Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[1]
Switched Capacitors Converters
Sam Ben-Yaakov
Power Electronics Laboratory
Department of Electrical and Computer Engineering
Ben-Gurion University of the Negev
P.O. Box 653, Beer-Sheva 84105, ISRAEL
Phone: +972-8-646-1561; Fax: +972-8-647-2949;
Email: sby@ee. bgu.ac.il; Website: www.ee.bgu.ac.il/~pel
APEC09, February 2009
Full set of slides:
http://www.ee.bgu.ac.il/~pel/seminars/APEC09.pdf
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[2]
OUTLINE
1. Introduction (30min)
Switched capacitors versus switched inductors converters
Charge Pumps and Switched Capacitors converters
Losses in switched capacitors converters – overview
Examples of SCC and charge pump topologies
2. Losses in Hard Switched SCC (50 min)
Target voltages
Equivalent resistance
Efficiency
Inherent power loss
Effect of switch resistances
Equivalent-circuit based average models – New Approach
Regulation
Examples
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[3]
OUTLINE
3 Losses in Soft Switched SCC - New Results (50 min)
Topologies
Waveforms of resonant networks
Losses in resonant networks
Parasitic
Equivalent-circuit based average models
Regulation
Examples
4 Self-Adjusting Binary SCC (50 min) - New Concept
The concept
The Extended Binary (EXB) numbers representation
Features of the EXB
Translating the EXB to SCC topologies
Proof of solution
Examples –simulation – experimental results
Efficiency – output resistance
Regulation
Examples
5. Q&A
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Power Conversion Objective
y Needed in all modern systems
y Except: incandescent lamps, heaters…
[4]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[5]
Linear Voltage Regulator
η=
Pout Vout ⋅ I out
=
Pin
Vin ⋅ I in
since
I out ≅ I in
η=
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Vout
Vin
[6]
Types of Switching DC-DC Converters
Switched inductor
Lossless process
Switched capacitor
Lossy process
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[7]
Inherent Energy Loss due to ΔV
For complete charge/discharge
Rp
Sw
V1 ≠ V2
C1
ΔV = V2 − V1
CV12
+ QV1
2
Q = C( V2 − V1) V1
E0 =
E1 =
CV22
2
Lossy
process
C (ΔV )2
E1 − E0 = ΔE = 1
2
C2
V1
E0 =
C1V12 C2V22
+
2
2
E1 =
(C1V1 + C2V2 )2
2 (C1 + C2 )
V2
C1C2 (ΔV )
C1 + C2 2
2
E1 − E0 = ΔE =
Independent of parasitic resistances
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[8]
Types of the Switching DC-DC Converters
Lossless Switching
Lossy switching
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[9]
Relevancy of the Switched Capacitor
Converters (SCC)
Advantages
☺ No magnetic elements
☺ Minimal EM interferences
☺ Can be fabricated as IC
Disadvantages
Inherent power losses
Relatively large number of switches
High inrush current at start-up
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
The concept of Equivalent Circuit
y The input voltage is divided or multiplied by k
y The losses are emulated by equivalent resistor Req
Target voltage
TR
Vout
= kVin
⇓
η=
Vout
TR
Vout
[10]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[11]
Output Resistance in Charge Transfer
(The switched capacitor approximate approach)
y The output capacitor is sufficiently large
y The output voltage is averaged to DC
y The charge/discharge process is completed
Iavg = C1
R eq =
Peq =
Vin − Vout C1 ΔV
=
T
T
ΔV T
1
=
=
Iavg C1 f ⋅ C1
(ΔV )2
R eq
= f ⋅ C1 (ΔV )
2
Independent of parasitic resistances
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Output Resistance- Doubler
[12]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[13]
Output Resistance- Doubler
What is going on???
To be completely deciphered in this seminar
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[14]
Target Voltages
No-Load No-Loss
Target Voltages
Vc1 = Vc 2 = Vc3
Vout = Vin − Vc1 − Vc 2 − Vc3
Vout = Vc1
1
Vout = Vin
4
TargetVoltage =
1
Vin
4
Vc1 = Vc 2 = Vc3
Vout = Vc1 + Vc 2 + Vc3
Vout = Vc1
3
Vout = Vin
4
Solution of algebraic equations
TargetVoltage =
3
Vin
4
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[15]
Multiple Target Voltage Ratios
y Number of target voltage ratios is limited
y Target voltage ratios are spread apart
N = 1;
Vout 1
=
Vin 2
N = 2;
Vout 1
=
Vin 3
N = 2;
Vout 2
=
Vin 3
N = 3;
Vout 1
=
Vin 4
N = 3;
Vout 3
=
Vin 4
N=number of capacitors
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Commercial SC Converters
y Maximum efficiency at the fixed voltage ratios:
2/3 and 1/2
Can it be improved ?
[16]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[17]
Soft Switched SCC
Sinusoidal rather than exponential currents
Claimed to be of low loss
Soft switching – does it help reduce losses?
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Classic Dickson’s charge pump
Using diodes
[18]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[19]
Dickson’s charge pump
Using MOSFETs as diodes
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[20]
Dickson’s charge pump
Using MOSFETs
as switches
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[21]
Dickson’s charge pump
Operational
modes
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Charge-pump/Switched-capacitor
The same operation principle
[22]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[23]
Charge-pump/Switched-capacitor
Many other modern charge pump topologies
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
2. Losses in Hard Switched SCC
Features of the new model presented here:
Average model
Relating the losses to the output current
Generic – applicable to practically any SCC
Can take into account output capacitor
Takes into account diode losses
Unified – applicable to hard and soft switched SCC
Has it’s own limitations….
[24]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[25]
The Generic Charging/Discharging Process
1:1 converter
ΔV≡ Voltage difference before
switch closure
t
ΔV − RC
i(t) =
⋅e
; τ = RC
R
PR = i(t) 2 ⋅ R =
t1
E R = ∫ PR dt =
0
R = R S1 + R ESR
ER =
ΔV 2 ⋅ e −2t/τ
R
t
ΔV 2 1 − 2t/τ
dt
⋅ ∫ e
R 0
(
ΔV 2 ⋅ C
1 − e − 2β
2
)
t
β= 1
τ
ER= Energy dissipated during switch closure time t1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[26]
The Generic Charging/Discharging Process
Energy Dissipated in each switching period
)
t
;β = 1
For β >> 1 → E R =
ΔV 2 ⋅ C
2
ER =
ex
(
ΔV 2 ⋅ C
1 − e − 2β
2
x →0
τ
= 1 + x ...
