# Low Voltage Floating Gate MOS Transistor Based Differential Voltage Squarer

Authors: Maneesha Gupta; Richa Srivastava; Urvashi Singh
Journal: ISRN Electronics
 (2014)

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## Abstract

This paper presents novel floating gate MOSFET (FGMOS) based differential voltage squarer using FGMOS characteristics in saturation region. The proposed squarer is constructed by a simple FGMOS based squarer and linear differential voltage attenuator. The squarer part of the proposed circuit uses one of the inputs of two-input FGMOS transistor for threshold voltage cancellation so as to implement a perfect squarer function, and the differential voltage attenuator part acts as input stage so as to generate the differential signals. The proposed circuit provides a current output proportional to the square of the difference of two input voltages. The second order effect caused by parasitic capacitance and mobility degradation is discussed. The circuit has advantages such as low supply voltage, low power consumption, and low transistor count. Performance of the circuit is verified at ±0.75 V in TSMC 0.18μm CMOS, BSIM3, and Level 49 technology by using Cadence Spectre simulator.


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## Body

## 1. Introduction

Technology scaling and growing demand of portable electronic equipments have motivated the researchers towards the design of low voltage and low power analog signal processing circuits. Low supply voltage increases the battery lifetime and hence reduces the power consumption of the portable equipment. Various low-voltage lowpower design techniques reported in literatures include subthreshold MOSFETs, level shifters, self-cascode, bulk-driven, and FGMOS techniques [1–10]. Among these, FGMOS concept has gained prime importance due to its ability to reduce or remove the threshold voltage requirement of the circuit. Scaling of transistor dimensions has motivated the designers towards the design of low voltage nonlinear CMOS circuits. Voltage squarer is one of the most versatile nonlinear blocks that find application in several fields like neural and image signal processing [11–18]. It can be used to implement various nonlinear circuits such as multipliers, balanced modulators, and phase comparators. Analog hardware implementation of these blocks offers advantage of reduced silicon area and low power consumption. CMOS squarer circuit based on cross-coupled differential pair has been proposed in [11] but the circuit is complex and has large supply voltage requirement. Squarer with low supply voltage and high rejection of common-mode variations has been proposed in [12] and [13], respectively, but again these circuits require large number of transistors. Recently, the squarer topology using NMOS transistor has been proposed in [16] but it requires positive and negative bias voltage generator for threshold voltage cancellation and can process only single ended input signal. This paper presents very simple and new FGMOS based differential squarer which is the combination of the FGMOS based squarer proposed in [17] and differential voltage attenuator proposed in [18]. The proposed squarer can process differential signals and has low supply voltage, low power consumption, and low circuit complexity.The operation of FGMOS transistor is described in Section2. FGMOS based differential squarer is proposed and analyzed in Section 3. In Section 4, the simulation results are given to verify theoretical results and to demonstrate the effectiveness of the proposed circuit. Finally, the conclusions are drawn in the last section.


