# On the Potential of CMOS Recharged Semi-Floating Gate Devices Used in Balanced Ternary Logic

**ABSTRACT** Most of the electronic circuits designed today use binary logic. However, will binary logic be the leading technology in the future, why not uses balanced ternary logic, imple-mented using recharged semi-floating gate (RSFG) devices, instead? This paper gives some measurements and analyzes novel applications using CMOS RSFG technology.

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**ABSTRACT:**This tutorial places the developments and potential of multiple-valued signals and logic in the relevant context of binary and two-valued signals. It covers: the role of multivalued logic (MVL) in the binary world; multivalued representation; binary-related radices; multivalued functions; storage techniques in MVL; and implementation issues. An overview of applications is included.< >Computer 01/1988; 21:17-27. · 1.44 Impact Factor -
**The Direct Measurment of the Napierian Base**. R Appleyard . 1913. Proc. Phys. Soc. London 26 178-182. -
**Ultra Lowvoltage Floating-gate (FGUVMOS) Amplifiers**. Y Berg, Ø Naess, M Høvin, H Gundersen . 2001. Analog Integrated Circuits and Signal Processing 26 63-73.

Page 1

On the Potential of CMOS Recharged Semi-Floating Gate Devices used in

Balanced Ternary Logic

Henning Gundersen

Department of Informatics, Nanoelectronic Systems Group, University of Oslo

Blindern, NO-0316, Oslo, Norway

Email: henningg@ifi.uio.no

Abstract

Most of the electronic circuits designed today use binary

logic. However, will binary logic be the leading technology

in the future, why not uses balanced ternary logic, imple-

mented using recharged semi-floating gate (RSFG) devices,

instead? This paper gives some measurements and analyzes

novel applications using CMOS RSFG technology.

1Introduction

Use of transistors has forced the developers to use binary

solutions based on the fact that a transistor usually has two

stable states; "on" or "off". And somewhere between the

two stable states, it is an analog state of behaviour. Donald

Knuth a famous computer scientist which wrote "The Art

of Computer Programming", said "If it would have been

possible to build reliable ternary architecture, everybody

would be using it" [17]. If we use balanced ternary num-

bers, we can add both negative and positive numbers with-

out any use of a sign bit. This is one of my motivations

to use Recharged Semi-Floating Gate technology to cope

with balanced ternary logic. By using this technology, we

are one step closer to find a possible solution to a reliable

ternary hardware. This paper presents some novel applica-

tions which hopefully can be used in a ternary logic design

in the future.

2Recharged Semi-Floating Gate CMOS

Technology

2.1 Volatile Floating Gate Circuits

Volatile Floating gate transistors are not genuine floating

gates since the gate node is not completely isolated. There

are several methods to maintain the charge on the floating

gate node; pseudo, semi and recharged. This paper focuses

on recharged semi-floating gate technology.

Floating gate circuits need to be initialized, either once

only or frequently. The once and for all initialization is syn-

onymous with programming. By recharging the floating

gate frequently, we avoid problems with any leakage cur-

rents and random or undesired disturbance of the floating-

gate charges. The reset or recharge scheme can be used to

overcome some problems associated with floating-gate cir-

cuit design [2]. In a modern CMOS processes thin gate ox-

ide makes it almost impossible to design a genuine floating

gate structure, because of the constant gate oxide current

leakage [13, 19].

2.2 Recharged Semi-Floating Gate

Devices

Recharged floating gate is also known as switched or

clocked floating gate transistors.

sented by Kotani et al. in 1998 [18]. It makes it easy to con-

trol the charge on the floating gate. However, the transmis-

siongateusedfortheclockingintroducesaconstantleakage

current. This makes the circuit more suitable to be used in

digital applications, with sufficient clock frequency.

This method was pre-

2.3 RSFG Binary Inverter

A simple binary single input gate, an inverter, is shown

in figure 1. By equalizing the transfer parameters βnand

βpof the N- and P-transistor, we obtain an equilibrium state

when the recharge clock is 1. The output and gate are driven

toward

distinct cases. Assuming that the input signal is initially

1 (Vdd), the SFG voltage can be expressed as (Vdd

+ ki), where ki = Ci/Ctand Ctis the total capacitance

seen by the SFG, and the output is equal to 0. The out-

put and the SFG will be forced towardVdd

The recharge current which will pull the SFG down towards

Vdd

2, is larger than the equilibrium current (Ibec). We define

Vdd

2. When the recharge clock is 1, we have two

2) x (1

2simultaneously.