ER β→0 =
ΔV 2
t1
R
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[27]
The Generic Charging/Discharging Process
Relating the losses to capacitor’s current
Average current through capacitor
t
t
ΔV 1 − RC
⋅ ∫ e
Qc =
dt
R 0
Qc ⋅ fs = IC1(avg)
(
Iout(avg) = fs ΔV ⋅ C ⋅ 1 − e −β
ΔV =
ER =
iC
ΔV
R
)
0
ΔV
R
IC1(avg)
τ
t1
t
t1
t
iC
fsC ⋅ (1 − e −β )
ΔV 2 ⋅ C
⋅ (1 − e − 2β )
2
0
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[28]
The Generic Charging/Discharging Process
Energy lost per switching period
2
⎛
⎞ C
IC(avg)
⎟ ⋅ ⋅ (1 − e − 2β )
ER = ⎜
⎜ f C ⋅ (1 − e −β ) ⎟ 2
⎝ s
⎠
(1 − e −2β )
(1 − e
−β 2
)
=
(1 − e −β )(1 + e −β )
(1 − e
−β
)(1 − e
−β
)
=
(1 + e −β )
(1 − e
−β
−x
⎛ x ⎞ 1+ e
coth⎜ ⎟ =
−
⎝ 2 ⎠ 1− e x
)
2
2
⎛ IC (avg) ⎞ C (1 + e −β ) ⎛ IC1(avg) ⎞ C
β
⎟ ⋅ coth( )
⎟ ⋅ ⋅
ER = ⎜⎜ 1
= ⎜⎜
⎟ 2
⎟ 2
−β
f
C
f
C
2
(1 − e ) ⎝ s
⎠
⎠
⎝ s
Taking into account deadtime
⎛ 1
⎞ 1
β = ⎜⎜
− DeadTime ⎟⎟ ⋅
2f
⎝ s
⎠ RC
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[29]
The Generic Charging/Discharging Process
Relating the losses to capacitor’s current
2
2
⎛ IC (avg) ⎞ C (1 + e −β ) ⎛ IC1(avg) ⎞ C
β
⎟ ⋅ coth( )
⎟ ⋅ ⋅
ER = ⎜⎜ 1
= ⎜⎜
⎟
⎟
β
−
2
⎝ fsC ⎠ 2 (1 − e ) ⎝ fsC ⎠ 2
P=
β1 =
E charging + E discharging
TS
t1
R1C
β2 =
t2
R 2C
R1 = Rs1 + RESR
R 2 = Rs2 + RESR
Energy lost per
switching cycle
IC1(avg) = IC2 (avg) = IC(avg)
⎧⎪ 1
2
PR(avg) = IC(avg)
⋅⎨
⎪⎩ 2Cfs
⎡ (1 + e−β1 ) (1 + e−β2 ) ⎤ ⎫⎪
⋅⎢
+
⎥⎬
−β
−β
⎢⎣ (1 − e 1 ) (1 − e 2 ) ⎥⎦ ⎪⎭
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[30]
SCC Equivalent Resistance
1:1 converter
Assuming
V TRG
out = Vin
β1 = β1 = β
⎡ 1
⎛ β ⎞⎤
2
PR(avg) = I out(avg)
⋅⎢
⋅ coth⎜ ⎟⎥
⎝ 2 ⎠⎦
⎣ fsC
R eq =
1
1 (1 + e −β )
⎛β ⎞
⋅ coth⎜ ⎟ =
⋅
fsC
⎝ 2 ⎠ f s C (1 − e − β )
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[31]
SCC Equivalent Resistance
Limits β→ ∞
Complete charge/discharge RC<<Ts
D2Ts
RC
⎧⎪ 1
Req = ⎨
⎪⎩ 2Cf s
β2 =
D1Ts
RC
⎡ (1 + e − β1 ) (1 + e − β2 ) ⎤ ⎫⎪
⋅⎢
+
− β1
− β2 ⎥ ⎬
⎣ (1 − e ) (1 − e ) ⎦ ⎪⎭
Re β>>1 =
β1 =
1
fsC
0
τ
t1
t
Independent of R
High losses (large rms currents)
The classical solution
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
SCC Equivalent Resistance
Incomplete charge/discharge RC>>Ts; β→0
For t1= t2=Ts/2
⎧⎪ 1
Req = ⎨
⎪⎩ 2Cf s
⎡ (1 + e − β1 ) (1 + e − β2
⋅⎢
+
− β1
− β2
⎣ (1 − e ) (1 − e
) ⎤ ⎫⎪
⎥⎬
) ⎦ ⎪⎭
⎡ 1 + e −β ⎤
⎥
β⎢
⎢ 1 − e −β ⎥
⎦
⎣
1 (1 + e −β )
R eq =
⋅
f sC (1 − e −β )
⎧⎪ 2R (1 + e −β ) ⎫⎪
(1 + e −β )
⋅
Re = ⎨
⎬ = 2Rβ ⋅
−
β
⎪⎩ 2fsCR (1 − e ) ⎪⎭
(1 − e −β )
Re β→0 = 4R
Why??