## 2. FGMOS Transistor

FGMOS is a multiple-input floating gate transistor whose threshold voltage can be controlled and tuned by the values of capacitors and bias voltage applied. The symbol ofn-input FGMOS transistor and its equivalent circuit are shown in Figures 1(a) and 1(b), respectively. The voltage on floating gate (FG) V
FG is given by [10]
(1)
V
FG
=
∑
i
=
1
N
C
i
C
T
V
i
+
C
GS
C
T
V
S
+
C
GD
C
T
V
D
+
Q
FG
C
T
,
where C
i is the set of capacitors associated with effective inputs and the floating gate.(a) Symbol of FMGOS; (b) FGMOS equivalent circuit.
(a)
(b)C
T
=
C
1
+
C
2
+
C
FG
S
+
C
FG
D
+
C
FG
B is the total floating gate capacitance. C
FG
D, C
FG
S, and C
FG
B are the overlap capacitances of floating gate with drain, source, and bulk, respectively, V
D is the drain voltage, V
S is the source voltage, V
B is the bulk voltage, and Q
FG is the residual charge trapped in the oxide-silicon interface during fabrication process. The trapped residual charges give rise to the problem of offset in threshold voltage of the device. The removal of the residual charge can be done by using the method suggested in [19, 20], in which the first polysilicon layer is connected to the metal-k (where k represents number of metals available in the technology). By this contact, the floating gate is not connected to any part of the circuit so it will not affect the operation of FGMOS transistor. Therefore, neglecting the residual charge (1) can be modified as
(2)
V
FG
S
=
∑
i
=
1
N
C
i
C
T
V
i
S
+
C
FG
D
C
T
V
DS
+
C
FG
B
C
T
V
BS
.The drain current (I
D) of the FGMOS transistor operating in saturation region is given by [10]
(3)
I
D
=
μ
0
Co
x
2
W
L
(
V
FG
S
-
V
T
)
2AssumingC
i
≫
C
FG
D
,
C
FG
B [10, 21], the drain current of FGMOS transistor in saturation region can be expressed as
(4)
I
D
=
β
2
(
∑
i
=
1
N
k
i
V
i
S
-
V
T
)
2
,
where k
i
=
C
i
/
C
T, β is the transconductance, and V
T stands for the threshold voltage. In (4), it can be seen that by choosing proper values of multiple input voltages along with capacitance ratio the threshold voltage term can be cancelled so as to get the perfect squarer equation. The proposed circuit utilizes this property of FGMOS transistor to implement the squarer function.


## 3. Proposed FGMOS Based Differential Squarer

The proposed differential squarer is shown in Figure2. It is constructed by FGMOS based squarer (M1, M2) and linear voltage attenuator (M3, M4). The squarer function is obtained by taking the advantage of FGMOS square law characteristics in saturation region. V
B and V
in are bias and signal voltages applied at the two inputs of FGMOS transistor M1 and M2. If the input voltage V
in is positive, M2 is off while M1 operates in saturation region and if the input voltage V
in is negative, M1 is off while M2 operates in saturation region.Figure 2
Proposed differential squarer.The output current of the squarer (neglecting the parasitic capacitances, channel-length modulation, mobility degradation, and the body effect) is given by(5)
I
out
=
β
1
2
(
k
1
V
inS
1
+
k
2
V
BS
1
-
V
T
1
)
2
,
if
V
in
>
0
(6)
I
out
=
β
2
2
(
k
1
V
BS
2
+
k
2
V
inS
2
-
V
T
2
)
2
,
if
V
in
<
0
.Ifβ
1
=
β
2
=
β, V
T
1
=
V
T
2
=
V
T, k
1
=
k
2
=
k, and k
V
B
=
V
T, then the output current of the squarer can be approximated as
(7)
I
out
=
β
2
(
k
V
in
)
2
.The input voltageV
in of the squarer is generated by voltage attenuator formed by FGMOS transistor M3 and M4. The output voltage of the attenuator is given by [18]
(8)
V
in
=
α
(
V
1
-
V
2
)
+
k
V
SS
-
k
B
V
C
k
+
k
B
,
where k
=
C
/
(
C
+
C
B
), k
B
=
C
B
/
(
C
+
C
B
) are the capacitive coupling ratio and α
=
k
/
(
k
+
k
B
) is the attenuation factor which can be adjusted by choosing proper values of k and k
B. From (8), it can be seen that the offset voltage term (
k
V
SS
-
k
B
V
C
)
/
(
k
+
k
B
) can be cancelled by choosing proper value of bias voltage V
B. For zero output offset, the bias voltage V
C must be equal to
(9)
V
C
=
k
k
B
V
SS
.Assuming zero-output offset for the voltage attenuator and combining (7) and (8), the output current of the squarer is modified as
(10)
I
out
=
β
2
{
k
α
(
V
1
-
V
2
)
}
2
.Ifk
α
=
k
eq, then (10) can be written as
(11)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
.From the above equation, it can be seen that the proposed squarer gives the output current proportional to the difference of input voltagesV
1 and V
2 and The voltage range of input signals can be determined by the factor k
eq.


