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A comprehensive, adjustable approach for linearizing and broadening the gain characteristic of variable gain amplifiers
Microelectronics Journal 45 (2014) 1079–1086
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Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
A comprehensive, adjustable approach for linearizing and broadening the gain characteristic of variable gain amplifiers H. Bameri a,n, H. Abdollahi b, A. Hakimi a a b
Electrical Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran Electrical and Computer Engineering Department, Tarbiat Modares University, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 16 October 2013 Received in revised form 26 April 2014 Accepted 28 April 2014
In this paper a comprehensive approach is presented to linearize and adjust gain characteristic of variable gain amplifiers (VGAs). It is also capable of increasing the output linear dynamic range of VGAs and modifying variation range of control voltage. The approach is able to change the voltage gain characteristic of an amplifier, even after fabrication, to a desired one by means of a digital control signal and a digital to analog converter. Using this approach, the gain of basic differential amplifier is controlled by two different predistorters, and adjustable dB-linear characteristics in range of greater than 60 dB are achieved. The approach, also, is applied to two conventional VGAs, the gain characteristic of first VGA is linearized, and in the second VGA, the linear dynamic range is expanded about 26 dB. The controller uses 1.2 V voltage supply, and simulations are done using 0.13 mm CMOS process model. The other characteristics of each mode of control are reported completely. & 2014 Elsevier Ltd. All rights reserved.
Keywords: VGA dB-linear Dynamic range Adjustable gain characteristic
1. Introduction The variable gain amplifier (VGA) is a significant analog block in wireless communication receivers. Applications of this block in Bluetooth, UWB, WLAN, and WSN (Wireless Sensor Network) indicate its substantial role [1–3]. In such wireless links, as a result of channel fading, the amplitude of received signal varies greatly. In conjunction with an automatic gain control (AGC) feedback loop, VGA is implemented to compensate variations and provide fairly constant amplitude for ADC in order that the dynamic range of receiver will be increased [4]. It is desired to a have dB-linear gain characteristic for VGA to achieve a constant settling time for AGC loop [1]. It may be possible to achieve this gain characteristic by using the exponential relation between collector current and base–emitter voltage in BJT or BiCMOS technology [3]. Nevertheless, an all-CMOS integration of digital and analog circuits is easier and more economical. The other solution is a subthreshold-biased MOS device, which has an exponential transconductance versus gate–source voltage [5]. However, it suffers from undesired effects such as poor noise performance and limited bandwidth. The final solution is a parasitic BJT in deep n-well CMOS technology which can imitate
n
Corresponding author. Tel.: þ 98 9132968531; fax: þ98 3413235900. E-mail address: [email protected] (H. Bameri).
http://dx.doi.org/10.1016/j.mejo.2014.04.041 0026-2692/& 2014 Elsevier Ltd. All rights reserved.
the function of an inherent BJT, but imperfect effects of lateral NPN/PNP transistors degrade the gain characteristic [6]. Based on mentioned difficulties, designers commonly use the square or linear characteristics of MOS devices, which inherently are not dB-linear. Several approaches have been proposed to realize a linear-in-decibel gain characteristic. These approaches are divided into two categories: the first approach is predistortion [6–8], and the second is topology modification and the combination of VGA elements characteristics [9–11]. In some cases a combination of both approaches is used [12–14]. In the first category, the transfer function of a predistorter block converts VC to a secondary voltage or current in such a way that if it is applied to the VGA, a dB-linear gain characteristic will be achieved. In the second category, the circuit topology of VGA is changed and modified to provide the desired gain characteristic. Examples of such categories are a differential amplifier modified by diode-connected loads (Gm-Ratioed amplifier) and Cherry– Hooper amplifier [15] with tunable resistive feedback, which are shown in Fig. 1. Among described categories, in the former, the application of the proposed predistorter block is limited to the VGA circuit topology, so it is not feasible to use the predistorter as the controller of other VGAs, which have different circuit topologies. In addition, the utilized predistorters dissipate significant amount of DC power that increases the total power consumption of VGA [7,8]. On the other hand, in the latter, the proposed approach is unique for a specific VGA, and cannot be applied to others.
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H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
diverse MOS amplifier formulas must be obtained so that it will be possible to devise an exhaustive controller.