[32]
β=
Ts
2 RC
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[33]
SCC Equivalent Resistance
Behavior
Re β→0 = 4R
PR =
For t1 = t2=Ts/2
β=
Ts
2 RC
(2I out )2 ⋅ Ts + (2I out )2 Ts
2
2 R
Ts
(2 * Io )2 R = (I0 )2 ⋅ 4R
Re β →0 = 4 R
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[34]
SCC Equivalent Resistance
Frequency dependence
⎧⎪ 1 ⎡ (1 + e − β1 ) (1 + e − β2 ) ⎤ ⎪⎫
⋅⎢
+
Req = ⎨
−β
− β ⎥⎬
⎪⎩ 2Cf s ⎣ (1 − e 1 ) (1 − e 2 ) ⎦ ⎪⎭
⎧ 1
R eq = ⎨
⎩ Cfs
β =
⎡ (1 + e −β ) ⎤ ⎫
⋅⎢
−β ⎥ ⎬
⎣ (1 − e ) ⎦ ⎭
10
Re
1
2 RCf s
1
fs
⎡ 1 + e(−1 fs ) ⎤
⎢
⎥
⎢ 1 − e(−1 fs ) ⎥
⎣
⎦
1
=1
2 RC
2
1
0.1
1
f
10
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
SCC Equivalent Resistance
Incomplete charge/discharge RC>>Ts; β→0
For t1= t2=Ts/2
−β
−β
⎧⎪ 2R (1 + e ) ⎪⎫
(1 + e )
⋅
Re = ⎨
⎬ = 2Rβ ⋅
⎪⎩ 2fsCR (1 − e −β ) ⎪⎭
(1 − e −β )
β=
Ts
2 RC
⎡ 1 + e −β ⎤
⎥
β⎢
⎢ 1 − e −β ⎥
⎦
⎣
Sizing C
β=
[35]
Ts
<1
2RC
C>
1
2Rfs
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[36]
SCC Equivalent Resistance
Effect of Duty Cycle
⎧⎪ 1 ⎡ (1 + e − β1 ) (1 + e − β2 ) ⎤ ⎪⎫
⋅⎢
+
Req = ⎨
−β
− β ⎥⎬
⎪⎩ 2Cf s ⎣ (1 − e 1 ) (1 − e 2 ) ⎦ ⎪⎭
Re
R eq =
1 ⎧
⎛β ⎞
⎛ β ⎞⎫
⋅ ⎨coth⎜ 1 ⎟ + coth⎜ 1 ⎟ ⎬
2fsC ⎩
⎝2⎠
⎝ 2 ⎠⎭
20
D
1- D
−
−
⎡
⎤
fs
1 ⎢1 + e
1 + e fs ⎥
+
D
1- D ⎥
−
−
fs ⎢
fs
⎢⎣1 − e
1 − e fs ⎥⎦ 1
=1
D=0.2
10
D=0.5
0.1
2 RC
D=0.3
D=0.4
1
f
10
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[37]
SCC Equivalent Resistance
Effect of duty cycle β→0
⎧⎪ 1
Req = ⎨
⎪⎩ 2Cf s
R eq =
⎡ (1 + e − β1 ) (1 + e − β2
⋅⎢
+
− β1
− β2
⎣ (1 − e ) (1 − e
) ⎤ ⎫⎪
⎥⎬
) ⎦ ⎪⎭
R eq =
1 ⎧
⎛β ⎞
⎛ β ⎞⎫
⋅ ⎨coth⎜ 1 ⎟ + coth⎜ 1 ⎟ ⎬
2fsC ⎩
2
⎝ ⎠
⎝ 2 ⎠⎭
β1 =
1 ⎧2
2⎫
⋅⎨ + ⎬
2fsC ⎩ β1 β 2 ⎭
D
f s RC
β2 =
1− D
f s RC
R ⎫
⎧R
R eq = ⎨ +
⎬
⎩D 1− D ⎭
Req
β →0
=
R
(1 − D)D
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[38]
Re as a function of Duty Cycle
Explanation
2
2
⎛I ⎞
⎛I
⎞
PR = ⎜ out ⎟ ⋅ D ⋅ R + ⎜ out ⎟ ⋅ (1 − D) ⋅ R
−
1
D
D
⎝
⎠
⎝
⎠
R ⎞ 2
2 ⎛R
PR = Iout
⎜ +
⎟ = IoutRe
D
1
−
D⎠
⎝
Re =
R
R
R
+
=
D 1 − D D(1 − D)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[39]
Simulation/Experimental Demonstartion
Mosfets S1, S2: IRF840, Rdson = 0.85Ω, C = 1μF
Vin = 24V; Vout_theoretical = 24V;
RL = 100Ω || 1K Ω ~ 91Ω or 1KΩ; Duty Cycle = 0.5;
Power Level: 6.3 Watts (max)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[40]
Simulation/Experimental Demonstartion
20KHz
200KHz
Vout= 22.58V
94% Efficiency
RL =91 Ω
Vout= 15.49V
64% Efficiency
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[41]
Simulation/Experimental Demonstartion
D = 0.5; Vout = 20.58V;
Vin = 24V; Vout_target = 24V
RL ~ 91Ω;
Rs = 3.35Ω; fs = 200KHz;
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[42]
Simulation/Experimental Demonstration
D = 0.7; Vout = 19.94V;
D = 0.9; Vout = 14.456V;
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[43]
SCC Equivalent Resistance
Generalization
⎧⎪ 1 ⎡ (1 + e −β1 ) ⎤ ⎡ (1 + e −β 2 ) ⎤ ⎫⎪
2
⋅
⋅⎢
PCi (avg) = IC
⎥+⎢
⎥⎬
⎨
−β
−β
i (avg)
⎪⎩ 2Cfs ⎣⎢ (1 − e 1 ) ⎦⎥ ⎣⎢ (1 − e 2 ) ⎦⎥ ⎪⎭
β1, 2 =
t1, 2
τ 1, 2
I Ci = ki I out (avg )
⎧⎪ 1
R eC = k 2 ⋅ ⎨
i
⎪⎩ 2Cfs
⎡ (1 + e −β1 ) ⎤ ⎡ (1 + e −β 2 ) ⎤ ⎫⎪
⋅⎢
⎥+⎢
⎥⎬
−β
−β
⎢⎣ (1 − e 1 ) ⎥⎦ ⎢⎣ (1 − e 2 ) ⎥⎦ ⎪⎭
PCi = (I o ) ⋅ ReCi
2
n
Re = ∑ ReCi
i=1
Ci = flying capacitor i
τ = time constant of charge/discharge loop
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[44]
SCC Equivalent Resistance
1/2 converter
C
VIn
COut
IOut
l o ut ( avg ) = lC1( avg ) + lC2 ( avg )
⎧⎪ 1 (1 + e −β ) ⎫⎪
⋅
Re = k 2 ⎨
⎬
⎪⎩ fsC (1 − e −β ) ⎪⎭
Re =
1 ⎧⎪ 1 (1 + e −β ) ⎫⎪
⋅
⎨
⎬
4 ⎪⎩ fsC (1 − e −β ) ⎪⎭
lC ( avg ) =
l o ut ( avg )
2
1
1
=
R1C1 R2C2
→ k = 1/ 2
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[45]
SCC Equivalent Resistance
Example
I C + 3I C = I out
k = 1/ 4
In general τ 1 ≠ τ 1
In practice τ 1 = τ 1
ReC i
1 ⎧ 1 (1 + e − β ) ⎫
= ⎨
⋅
⎬
16 ⎩ f sC (1 − e − β ) ⎭
⎡ 1 ⎧⎪ 1 (1 + e −β ) ⎫⎪⎤
ReT = 3 ⎢ ⎨
⋅
⎬⎥
⎢⎣ 16 ⎪⎩ fsC (1 − e −β ) ⎪⎭⎥⎦
Assuming equal size capacitors
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[46]
Including Finite Output Capacitor
RS1 S1 i(t) RS2 S2
Vin
β1,2 =
C1
Co
R ESR
R ESR(out)
