1-s1+1R1C1es1t1s2-s1

(10)

 

  1. Second topology

The switches S2 and S4 in Fig. 1(a) are turned on during nT+t1t<(n+1)T, such that in the second topology (Fig. 3) R2=r2+r4.

 

Fig. 3: Second topology of the considered SCC.

 

The KVL and KCL equations for this circuit are:

 VC1=-i1R2+VCo-i1=i2+i3

or

 VC1=-C1dVC1dtR2+VCo-C1dVC1dt=CodVCodt+VCoRo

(11)

Using (2), we write (11) in the Laplace domain:

 (sR2C1+1)VC1-VCo=R2C1VC1(n)sC1VC1+sCo+1RoVCo=C1VC1(n)+CoVCo(n)

(12)

The solution of (12) is:

 VC1=(s-p1)(s-s1)(s-s2)VC1(n);VCo=(s-p2)(s-s1)(s-s2)VCo(n)

(13)

where

 p1=-VC1(n)CoVCo(n)+1C11R2; 

 p2=-1Ro+1R21Co-1R2C1VCo(n)VC1(n)

(14)

and

 s1,2=-α±(α2-ω02); α=121C1+1Co1R2+1RoCo; ω02=1R2C1RoCo

(15)

The inverse Laplace transform of (13) is:

 L-1VC1=(s2-p1)es2t-(s1-p1)es1ts2-s1VC1(n) 

L-1VCo=(s2-p2)es2t-(s1-p2)es1ts2-s1VCo(n)

(16)

Since for the second topology t1tt2, the initial conditions will be VC1(n)=VC1(t1) and VCo(n)=VCo(t1), whereas t=t2-t1.

 

Difference equations

Substituting (14) and (15) into (16), we obtain the recurrent equations:

 VC1(n+1)=fVC1(t1)+gVCo(t1)

 VCo(n+1)=hVC1(t1)+kVCo(t1)

(17)

where

 f=s2+1Ro+1R21Coes2(t2-t1)-s1+1Ro+1R21Coes1(t2-t1)s2-s1

g=es2(t2-t1)-es1(t2-t1)s2-s11R2C1; h=es2(t2-t1)-es1(t2-t1)s2-s11R2Co

k=s2+1R2C1es2(t2-t1)-s1+1R2C1es1(t2-t1)s2-s1

(18)

Now, substituting (9) into (17), we obtain the difference equations:

 

 VC1(n+1)=A+BVCo(n)+GVC1(n)VCo(n+1)=E+FVCo(n)+HVC1(n)

(19)

where

 A=(fb+gd)Vin+fa;B=ge-fb;G=fc-gdE=(hb+kd)Vin+ha;F=ke-hb;H=hc-kd

(20)

Before we proceed to the solution of (19), let us consider the steady-state operation, where

 VC1(n+1)=VC1(n)

 VCo(n+1)=VCo(n)

(21)

substituting (21) into (19), we have

 

 (G-1)VC1(n)+BVCo(n)=-AHVC1(n)+(F-1)VCo(n)=-E

(22)

The solution of (22) is:

 VC1(n)=BE-A(F-1)(F-1)(G-1)-BH

 VCo(n)=HA-E(G-1)(F-1)(G-1)-BH

(23)

Let us write (19) in the Z-domain using

 ZK=Kzz-1

and

 Zfn+1=zF(n)-f(0)

(24)

such that

 (z-G)VC1(z)-BVCo(z)=Azz-1+zVC1(0)-HVC1(z)+(z-F)VCo(z)=Ezz-1+zVCo(0)

(25)

The solution of (25) is:

 VC1(z)=(z-F)VC1(0)+BVCo(0)(z-z1)(z-z2)z+(z-F)A+BE(z-z1)(z-z2)zz-1VCo(z)=HVC1(0)+(z-G)VCo(0)(z-z1)(z-z2)z+HA+(z-G)E(z-z1)(z-z2)zz-1

(26)

where

 

 z1,2=G+F±((G+F)2-4(GF-BH))2

(27)

Since (26) is represented in the form of partial fractions, we can apply the inverse Z-transform to each term separately:

 

 VC1(n)=A(z2-F)+BEz2-z1z2n-1z2-1-A(z1-F)+BEz2-z1z1n-1z1-1+(z2-F)z2n-(z1-F)z1nz2-z1VC1(0)+z2n-z1nz2-z1BVCo(0)

VCo(n)=E(z2-G)+AHz2-z1z2n-1z2-1-E(z1-G)+AHz2-z1z1n-1z1-1+(z2-G)z2n-(z1-G)z1nz2-z1VCo(0)+z2n-z1nz2-z1HVC1(0)

(28)

 

Simulation results

To verify the obtained analytical expressions (9), (17) and (28), the circuit shown in Fig. 4 was simulated in PSIM 9.0. Since the PSIM bidirectional switches have zero on-resistance, two external resistors corresponding to R1 and R2 in Fig. 2 and Fig. 3 were added. The parameters of the circuit in Fig. 4 are as follows: Vin=10V, VC1(0)=VCo(0)=0V, t1=t2=5μs,R1=R2=1Ω, C1=10μF, Ro=100Ω,  and Co=100μF.