Page 2

??

????

Ci

Ne

Ci

VVV

+ CLK

VININOUTOUT

+ CLK

Pe

Figure 1. A typical recharged semi-floating gate bi-

nary inverter

the recharge rise rime tras the time required to recharge the

output from 0 toVdd

input signal is initially 0, the SFG voltage is (Vdd

and the output is 1. The recharge current will be reduced

compared to the former case due to the body effect of the

n-channel recharge transistor. In order to achieve a correct

recharge to the equilibrium state in a chain of gates, we need

to recharge all gates and all inputs simultaneously. In addi-

tion, we need to develop a synchronization scheme for the

recharge. We define the recharge fall time tf as the time

required to recharge the output from 1 toVdd

frequency is twice the frequency of the input signal.

2(and the SFG simultaneously). If the

2) x (1 - ki)

2. The recharge

2.4The Auto-Zero Element

The Auto-Zero Element (AZE) can be seen as a signal

converter. By using the AZE we are able to use DC signals

in addition to a conventional binary signal as input to the

RSFG circuits. When interfacing with a binary signal, the

clock frequency has to be twice the input frequency of the

binary signal. An example of AutoZero elements is seen

in figure 2. Figure 2 (a) consist of two Pass Gate circuits

which is opposite clocked. The upper hasVdd

lower Pass Gate circuit has the input signal Vinas input.

Measured typical output characteristics of the circuit in

figure 2 (a) is presented in figure 12, 13 and 14. The upper

signal signifies the clock signal (1Vpp), the lower is the out-

put signal. The input signals are DC-signals: ’-1’ (100mV),

’+1’ (900mV) and ’0’ (500mV). The clock frequency is 200

KHz. This shows a typical recharge signal, with three sig-

nificant levels. The refresh or recharge period is when the

clock is high, the evaluate period is when the clock is set

low.

Figure 2 (b) shows another solution. It has stacked tran-

sistors, and therefore has limitations on use in very high

frequency applications due to the body effect. The N and

P transistors connected to the rails, Vddand Vss, has to have

matched transfer parameters (βnand βp). The two solutions

2as input, the

uses an equal amount of transistors, because theVdd

Figure 2 (a) is made using a diode couple N and P transistor.

2input in

??

?

b)

+ CLK

V

− CLK

VVV

VIN

INOUTOUT

a)

− CLK

+ CLK

+ CLK

DD/2

Figure 2. Typical auto-zero elements

3 Ternary Logic vs. Multiple Valued Logic

in the last few decades Multiple-Valued logic (MVL)

has been proposed as a possible substitute to binary logic.

While binary logic is limited to only two states, "true" or

"false", multiple-valued logic can replace these with finitely

or infinitely numbers of values. A MVL system is defined

as a system operating on a higher radix than two [20]. A

radix-n set has n elements, {0, 1, ...., n-1}. The practica-

bility of MVL depends on the accessibility of the devices

constructed for MVL operations [6]. The devices should

be able to switch between the different logical levels, and

preferably be less complex than the binary counterparts.

Ternary logic is MVL compliant. However, it only uses

three logic states, "0","1" and "2". Higher radix gives more

complexity, but is there an optimal radix? If we look at

the r ∗ w product, where r is the radix and w is the width

of the word, this product is said to be a good estimate for

optimal hardware complexity. The derivation of the func-

tion r ∗ w gives a minimal point at 2.71828; the radix and

width is treated as contentious variables. This is remark-

ably the napierian base1[1]. The minimal point is closer

to radix 3 than to radix 2, hence base 3 is the most opti-

mal numbering system. Here is an example for the r ∗ w

product for the decimal number 1024. By using the deci-

mal numbering system with radix=10, the r ∗ w product is

given as 10 ∗ 4 = 40, binary with radix=2 (10000000000)

gives 2 ∗ 11 = 22, ternary with radix=3 (1101221) gives

3 ∗ 7 = 21.