t1,2
τ 1,2
τ1 = (RS2 + RESR )C1
Ro
Vo
⎧⎪ 1 ⎡ (1 + e −β1 ) ⎤ ⎡ (1 + e −β 2 ) ⎤ ⎫⎪
⋅⎢
R eC = k 2 ⋅ ⎨
⎥+⎢
⎥⎬
−β
−β
i
⎪⎩ 2Cfs ⎣⎢ (1 − e 1 ) ⎦⎥ ⎣⎢ (1 − e 2 ) ⎦⎥ ⎪⎭
⎧ CC ⎫
τ2 = (RS1 + RESR + RESRo )⎨ 1 o ⎬
⎩ C1 + Co ⎭
n
Re = ∑ ReCi
i=1
Including Co (could be neglected in practical cases)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[47]
Including diodes
Step up X3
IC=ID =Iout(avg)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[48]
Including diodes
Step up X3
VTRG = 3 × Vin
Req =
⎡
⎛ β ⎞⎤
⎛β ⎞
⎛β ⎞
⎛ β ⎞ C +C
1
⋅ coth⎜⎜ (2) ⎟⎟ + coth⎜⎜ (3) ⎟⎟ + 2 ⋅ coth⎜⎜ (4) ⎟⎟⎥
⋅ ⎢2 ⋅ coth⎜⎜ (1) ⎟⎟ + out
2f S ⋅ C ⎣
2
C
2
2
⎝ 2 ⎠⎦
⎠
⎝
⎠
⎝
⎠
⎝
out
β (1) =
t2
(R1 + 2ESR) ⋅ C/2
β (3) =
t1
(R1 + ESR) ⋅ C
β (2) =
t2
⎛ C ⋅C ⎞
⎟⎟
(R1 + ESR + ESR out ) ⋅ ⎜⎜ out
⎝ C out + C ⎠
β (4) =
t1
(R1 + 2ESR) ⋅ C/2
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[49]
Including diodes/Simulation
Step up X3
PSIM Parameters file
Rout
= 1000
Cout
= 100u
C_init
= 23.22
Vin
= 10
C_fly
= 1uF
R1
= 1m
ESR
= 10m
ESR_out = 100m
V_forward = 1
DeadTime = 10n
f_s
= 100k
k_2
= ((Cout + C_fly) / Cout)
beta _1 = (1 / (2 * f_s) - DeadTime) * (1 / ((R1+ 2*ESR) * (C_fly/2)))
beta _2 = (1 / (2 * f_s) - DeadTime) * (1 / ((R1 + ESR + ESR_out) * (Cout * C_fly /(C_fly + Cout)) ))
beta _3 = (1 / (2 * f_s) - DeadTime) * (1 / ((R1+ ESR) * C_fly))
beta _4 = (1 / (2 * f_s) - DeadTime) * (1 / ((R1+ 2*ESR) * (C_fly/2)))
coth_1 = (1 + exp(-beta_1)) / (1- exp(-beta_1))
coth_2 = (1 + exp(-beta_2)) / (1- exp(-beta_2))
coth_3 = (1 + exp(-beta_3)) / (1- exp(-beta_3))
coth_4 = (1 + exp(-beta_4)) / (1- exp(-beta_4))
Req = ((1 / (2 * f_s * C_fly)) * (2 * coth_1+ k_2 * coth_2 + coth_3 + 2 * coth_4))
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[50]
Including diodes
Step up X3
Rout
[Ω]
Cout
[F]
Vin
[V]
Cfly
[F]
R1
[Ω]
ESR
[Ω]
ESRout
[Ω]
1000
100μ
10
1μ
100m
10m
10m
time
Vforward
[V]
1
Dead
Time
[sec]
10n
fS
[Hz]
100k
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[51]
Complications
Rout
[Ω]
1000
Cout
[F]
100μ
Vin
[V]
10
Cfly
[F]
1μ
R1
[Ω]
10
ESR
[Ω]
ESRo
Vforwa
ut
rd
10m
[Ω]
[V]
Dead
Time
[sec]
10m
1
10n
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Coupled Loops
R1 is shared by two loops
Small effect in practical cases
fS
[Hz]
100k
[52]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[53]
Output Voltage Regulation
β1 =
⎧⎪ 1
R eC = k 2 ⋅ ⎨
i
⎪⎩ 2Cfs
D1Ts
RC
β2 =
D2Ts
RC
⎡ (1 + e −β1 ) ⎤ ⎡ (1 + e −β 2 ) ⎤ ⎫⎪
⋅⎢
⎥+⎢
⎥
−β1
−β 2 ⎬
⎣⎢ (1 − e ) ⎦⎥ ⎣⎢ (1 − e ) ⎦⎥ ⎪⎭
Variable frequency control (increases output voltage ripple)
Frequency dithering (increases output voltage ripple)
Variable loop resistance
Duty Cycle control (high operating frequency)
Global PWM (increases output voltage ripple)
Switching VTRG (increases complexity)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Output Voltage Regulation
Up converter: x2, x1.5
[54]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[55]
Interim Summary
Expressing losses as a function of capacitor's current
Generic average model
Limits
Duty cycle effect
Generalization
Regulation
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
3. Losses in Soft Switched SCC
Model building approach
Analysis follows that of hard switched SCC
Expressing losses as a function of output current
Generalizing
Comparison to hard switched SCC
[56]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[57]
Generic Resonant Charging/Discharging Process
ΔV≡ Voltage difference before switch closure
i(t) =
ΔV
⋅ e − α ⋅ t ⋅ sin(ω d t)
ωd ⋅ L
Q>
1
2
2
⎛ ΔV ⎞
⎟⎟ ⋅ e − 2α ⋅ t ⋅ sin 2 (ωd t) ⋅ R
PR = i(t) 2 ⋅ R = ⎜⎜
⎝ ωd L ⎠
E R(res) =
π/ω d
π/ω d
ΔV 2 ⋅ R
0
0
ω d2 ⋅ L2
E R(res) =
∫ PR dt =
∫
α
⎛
− 2π
ΔV 2 ⋅ C ⎜
ωd
⋅ ⎜1 − e
2
⎜
⎝
⋅ e − 2α ⋅ t ⋅ sin 2 (ω d t)dt
⎞
⎟
⎟
⎟
⎠
α=
R
2L
ω0 =
1
LC
ω d2 = ω 02 − α 2
R = R S1 + R ESR + R Ind
ER(res)= Energy dissipated per switch closure
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[58]
Generic Resonant Charging/Discharging Process
Energy Dissipated in each switching period
α
⎛
− 2π
ΔV 2 ⋅ C ⎜
ωd
E R(res) =
⋅ ⎜1 − e
2
⎜
⎝
⎞
⎟
⎟
⎟