 

Fig. 4: Simulation circuit for the voltage-halving SCC.

 

Since R1=R2 and t1=t2,, the expressions (10) and (18) are reduced to:

 a=1+s1es2t1-s2es1t1s2-s1Vin;b=g=es2t1-es1t1s2-s11R1C1;

c=f=s2+1R1+1Ro1Coes2t1-s1+1R1+1Ro1Coes1t1s2-s1;

e=k=s2+1R1C1es2t1-s1+1R1C1es1t1s2-s1;d=h=bC1Co

(29)

These constants are substituted into (20) and then into (28), which sets the initial conditions for (9) and (17). The voltages across C1 and Co during the first ten periods ( 0t10T) is shown Fig. 5, which compares the MathCAD calculation and PSIM simulation.

 

Fig. 5: Voltages across C1 and Co for 0t10T0t10T, MathCAD (a) and PSIM (b). The horizontal scale is 10μsdiv, i.e. each division is T.

 

The voltages in Fig. 5 were measured at the points  and are given in Table I along with the relative error, ε.

 

Table I: Measured values of the voltages in Fig. 5.

  

[%]

 

[%]

Calc.

Simul.

Calc.

Simul.

1

2.5146

2.5046

0.398

0.5174

0.5119

1.074

2

3.4986

3.4937

0.140

0.9542

0.9449

0.984

3

3.9147

3.9144

0.008

1.3381

1.3259

0.920

4

4.1172

4.1193

0.051

1.6815

1.6670

0.870

5

4.2366

4.2397

0.073

1.9909

1.9747

0.820

6

4.3211

4.3254

0.010

2.2706

2.2531

0.777

7

4.3890

4.3943

0.121

2.5236

2.5054

0.726

8

4.4474

4.4534

0.135

2.7526

2.7339

0.684

9

4.4991

4.5055

0.142

2.9601

2.9410

0.649

10

4.5454

4.5455

0.002

7.1478

7.1282

0.627

 

Fig. 6 compares the MathCAD calculation and PSIM simulation for 0t100T. Note that at t60T the SCC reaches the steady-state.

 

Fig. 6: Voltages across C1 and Co for 0t100T, MathCAD (a) and PSIM (b). The horizontal scale is 100μsdiv, i.e. each division is 10T.

 

The steady-state voltages are shown Fig. 7. Their discrete values were calculated by (23) and then substituted as the initial conditions into (9) and (17).

 

Fig. 7: Steady-state voltages across C1 and Co for 95Tt100T.

 

Conclusion

Based on the method of difference equations the closed form expressions for the voltages across the capacitors in the voltage-halving SCC were derived. The solution of these equations allows us to predict the SCC behavior in both the transient and steady-state operation. That is for a given period, , we know the initial values of the voltages and functions according to which these voltages change. The obtained expressions were verified by simulations. As evident from Table I, the deviation between the theoretical and simulation results does not exceed 1.1%. The used method however, is very complex even in the case of the two-phase SCC and its extension to the multi-phase SCC will apparently require some special assumptions.

About the authors

Alexander Kushnerov

Ben-Gurion University of the Negev

Author for correspondence.
Email: kushnero@ee.bgu.ac.il

Israel, Beersheba

Department of Electrical and Computer Engineering

References

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  6. N. Krihely, M. A. Slonim and S. Ben-Yaakov, "Transient and steady-state analysis of three-phase PWM buck rectifier," IET Power Electronics, vol. 5, no. 9, pp. 1764-1775, 2012.
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  9. M. E. Karagozler, S. C. Goldstein and D. S. Ricketts, "Analysis and modeling of capacitive power transfer in microsystems," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 7, pp. 1557-1566, 2012.
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Supplementary files

Supplementary Files Action
1.
Fig. 1: Considered SCC (a) and two non-overlapping clocks and (b).

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2.
Fig. 2: First topology of the considered SCC.

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3.
Fig. 3: Second topology of the considered SCC.

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4.
Fig. 4: Simulation circuit for the voltage-halving SCC.

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5.
Fig. 5: Voltages across and for , MathCAD (a) and PSIM (b).

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6.
Fig. 6: Voltages across and for , MathCAD (a) and PSIM (b).

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7.
Fig. 7: Steady-state voltages across and for .

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