1Named after the Scottish mathematician John Napier (1550-1671), he

developed the concept of the logarithm and also effectively introduced the

modern notation of decimal fractions

Page 3

3.1The

System

BalancedTernaryNumbering

Balanced ternary is a non-standard positional numeral

system (a balanced form), useful for comparison logic. It

is a ternary system, but unlike the standard (unbalanced)

ternary system, the digits have the values -1, 0, and +1.

This combination is especially usable for comparison be-

tween two values, where the three possible relationships are

less-than, equals, and greater-than. Balanced ternary can

represent all integers without resorting to a separate minus

sign. As early as 1840, Thomas Fowler, a self-taught En-

glish mathematician, invented a ternary mechanical calcu-

lating machine which used balanced ternary notation. All

details on the calculating machine were lost, until recently.

A research project, started in 1997, have managed to get

enough information needed to create a complete historical

replica [7].

The balanced ternary radix notation has some beneficial

properties:

a) ’Ternary inversion’is easy, change -1 with +1, and vice

versa [21]. This is a simpler than the rule for the twos com-

plement in binary.

b) The sign of a number is given by its most significant

nonzero ’trit2’

c) The operation of rounding to the nearest integer is iden-

tical to truncation.

d) Addition and subtraction are essentially the same opera-

tion; you merely add the digits using the rules for addition

of digits.

e) Carry occurs less often because only 2/9 of the possible

digitssumsresultsincarry, comparedto1/4inbinary. Carry

ripples tend to be shorter in balanced ternary than what is

the case in binary, due to a zero result from a plus carry into

a minus digit or vica versa.

There has been several attempts to realize arithmetic ap-

plications by using the ternary numbering system, but cur-

rently they lack any commercial success [12, 5].

3.2The MAX and MIN Circuit

MAX and MIN functions are fundamental functions in

multiple valued logic and it is analogous to the OR- and

AND-function in the binary world [16, 14]. M. Inaba et al.

have presented the MAX and MIN function implemented

using Neuron-MOS Transistors in floating gate technology

[15, 14, 16]. However, another way of realizing these func-

tions is by using recharged Semi-Floating Gate Logic. This

makes it possible to implement a low-power digital system

with reduced dynamic power dissipation without any post-

production of the chips. The advantage of this technology

2One trit has 3 values -1, 0, and +1, it is analogous to bit in the binary

world (0 , 1).

is presented in [2] and the circuits are presented in figure 3

and figure 4. As the figure shows, the circuit use an analog

inverter (MVL inverter) and a down literal circuit to make a

voltage comparator [3, 4, 8]. The output of the voltage com-

parator gives a selection signal to the pass gate circuitry,

which consists of a pPassGate (pPG), a nPassGate (nPG)

and a inverter. The pPG let the signal through when the out-

put of the comparator is "0", and the nPG let the signal pass

when the output of the comparator is "1". To obtain a MIN

function, the nPG can swap place with the pPG, as is shown

in figure 4.

Measurements of the MAX circuit is provided in fig-

ure 11. Figure 11 verifies the operation of the circuit show-

ing satisfactory noise margins. The output signal of the

MAX circuit is given in balanced ternary notation (-1, 0,

+1).

4 Balanced Ternary Arithmetic

4.1Balanced Ternary Adders

My contribution to the balanced ternary arithmetic is

the hardware implementation of a Balanced Ternary Adder

(BTA) [9] and balanced ternary counters (BTC) [11, 10]. A

BTA is an essential component in arithmetic circuits, and

is used in multiplication and division structures. The first

implementation of a one trit adder, which was presented

at ISMVL in 2006, takes two Ternary inputs (X and Y)

and generate the SUM output (S0and S1). The complete

schematic diagram of the first Balanced Ternary Adder cir-

cuit is shown in figure 5. Supplementary measurement re-

sults are provided in figure 15 and 16.

4.2Balanced Ternary Counters

A balanced ternary counter (BTC) is comparable with a

ternary full adder, where the carry signal can have all logic

values (1,0 and 1). A balanced ternary counter sums up the

inputs Xi, where i is number of trits of the same weight,

and gives an output in balanced ternary notation.