⎠
α=
R
2L
ω0 =
ω d2 = ω 02 − α 2
Qd =
E R(res) =
ΔV 2 ⋅ C
⋅ (1 − e − 2π⋅ζ d )
2
1
LC
ζd =
ωd L
R
1
R
=
2Q d 2ω d L
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[59]
Generic Resonant Charging/Discharging Process
Relating the losses to the output current
Average current through a capacitor
Qc =
ΔV π/ω d − α ⋅ t
⋅ sin(ω d t)dt
∫ e
ωd ⋅ L 0
ΔVQc = ΔVC ⋅ (1 + e − πζ d )
Qc
ΔV =
C ⋅ (1 + e − πζ d )
Q c ⋅ f s = I C1 (avg)
I out(avg) = f s ⋅ ΔV ⋅ C ⋅ (1 + e − πζ d )
ΔV =
I C1 (avg)
f s C ⋅ (1 + e − πζ d )
E R(res) =
ΔV 2 ⋅ C
⋅ (1 − e − 2π⋅ζ d )
2
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[60]
Generic Resonant Charging/Discharging Process
Average current through a capacitor
2
⎛
⎞ C
I C1 (avg)
⎟ ⋅ ⋅ (1 − e − 2πζ d )
E R(res) = ⎜
⎜ f C ⋅ (1 + e − πζ d ) ⎟ 2
⎝ s
⎠
(1 − e −2 πζ d )
(1 + e − πζ d ) 2
=
2
(1 − e − πζ d )(1 + e − πζ d )
(1 + e − πζ d )(1 + e − πζ d )
=
(1 − e − πζ d )
(1 + e − πζ d )
2
⎛ I out(avg) ⎞ C (1 − e − πζ d ) ⎛ I out(avg) ⎞ C
πζ
⎟ ⋅ ⋅
⎟ ⋅ ⋅ tanh⎛⎜ d ⎞⎟
= ⎜⎜
E R = ⎜⎜
⎟ 2
⎟ 2
− πζ d
f
C
f
C
2 ⎠
⎝
(1 + e
) ⎝
s
s
⎝
⎠
⎠
−x
⎛ x ⎞ (1 − e )
tanh ⎜ ⎟ =
−
⎝ 2 ⎠ (1 + e x )
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[61]
Generic Resonant Charging/Discharging Process
Average current through a capacitor
2
⎛ I out(avg) ⎞ C
πζ
⎟ ⋅ ⋅ tanh⎛⎜ d ⎞⎟
E R = ⎜⎜
⎟ 2
f
C
⎝ 2 ⎠
s
⎝
⎠
E charging + E discharging
P=
ζ d(1) =
TS
R1
2ωd1L
ζ d(2) =
R2
2ω d2 L
R 1 = R S1 + R ESR + R Ind
I C1 (avg ) = I C 2 (avg ) = I C(avg )
R 2 = R S 2 + R ESR + R Ind
ω d(1,2) =
1 ⎛ R(1,2)
−⎜
LC ⎜⎝ 2 L
⎞
⎟
⎟
⎠
2
⎧
⎪ 1
2
PR(avg) = I C(avg)
⋅⎨
⎪⎩ 2Cf s
⎡ (1 − e −πζ d (1) ) (1 − e −πζ d ( 2) ) ⎤ ⎫⎪
⎥⎬
⋅⎢
+
⎢ (1 + e −πζ d (1) ) (1 + e −πζ d ( 2) ) ⎥ ⎪
⎣
⎦⎭
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[62]
Resonant SCC Equivalent Resistance
1:1 converter
In steady state
Q charging = Q discharging
IC1(avg) = IC2 (avg) = Iout(avg)
V TRG
out = Vin
⎧ 1
⎛ πζ ⎞⎫
2
PR(avg) = I out(avg)
⋅⎨
⋅ tanh⎜ d ⎟⎬
⎝ 2 ⎠⎭
⎩fsC
For ζ d(1) = ζ d(2)
R eq =
1
1 (1 − e −πζ d )
⎛ πζ ⎞
⋅ tanh⎜ d ⎟ =
⋅
fsC
⎝ 2 ⎠ f s C (1 + e − πζ d )
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[63]
Resonant SCC Equivalent Resistance
1:1 converter Dependence on fs
R eq =
1
1 (1 − e −πζ d )
⎛ πζ ⎞
⋅ tanh⎜ d ⎟ =
⋅
fsC
⎝ 2 ⎠ f s C (1 + e − πζ d )
Losses decrease as 1/fs up to the limit: fs=ωd/2 π
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[64]
Comparing Hard and Soft SCC Losses
Soft
Req =
ζ d(1) =
1 (1 − e − πζ d )
1
⎛ πζ ⎞
⋅
=
⋅ tanh⎜ d ⎟
fsC (1 + e − πζ d ) fsC
⎝ 2 ⎠
ω
fs ≤ d
2π
Hard
Req =
1 (1 + e −β )
1
⎛β⎞
⋅
=
⋅ coth⎜ ⎟
fsC (1 − e −β ) fsC
⎝ 2⎠
β=
Unified
R eq
soft
Req
hard
=
=
⎛ α ⎞
1
tanh⎜⎜
⎟⎟
fsC
⎝ 2fs ⎠
⎛ α ⎞
1
⎟⎟
coth⎜⎜
fsC
⎝ 2fs ⎠
Soft → α =
R
4L
Hard → α =
1
2RC
R1
2ωd1L
ω d(1,2) =
1
2fsRC
ζ d(2) =
R2
2ω d2 L
1 ⎛ R(1,2)
−⎜
LC ⎜⎝ 2 L
⎞
⎟
⎟
⎠
2
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[65]
Comparing Hard and Soft SCC Losses
Limits
For high Q soft switched SCC
Req
soft
=
R
8(fs )2 LC
=
(ζd >> 1)
π2R
≈ 5R
2
R→ 0
Req
soft
→0
Req
hard
→
1
fsC
5
4
( 1+e− β )
3
( 1−e− β )
fs→ ∞
Req
hard
= 4R
π2R
Req
=
≈ 5R
soft
2
2
1
0
2
4
6
β
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[66]
Simulation/Experimental Demonstration
RS1 S1
RS2 S2
i(t)
L
Vin
RInd
C
Co
Ro
Vo
R ESR
Mosfets S1, S2: IRF840, Rdson = 0.85Ω, C = 1μF
Vin = 24V; Vout_theoretical = 24V;
RL = 100Ω || 1K Ω ~ 91Ω; Duty Cycle = 0.5;
Power Level: 5.5 Watts (max)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[67]
Simulation/Experimental Demonstration
20 KHz
100 KHz
RL = 91Ω
L = 2.3 μHy
Q = 1.8
Vout = 22V
Vout = 16.1V
Efficiency = 91.7%
Efficiency = 67%
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[68]
Simulation/Experimental Demonstration
20 KHz
100 KHz
RL = 91Ω
L = 0.5 μHy
Q = 0.9
Vout = 21.2V
Vout = 15.4V
Efficiency = 88.3%
Efficiency = 64.2%
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[69]
Resonant SCC Equivalent Resistance
Generalization
⎧
⎪ 1
2
⋅⎨
PC i (avg) = I C
i (avg)
⎪⎩ 2Cf s
⎧
⎪ 1
R eC i = k 2 ⋅ ⎨
⎪⎩ 2Cf s
⎫
⎡ (1 − e
) (1 − e
) ⎤⎪
⎥⎬
⋅⎢
+
⎢ (1 + e − πζ d(1) ) (1 + e − πζ d(2) ) ⎥ ⎪
⎣