### 3.1. Second Order Effects

The operation of squarer has been analyzed by neglecting the deviations from ideal square-law characteristics due to parasitic capacitance and mobility degradation. These nonideal effects are the basic source of discrepancy between the ideal and simulated output currents of the proposed squarer.


#### 3.1.1. Parasitic Capacitance

Parasitic capacitances have a minor effect on the squarer operation. The modified current equation after considering the parasitics is given by(12)
I
out
=
β
2
(
k
eq
(
V
1
-
V
2
)
+
C
GD
C
T
V
DS
+
C
GB
C
T
V
BS
)
2
.The ratiosC
GD
/
C
T and C
GB
/
C
T can be neglected if the transistors are operating in saturation mode [21]. Therefore, the parasitic capacitances do not contribute significantly to the squarer operation and have only a small effect on the input range.


#### 3.1.2. Mobility Degradation

Considering the mobility degradation effect, theI-V characteristic of NMOS transistor can be modelled by
(13)
I
D
=
(
β
/
2
)
(
V
GS
-
V
T
)
2
1
+
θ
(
V
GS
-
V
T
)
,
where θ is mobility degradation parameter which has a value of about 0.1~0.001 V−1. According to the above equation, the output current of the squarer can be modified as
(14)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
[
{
k
eq
(
V
1
-
V
2
)
}
2
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
θ
2
{
k
eq
(
V
1
-
V
2
)
}
2
]
(15)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
ε
,
where the output current error of the squarer can be given by
(16)
ε
=
-
β
2
θ
{
k
eq
(
V
1
-
V
2
)
}
3
[
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
]
.
From the above equation, it can be seen that total harmonic distortion due to mobility degradation will be negligible because of small value of mobility degradation parameter.


## 3.1. Second Order Effects

The operation of squarer has been analyzed by neglecting the deviations from ideal square-law characteristics due to parasitic capacitance and mobility degradation. These nonideal effects are the basic source of discrepancy between the ideal and simulated output currents of the proposed squarer.


### 3.1.1. Parasitic Capacitance

Parasitic capacitances have a minor effect on the squarer operation. The modified current equation after considering the parasitics is given by(12)
I
out
=
β
2
(
k
eq
(
V
1
-
V
2
)
+
C
GD
C
T
V
DS
+
C
GB
C
T
V
BS
)
2
.The ratiosC
GD
/
C
T and C
GB
/
C
T can be neglected if the transistors are operating in saturation mode [21]. Therefore, the parasitic capacitances do not contribute significantly to the squarer operation and have only a small effect on the input range.


### 3.1.2. Mobility Degradation

Considering the mobility degradation effect, theI-V characteristic of NMOS transistor can be modelled by
(13)
I
D
=
(
β
/
2
)
(
V
GS
-
V
T
)
2
1
+
θ
(
V
GS
-
V
T
)
,
where θ is mobility degradation parameter which has a value of about 0.1~0.001 V−1. According to the above equation, the output current of the squarer can be modified as
(14)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
[
{
k
eq
(
V
1
-
V
2
)
}
2
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
θ
2
{
k
eq
(
V
1
-
V
2
)
}
2
]
(15)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
ε
,
where the output current error of the squarer can be given by
(16)
ε
=
-
β
2
θ
{
k
eq
(
V
1
-
V
2
)
}
3
[
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
]
.
From the above equation, it can be seen that total harmonic distortion due to mobility degradation will be negligible because of small value of mobility degradation parameter.


## 3.1.1. Parasitic Capacitance

Parasitic capacitances have a minor effect on the squarer operation. The modified current equation after considering the parasitics is given by(12)
I
out
=
β
2
(
k
eq
(
V
1
-
V
2
)
+
C
GD
C
T
V
DS
+
C
GB
C
T
V
BS
)
2
.The ratiosC
GD
/
C
T and C
GB
/
C
T can be neglected if the transistors are operating in saturation mode [21]. Therefore, the parasitic capacitances do not contribute significantly to the squarer operation and have only a small effect on the input range.