4.3The Balanced Ternary (4,2) Counter

The balanced ternary (4,2) counter (figure 6) is a modi-

fication of the BTA presented at ISMVL2006, and has sim-

ilar functionalities. A (4,2) ternary counter has 4 balanced

ternary inputs (X1..X4) and two balanced ternary outputs

(S0, S1). A (4,2) is also known as a 4 to 2 reducers [22].

4.3.1 The Carry Detect Stage

Thecarrydetectstagegeneratesaternarycarrysignal, when

the input signals?4

2.

i=1Xiis less than −2 and greater than

Page 4

C4

C5

nPG

pPG

C1

C2C3

+ CLK + CLK

+ CLK

BINOUT

INPUT 2

INPUT 1

OUTPUT

MVL INVERTERDLC

VOLTAGE COMPARATOR

?

?

Figure 3. A recharged semi-floating gate MAX circuit

C4

C5

pPG

nPG

C1

C2C3

+ CLK + CLK

+ CLK

BINOUT

INPUT 2

INPUT 1

OUTPUT

MVL INVERTER DLC

VOLTAGE COMPARATOR

?

?

Figure 4. A recharged semi-floating gate MIN circuit

S1

S0

S1

+ CLK

C1

C2

C3

C4

C5

C6

C9

X

Y

C10

C14

C17

C15

C21

C20

C19

C18

C16

C11

i 3i 6

i 10

i 9

i 7

i 5i 2

i 1

i 8

i 4C−HIGH

C−LOW

C12

C13

C8

C7

+ CLK

+ CLK

+ CLK+ CLK

+ CLK

+ CLK+ CLK

+ CLK+ CLK

AZC

AZC

AZC

AZC

+ CLK

+ CLK

+ CLK

+ CLK

− CLK

− CLK

− CLK

− CLK

___

CARRY DETECT

PRE−ADDER

Figure 5. A recharged semi-floating gate balanced ternary adder

Page 5

C13

C14

C15

C16

C17

S1

S1

??

??

??

??

??

??

??

?

??

?

??

?

??

?

??

??

C1

C2

C3

C24

i 6i 5

C4

C5

C6

C7

C8

C9

C10

C23

C25

C18

C19

C−LOW

C−HIGH

− CLK

+ CLK

i 1

i 3

i 2

i 4

S

C20

C21

C22

i 7i 8

+ CLK+ CLK

+ CLK+ CLK

+ CLK

+ CLK+ CLK

+ CLK

0

4

X

X

X

2

1

X

3

__

CARRY DETECT

Figure 6. A recharged semi-floating gate balanced ternary (4,2) counter

This is achieved by using a 5 input threshold circuit (i1 and

i3). Lookingatfigure6, weseethatthe4inputsX1..X4, are

compared to the clock pulse ’+Clk’ and ’-Clk’, connected

respectively to input capacitors C5and C6.

The upper circuits i1 and i2 generates the binary ’C-

HIGH’ signal, which is set to the logic level ’1’ when the

sum of the inputs Xi, is greater than 2. The input capacitors

C1..C4are equal. Capacitor C5determines the threshold of

i1, by comparing the?4

’+Clk’ signal is in phase with reference clock).

’C-LOW’ is set in a similar way, except it will be set

to logic level ’−1’ when the sum of the inputs Xiis less

than -2. The threshold of i3 is determined by the capacitor

C6, which is connected to the ’-Clk’ signal (The ’-Clk’ is in

opposite phase with the reference clock).

The inverter i5 is a binary to ternary converter. It con-

verts the two binary carry signals (C-LOW, C-HIGH) to a

balanced ternary carry signal (S1).

i=1Xiwith the ’+Clk’ signal (The

4.3.2 The Output Stages, S0and S1

To generate the correct ’S1’ signal, we need to do a ternary

inversion of the output signal of inverter i5 (S1). This is

done by using the ternary inverter i6. The adder circuits i7

and i8 generates the ’S0’ signal. This is done by weighting

the inverted ’S1’ three times the input signals Xi, which in

turn is done by choosing C13three times larger than each of

the input capacitors C14..C17.

A 4 trit balanced ternary adder can be realized by us-

ing four (4,2) counters as shown in figure 7, compared to

a binary solution, it uses less active devices, hence reduced

power consumption [10].