⎦⎭
− πζ d(1)
− πζ d(2)
⎡ (1 − e − πζ d(1) ) (1 − e − πζ d(2) ) ⎤ ⎫⎪
⎥⎬
⋅⎢
+
⎢ (1 + e − πζ d(1) ) (1 + e − πζ d(2) ) ⎥ ⎪
⎣
⎦⎭
1 ⎛ R(1, 2)
−⎜
LC ⎜⎝ 2 L
R
ζ d(1,2) =
2ω d(1,2) L
ω d(1,2) =
⎞
⎟
⎟
⎠
2
2
PC i = I out
⋅ R eC i
n
Re = ∑ ReCi
i=1
Ci = flying capacitor i
ζd = damping ratio of charge/discharge loop
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[70]
Including Finite Output Capacitor
RS1 S1
RS2 S2
i(t)
L
RInd
Vin
C
Co
RESR(out)
Ro
Vo
R ESR
ζ d(1,2) =
R
2ω d(1,2) L
R S1 + R ESR + R Ind
ζ (1) =
2L
1 R S1 + R ESR + R Ind
−
LC
2L
⎧
⎪ 1
R eC i = k 2 ⋅ ⎨
⎪⎩ 2Cf s
ζ (2) =
⎡ (1 − e − πζ d(1) ) (1 − e − πζ d(2) ) ⎤ ⎫⎪
⎥⎬
⋅⎢
+
⎢ (1 + e − πζ d(1) ) (1 + e − πζ d(2) ) ⎥ ⎪
⎣
⎦⎭
R S 2 + R ESR + R ESR(out) + R Ind
R S 2 + R ESR + R ESR(out) + R Ind
1
−
2L
C out ⋅ C
2L
L⋅
C out + C
Including Co (could be neglected in practical cases)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[71]
Including Diodes
Step up x3
IC=ID =Iout(avg)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[72]
Including Diodes
Step up x3
VTRG = 3 × Vin
Re =
⎡
⎛ πζ (1)
1
⋅ ⎢2 ⋅ tanh⎜⎜
2f s ⋅ C ⎣⎢
⎝ 2
⎞ C out + C
⎛ πζ (2)
⎟+
⋅ tanh⎜⎜
⎟
C
out
⎠
⎝ 2
R S1 + 2R ESR + R Ind
ζ (1) =
2L
R S + 2R ESR + R Ind
1
− 1
L(C/2)
2L
IC=ID =Iout(avg)
ζ (3) =
ζ (2) =
⎞
⎛ πζ ⎞
⎛ πζ
⎟ + tanh⎜ (3) ⎟ + 2 ⋅ tanh⎜ (4)
⎟
⎜ 2 ⎟
⎜ 2
⎠
⎝
⎠
⎝
⎞⎤
⎟⎥
⎟
⎠⎦⎥
R S1 + R ESR + R ESR(out) + R ind
R S + R ESR + R ESR(out) + R Ind
1
2L
− 1
C ⋅C
2L
L ⋅ out
C out + C
R S 2 + R ESR + R Ind
1 R S 2 + R ESR + R Ind
2L
−
LC
2L
ζ (4) =
R S 2 + 2R ESR + R Ind
R S 2 + 2R ESR + R Ind
1
2L
−
L(C/2)
2L
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[73]
Including Diodes
Step up x3
Rout
[Ω]
Cout
[F]
Vin
[V]
Cr
[F]
Rr
[Ω]
1K
100μ
10
1μ
3m
Vforward ESR ESRout Lr
[Ω]
[Ω]
[V]
[Hy]
0
1m
1m
1μ
Dead
Time
[sec]
fS
[Hz]
10n
100k
Q ~ 200
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[74]
Including Diodes
Step up x3
Rout
[Ω]
Cout
[F]
Vin
[V]
Cr
[F]
Rr
[Ω]
Vforward
[V]
ES
R
[Ω]
ESRout
[Ω]
Lr
[Hy]
Dead
Time
[sec]
fS
[Hz]
1K
100μ
10
2μ
150m
0
10m
1m
1μ
10n
100k
Q~5
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[75]
The case of Coupled Loops
Rout
[Ω]
Cout
[F]
Vin
[V]
C
[F]
R1
[Ω]
1K
100μ
10
1μ
1
Vforward ESR ESRou Lr
[Ω]
[V]
[Hy]
t
[Ω]
0
1m
1m
Dead
Time
[sec]
fS
[Hz]
10n
100k
1μ
Q~1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Output Voltage Regulation
Regulation → Losses
ζ d(1,2) =
⎧
⎪ 1
R eC i = k 2 ⋅ ⎨
⎪⎩ 2Cf s
R
2ω d(1,2) L
⎡ (1 − e − πζ d(1) ) (1 − e − πζ d(2) ) ⎤ ⎫⎪
⎥⎬
⋅⎢
+
⎢ (1 + e − πζ d(1) ) (1 + e − πζ d(2) ) ⎥ ⎪
⎣
⎦⎭
Variable frequency control
Frequency dithering
Variable loop resistance
Global PWM (increases output voltage ripple)
[76]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[77]
2nd Interim Summary
Expressing losses as a function of capacitor's current
Generic average model
Limits
Generalization
Regulation
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[78]
Self Adjusting Binary SCC
Objective: To increase the number of target voltage ratios
y More target voltage ratios with same number of capacitors
y Small ΔV between adjacent target voltage ratios
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[79]
The Approach
y Developing a novel Extended Binary
(EXB) number representation for
increased resolution
y Translating the EXB sequences into
switched capacitor topologies
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Theoretical Foundation of novel SCC
Binary Fractions
y Negative powers of two are used
y Resolution is defined by the LSB 1·2-n
n
Bn = ∑ A j 2 − j
j= 0
3 Aj = {0, 1}
3 n – is the resolution
For example Bn = 5/8
5 8 = 0 ⋅ 20 + 1 ⋅ 2−1 + 0 ⋅ 2−2 + 1 ⋅ 2 −3 → {0 1 0 1}
[80]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[81]
Signed Binary Number Representation
3 Aj = {-1, 0, 1}
n
Zn = ∑ A j 2 j
3 n – is the resolution
j= 0
y More than one code for a given ZN
For example:
5 = 0 + 4 + 0 + 1 → {0 1 0 1}
5 = 0 + 4 + 2 − 1 → {0 1 1-1}
5 = 8 + 0 − 2 − 1 → {1 0-1-1}
5 = 8 − 4 + 0 + 1 → {1-1 0 1}
5 = 8 − 4 + 2 − 1 → {1-1 1-1}
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[82]