## 3.1.2. Mobility Degradation

Considering the mobility degradation effect, theI-V characteristic of NMOS transistor can be modelled by
(13)
I
D
=
(
β
/
2
)
(
V
GS
-
V
T
)
2
1
+
θ
(
V
GS
-
V
T
)
,
where θ is mobility degradation parameter which has a value of about 0.1~0.001 V−1. According to the above equation, the output current of the squarer can be modified as
(14)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
[
{
k
eq
(
V
1
-
V
2
)
}
2
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
θ
2
{
k
eq
(
V
1
-
V
2
)
}
2
]
(15)
I
out
=
β
2
{
k
eq
(
V
1
-
V
2
)
}
2
+
ε
,
where the output current error of the squarer can be given by
(16)
ε
=
-
β
2
θ
{
k
eq
(
V
1
-
V
2
)
}
3
[
1
-
θ
{
k
eq
(
V
1
-
V
2
)
}
]
.
From the above equation, it can be seen that total harmonic distortion due to mobility degradation will be negligible because of small value of mobility degradation parameter.


## 4. Simulation Results

The designed circuits are simulated using Cadence Spectre simulator in TSMC 0.18μm CMOS technology using ±0.75 V power supply. The aspects ratios of the transistors of the proposed circuits are given in Table 1. Since the floating gate (FG) of FGMOS does not have any connection to ground, the simulator cannot understand the floating gate and reports dc convergence problem during simulation. To avoid dc convergence error during simulation model suggested in [10] has been used in this work. This model is based on connecting large value resistors in parallel with the input capacitors as shown in Figure 3. In this model, the relation between resistances and capacitances can be given as follows: R
i
=
1
/
k
C
i
=
1000
G
Ω.Table 1
Aspect ratios of the transistor of the proposed circuit.
Transistor
W (µm)
L (µm)
M1-M2
4.4
0.18
M3-M4
0.54
0.18Figure 3
Simulation model of FGMOS.The DC response of the attenuator and squarer againstV
1 with V
2 varying from −0.75 V to 0.75 V is shown in Figures 4(a) and 4(b), respectively. It can be seen from the curves that the output voltage of attenuator varies from −320 mV to 320 mV and the maximum value of the output current of the squarer is approximately 12 μA. The proposed squarer operates at low supply voltage with total power consumption of 15 μW only. The transient response of the squarer is shown in Figure 5. Figure 5(a) shows that V
1 and V
2 are the two input sinusoidal signals with amplitude 0.75 V and 0.25 V peak-to-peak, respectively, and frequency 5 kHz and the output current is shown in Figure 5(b). The frequency response of the proposed squarer with V
1
=
1 mV is shown in Figure 6. It can be seen from the figure that the proposed squarer exhibits the bandwidth of 199.426 MHz. The performance parameters of the proposed circuit and various conventional circuits are compared in Table 2. It can be seen that the proposed configuration has the lowest transistor count, operates at low supply voltage, and also has low DC power consumption.Table 2
Comparison of various conventional and proposed squarers.
Parameters
[12]
[13]
Proposed
Technology (μm)
1.2
0.5
0.18
Supply voltage (V)
1.5
±2.5
±0.75
Number of transistors
11 MOS
6 MOS
4 FGMOS
Power dissipated (μW)
NA
NA
15
Input signal range (V
PP)
0.26
1.5
0.75
Attenuation factor (α)
NA
NA
1/4
FGMOS capacitances (fF)
NA
NA
C
B
=
432, C
=
144
C
1
=
C
2
=
100
Bias voltages (V)
NA
NA
V
C
=
-
0.25
V
B
=
0.75(a) DC response of the attenuator; (b) DC response of the proposed squarer.
(a)
(b)(a) Waveform of input signalsV
1 and V
2; (b) transient response.
(a)
(b)Figure 6
Frequency response of the proposed squarer.


## 5. Conclusions

In this paper, novel differential voltage squarer based on simple FGMOS squarer and voltage attenuator has been proposed. The proposed circuit operates at ±0.75 V with maximum power consumption of 15μW and bandwidth of 199.426 MHz. The circuit can process differential signal and hence it can be useful in various low voltage lowpower analog signal processing/generating applications.



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*Source: 357184-2014-02-09.xml*