BTC

(4,2)

BTC

(4,2)

BTC

(4,2)

BTC

(4,2)

S0

S1

S2

S3

S4

S1

S0

S2

S1

S3

S2

S4

S3

ZYXZYXZYXZYX

C0

000111222333

00000000011−1−1

000011−1−1

Figure 7. An example of a 4 trit parallel balanced

ternary adder using (4,2) counters

5Measurements and Results

Two prototype chips where produced. They where made

by STMicroelectronics using a 90nm CMOS process. A

layout of the second chip is shown in figure 8; the areal of

the chip is 1mm3. The first chip contained some simple

RSFG structures: Auto-zero, MAX, Ternary-NOT and the

Page 6

simple balanced ternary adder. The second chip contained

a balanced ternary (4,2) counter, a 4 trit parallel balanced

ternary adder and a (13,3) counter. The lower part of the

chip in figure 8 is the MVL applications. It shows ten (4,2)

counters; the four leftmost are used in the 4 trit parallel

balanced ternary adder, the five rightmost counters utilize

the (13,3) counter. The one in the middle is a single (4,2)

counter. This paper does not support measurement results

from the second chip; they will be made available later.

Figure 9 shows the measurement set-up for the first

chip. The measurement was done using the Tektronix TDS

3052 digital oscilloscope, with a 10x probe with load of

8pF,10MΩ. This limits the output frequency, since none

of the circuits has an output buffer. All inputs are connected

to an auto-zero element, which makes it possible to use DC

signals as input to the chip.

A measurement of a simple binary RSFG inverter is sup-

ported in figure 10. The DC gain, ADCis ≈ 1. The DC-

characteristic is interpolated from the recharged signal mea-

sured on the oscilloscope.

The poor open-loop gain makes it difficult to maintain

logic depths, hence it is difficult to make complex logi-

cal gates. This is the problem measured for the balanced

ternary adder in figure 15 and 16, which has a limited signal

amplitude of120mV for logical’-1’and 176mV for logical

’+1’. A correct operation would have given +/ − 400mV .

For a simple structures the functionality is maintained, as

measured for the auto-zero element, figure 12, 13 and 14

and the MAX circuit, figure 11.

Figure 8. Chip layout of the second MVL chip, de-

signed by using a 90nm CMOS process from STMi-

croelectronics

Figure 9. The instruments set-up for the measure-

ment of the first MVL chip.

6 Conclusions

This paper presents some useful implementations for re-

alizing balanced ternary arithmetic circuits, which can be

used in future VLSI/ULSI circuits. The balanced ternary

counters can be implemented in arithmetic applications,

for instance in multiplication and division circuits. They

are compatible with their binary counterparts, and can re-

place any binary full adder structure with a similar balanced

ternary full adder structure. This a great leap toward realiz-

ing a fully ternary ALU in the future. The drawback is that

the measurement results show the RSFG circuits need some

further development to work properly.

A floating gate structure will generally generate a larger

chip area compared to a conventional design, because of the

capacitors used in the design. However, by using very small

metal-capacitors and stacking, we are able to minimize the

area needed. The 90nm CMOS process from STMicroelec-

tronics that was used, has 7 metal layers.

Inthispaper, balanced ternaryadderstructuresusingbal-

anced ternary counters have been presented. It has the fol-

lowing beneficial properties:

1. There is no reason to worry about the sign bit, since

the structures use balanced ternary notation.

2. The resolution compared to number of transistors is

higher than with a typical binary solution [10]

3. It is possible to build fast addition structures, using

the theory from the binary world.

Page 7

0 0.10.20.3 0.40.50.60.70.80.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input (volt)

Output (volt)

In

Out

Figure 10. Measured DC response of the RSFG bi-

nary inverter

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0

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INPUT 1

−10+1−10+1 −10+1

0

0.5

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INPUT 2

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"−1"

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Figure 14. Measured output characteristics of the

auto-zero element in figure 2(a) with input signal +1

Figure 15. Measured output characteristics (S0) of

the BTA circuit, with input signal (+1,+1)

Figure 16. Measured output characteristics (S1) of

the BTA circuit, with input signal (+1,+1)

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