Extended Binary (EXB) Representation
developed in this study
y For numbers Mn in the range from 0 to 1
n
Mn = A 0 + ∑ A j 2
−j
j=1
3 A0 = {0, 1}
3 Aj = {-1, 0, 1}
3 n – is the resolution
y More than one code for a given Mn
Number
For example
5/8 = {0 1 0 1},
so that n = 3
EXB code
5 8 = 0 + 2-1 + 2-2 − 2-3 → {0 1 1 -1}
5 8 = 1 − 2-1 + 0 + 2-3 → {1 -1 0 1}
5 8 = 1 + 0 − 2-2 − 2-3 → {1 0 -1 - 1}
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[83]
Spawning the EXB Representation Sequences
The proposed algorithm
y Adding and subtracting 2-j keeps Mn unchanged
↓
20 2-1 2-2 2
+
+
-3
0 1 0 1
0 0 0 1
0 1 1 0
0 0 0 -1
0 1 1 -1
Example: 5/8 ⇒ {0 1 0 1},
j=3
binary addition
replace the original 1 with -1
{0 1 0 1}
↓
0
2 2 2-2 2
0
0
1
+
0
1
+
-1
-3
1 1 -1
0 1 0
0 0 -1
0 -1 0
0 -1 -1
0 1 1 -1
Result: 1 0 -1 -1
1 -1 1 -1
1 -1 0 1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[84]
Properties of the EXB representations
Corollary 1:
For any EXB number Mn in the range of 0 to 1 of
resolution n, the minimum number of EXB representations is
n+1.
This is because each of the Aj =1 (j>0) in the original binary
representation will generate a new representation.
Furthermore, each Aj =0 will turn into Aj =1 that can spawn
a new representation.
Example: n= 3, Mn= 5/8
{0 1 0 1}
0 1 1 -1
1 0 -1 -1
1 -1 1 -1
1 -1 0 1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[85]
Properties of the EXB representations
Corollary 2:
For each Aj=1 (j>0) in an EXB representations of a
given number Mn , there will be at least one Aj=-1 in another
sequence of same Mn.
This is because the generation process involves
replacing “1” by a “-1”.
Example: n= 3, Mn= 5/8
{0 1 0 1}
0 1 1 -1
1 0 -1 -1
1 -1 1 -1
1 -1 0 1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[86]
Sequences of the EXB representations
y Example for n = 3
y The number of the EXB representations is at least n+1
y There are “1” and “-1” placed in the same columns
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[87]
Back to Switched Capacitor DC-DC Converter
y The EXB representation attributes were used to develop
a new family of SC converters
y Capacitors’ interconnections follow the EXB codes
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[88]
Translating the EXB to Capacitor Connections
n
Mn = A 0 + ∑ A j 2 − j
j
y Mn represents the desired output target voltage
ratio
y Each EXB sequence of Mn is associated with a
switched capacitors topology
y A0 is associated with the input voltage
y Each Aj (j>0) is associated with a flying capacitor Cj
y Polarity of Aj (j>0) indicates polarity of Cj connection
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[89]
Translating the EXB to Capacitor Connections
n
Mn = A 0 + ∑ A j 2 − j
j
y The capacitors are always serially connected to the load
y The source is connected in series with the load (and
capacitors) in opposite polarity
3 A0 – the voltage source
1: connected
0: disconnected
3 Aj – capacitor connection
-1: charging
0: disabled
1: discharging
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[90]
All Topological Constraints for Vout = 3/8·Vin
y
For VC1=1/2Vin, VC2=1/4Vin, VC3=1/8Vin, the system is in
steady state
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[91]
All Topological Constraints for Vout = 3/8·Vin
n=3
n
Mn = A 0 + ∑ A j 2 − j
j
y
There are at least n+1 topologies for each ratio Mn.
(Corollary 1)
y
The capacitors are charged and discharged (Corollary 2)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[92]
The Self-Adjusting Property
The perpetual EXB sequences of the converter
But…
y Is there an unique steady state solution?
y Convergence from start up (zero voltage across the
capacitors)?
y Recovery from load step transient?
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[93]
The EXB as a System of Linear Equations
n
Mn = A 0 + ∑ A j 2
⎧
⎪
⎪⎪
⎨
⎪
⎪
⎪⎩
−j
j
For Mn= 3/8
Vin - V1 + 0 - V3 = Vo
0 + V1 + 0 - V3 = Vo
Vin - V1 - V2 + V3 = Vo
0 + V1 - V2 + V3 = Vo
0 + 0 + V2 + V3 = Vo
Solution of linear equation by hardware!
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[94]
The EXB as a System of Linear Equations
From Corollary 1:
Number of equations at least n+1
Number of unknowns n+1
For n=3
4 unknowns: V0, V1, V2, V3
⎧
⎪
⎪⎪
⎨
⎪
⎪
⎩⎪
- V1 - V2 + V3 - Vo = -Vin
V1 - V2 + V3 - Vo = 0
- V1 + 0 - V3 - Vo = -Vin
V1 + 0 - V3 - Vo = 0
0 + V2 + V3 - Vo = 0
Divide the system by Vin:
x1 = V1 Vin
x2 = V2 Vin
x3 = V3 Vin
x 4 = Vo Vin
⎧ - 1 ⋅ x1 - 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4
⎪
⎪ 1 ⋅ x1 - 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4
⎪
⎨ - 1 ⋅ x1 + 0 ⋅ x2 - 1 ⋅ x3 - 1 ⋅ x 4
⎪ 1⋅ x + 0 ⋅ x -1⋅ x -1⋅ x
1
2
3
4
⎪
⎪ 0 ⋅ x1 + 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4
⎩
= -1
= 0
= -1
= 0
= 0
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[95]
The EXB as a System of Linear Equations
⎧ - 1 ⋅ x1 - 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4 = -1
⎪
⎪ 1 ⋅ x1 - 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4 = 0
⎪
⎨ - 1 ⋅ x1 + 0 ⋅ x2 - 1 ⋅ x3 - 1 ⋅ x 4 = -1
⎪ 1⋅ x + 0 ⋅ x -1⋅ x -1⋅ x = 0
1
2
3
4
⎪
⎪ 0 ⋅ x1 + 1 ⋅ x2 + 1 ⋅ x3 - 1 ⋅ x 4 = 0
⎩
In the matrix form, vector X is the unknown weighted
voltages
⎡- 1⎤
⎡- 1 - 1 1 - 1 ⎤
AX = B, where
⎡ x1 ⎤
⎢ 0⎥
⎢ 1 -1 1 -1 ⎥
⎢x ⎥
⎢ ⎥
⎥
⎢
2
A = ⎢- 1 0 - 1 - 1 ⎥ X = ⎢ ⎥ B = ⎢- 1⎥
⎢x3 ⎥
⎢ ⎥
⎥
⎢
⎢⎣ x 4 ⎥⎦
⎢ 0⎥
⎢ 1 0 -1 -1 ⎥
⎢⎣ 0⎥⎦
⎢⎣ 0 1 1 - 1 ⎥⎦
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[96]
Solvability of the EXB Linear Equations
yThe Kronecker-Capelli theorem:
A system has at least one solution
if and only if rank(A) = rank(A1)
A solution is unique if and only if
rank(A) = rank(A1) = the number of unknowns
For the voltage ratio Mn= 3/8
the augmented matrix
⎡- 1 - 1 1 - 1
⎢ 1 -1 1 -1
⎢
A1 = ⎢- 1 0 - 1 - 1
⎢
⎢ 1 0 -1 -1
⎢⎣ 0 1 1 - 1
-1 ⎤
⎥
⎥
- 1⎥
⎥
0⎥
0⎥⎦
0
3 Number of unknowns is 4
3 rank(A) = rank(A1) = 4
⎡1 2 ⎤
⎢1 4 ⎥
X = A -1B = ⎢
⎥
⎢1 8 ⎥
⎢⎣3 8⎥⎦
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[97]
Explicating of the Obtained Solution
y At the steady state the capacitors keep the binary
weighted voltages:
VC1 = 1/2∙Vin
VC2 = 1/4∙Vin
VC3 = 1/8∙Vin
Vout = 3/8∙Vin
y The steady state solution follows the EXB sequence
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[98]
Step up conversion
⎧
⎪
⎪⎪
The step-down case: ⎨
⎪
⎪
⎩⎪
The step-up case:
⎧
⎪
⎪⎪
⎨
⎪
⎪
⎩⎪
Vin - V1 + 0 - V3 = Vo
0 + V1 + 0 - V3 = Vo
Vin - V1 - V2 + V3 = Vo
0 + V1 - V2 + V3 = Vo
0 + 0 + V2 + V3 = Vo
Vo - V1 + 0 - V3 = Vin
0 + V1 + 0 - V3 = Vin
Vo - V1 - V2 + V3 = Vin
0 + V1 - V2 + V3 = Vin
0 + 0 + V2 + V3 = Vin
y Step up by replacing the functions of input and output
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[99]
Implementation
y The capacitors need to have
3 types of connections -1, 0, 1
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[100]
Experimental evaluation board
Microcontroller PIC18F452 (MICROCHIP)
Quad bilateral CMOS switches MAX4678 (MAXIM)
Ceramic Z5U dielectric capacitors 4.7μF (KEMET)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[101]
Simulation and Experimental Results
for Starting Up Circuit, Vout = 3/8·Vin
Simulation
Experiment
Ch.1: Output voltage
Ch.2: Input voltage
Vertical scale: 1 V/div
Horizontal scale: 10 ms/div
Vin=8V, Load resistor: 3.9 kΩ
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[102]
Efficiency at different Mn vs. Load resistor
Mn= 3/8
Mn= 5/8
Î
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[103]
Efficiency at target voltage
yThe losses are determined by an equivalent resistor Req
T
Vout
= Target voltage
η=
Vout
T
Vout
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[104]
Testing the Req concept
Vo =
VS
Ro
Vo
RoVS
Req + Ro
VS
Ro = Ro + Req
Vo
y = x+b
Req = − x
− Req
y =0
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[105]
Experimental results for Mn=3/8
y Req= 7.35 Ω
Zoom in
Rds(on)=1.2 Ω (each switch)
y Supported by theoretical analysis
Î
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Voltage Ripple Reduction
[106]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[107]
Output voltage Regulation
y Voltage ratios outside the target voltages
y Two approaches examined
y Dithering
y Linear regulator (increasing Req)
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Output voltage Regulation by Dithering
y Repetitive change of target voltage ratios
y The output voltage is given by a “duty cycle” of dither
Vout 4 3 1 4 2
= ⋅ + ⋅ = = 0.4
Vin 5 8 5 8 5
[108]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[109]
Output voltage ripple
Constant 3/8 ratio
Output ripple for Vin = 8V,
Vertical scale: 10 mV/div,
Dithering between
3/8 and 4/8 in 2:1 ratio
Load resistor = 437 Ω
Horizontal scale: 100 μS/div
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
Using a Linear Regulator for the LSB
y Small power loss due to close target voltage ratios
y Lower output voltage ripple
[110]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[111]
Generalization to Other Number Systems
n
Nn (r ) = A0r 0 + ∑ A jr − j
j=1
r= radix
Aj = jth digit
Digit values= {-(r-1)…-1, 0, 1,…(r-1)}
r-1 capacitors per digit
Example radix 3, 3 bits
N3 (3) = 1⋅ 3 −0 + 0 ⋅ 3 −1 − 2 ⋅ 3 −2 + 1⋅ 3 −3 = 22 ⋅ 3 −3 = 22 / 27
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[112]
Generalization to Other Number Systems
N1( 4)
r= 4, one digit
1/ 4 = 0 ⋅ 4−0 + 1⋅ 4−1
{0
1}
1/ 4 = 1 ⋅ 4 −0 − 3 ⋅ 4 −1
{1
− 3}
3 / 4 = 0 ⋅ 4 −0 + 3 ⋅ 4 −1
3 / 4 = 1 ⋅ 4 −0 − 1 ⋅ 4 −1
{0
{1
− 1}
3}
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[113]
Generalization to Other Number Systems
Applying r=2 (binary) and 2 bits, only 2 capacitors are
required (instead of 3)
Last printed slide
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
3rd Interim Summary
Binary SCC
High efficiency in wide range of output to input
voltage ratios
2n-1 target voltage ratios with n capacitors
Binary resolution for the adjacent voltage ratios
Relatively large number of switches
Proposed representation by number system
Could help optimizing SCC topologies
[114]
Prof. Sam Ben-Yaakov, Switched Capacitors Converters, © S. Ben-Yaakov 2009
[115]
Thanks for your attention
Losses in hard-switched SCC
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