VDD
2.1. Gain analysis
M4
M5 CMFB
+ V OUT + VIN -
M2
VC1
M7
M3
M1
VC2
M8
M6
VDD Rb
Rd
Rd
Rb
VC
Mf1
+ V OUT M3
+
M1
Mf2
M4
ISS2
M2
VIN
A variable gain could be realized by changing the transfer function of the transconductor (Gm) or the load resistance by control voltage. Depending on the nodes where this voltage is applied (transconductor, active load, or tail current source), the operation and the characteristics of VGA will be different. According to the intrinsic properties of MOS devices, and regardless of how it operates in the circuit (saturation or triode), the relation between the logarithm of the voltage gain and control voltage will be nonlinear. As shown in Fig. 2, the control voltage can be applied to three different nodes of circuit. The analysis of each control voltage effect on the gain characteristic enables us to have a deep insight into how the circuit operates and how its gain depends on the control voltages, and as a result, a general form for predistorter operation will be obtainable. Moreover, more complex VGAs could be designed by applying a couple of the control voltages as shown in Fig. 2, and there will be several scenarios. It must be noted that active loads (M4 and M5) can be replaced by resistive loads occasionally. In Fig. 2, the input small signal is applied to the gate of transistors M2 and M3, and the output is obtained from drains of them. In the simplest scenario, each control signal is applied individually while others are fixed. According to the large signal relation between drain current and gate–source voltage in a MOS device, the amplifier gain can be calculated in terms of VCi (i¼1:3). First, when VC1 is applied to the gate of the current source of the amplifier, the gain is calculated by (1) and (2). RL and AL suffixes depict whether the load of the amplifier is a resistive load or an active load respectively rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ C ox W 1 W 2 Av1;RL ¼ n ð1Þ ðV C1 V TH ÞRL L 2
ISS1
Fig. 1. (a) Modified Cherry–Hooper gain cell and (b) Gm-Ratioed amplifier.
In this paper, a comprehensive approach is proposed to realize a dB-linear VGA. In addition to low power consumption, by some simple modifications, the proposed controller is capable to be applied to other VGAs. Using this approach, the slope of the linear gain characteristic as well as control voltage range can be adjusted arbitrarily even after fabrication to facilitate the AGC loop design. Moreover, this approach has the ability of linearization and linear dynamic-range extension of traditional VGAs. The remainder of this paper is organized as follows. In Section 2, the gain equations and general forms of fundamental amplifiers are calculated to find the optimum mathematical function for the predistorter based on the circuit of VGA. Section 3 describes advantages of this approach and how the predistorter of the proposed approach can be implemented. In Section 4, simulation results are presented, and Section 5 concludes the paper.
Av1;AL ¼
sffiffiffiffiffiffiffiffiffiffiffi W2 1 2 W 1 λðV C1 V TH Þ
ð2Þ
where Wi, L and VTH are the widths, length and threshold voltage of the MOS devices, i¼1, 2, and RL is the resistive load of the amplifier. Second, in the case of applying VC2, (3) and (4) are obtained for the voltage gain of the amplifier Av2;RL ¼ μn C ox
W2 ðV GS2 V TH ÞRL L
ð3Þ
VDD W3/L
VC3
W3/L
M4
B M2
VC2
To obtain a general solution for the controller that can convert the VGA to a dB-linear high dynamic range one, the total form of
M3
A
W2/L
VC2 W2/L
M1
VC1 2. Systematic analysis of control scheme
VC3
M5
W1/L Fig. 2. Differential amplifier with current-source and three nodes which control signals can be applied.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
Av2;AL ¼ 2
W2 ðV GS2 V TH Þ W 1 λðV GS1 V TH Þ2 ð1 þ λðV C2 V GS2 ÞÞ
1081
ð4Þ
Finally, the general form of the voltage gain equation after applying VC3 will be similar to the case of applying VC1. In (3) and (4), VC2 is not equal to VGS2, and VGS2 must be calculated as a function of it. Eqs. (13)–(17) show how VGS2 is dependent on VC2: I D1 ¼ 2I D2
ð5Þ
W W μn C ox ðV GS1 V TH ÞðV C2 V GS2 Þ ¼ μn C ox ðV GS2 V TH Þ2 L 1 L 2 ð6Þ Therefore, VGS2 will be V GS2 ¼ V TH
W1 1 ðV GS1 V TH Þ 7 2W 2 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W1 W1 ðV GS1 V TH Þ ðV GS1 V TH Þ þ 4ðV C2 V TH Þ W2 W2
ð7Þ The negative sign of the radical in (7) is not the solution because it results in a VGS2 less than VTH. Replacing VGS2 in (3) and (4) by (7), the voltage gains in (3) and (4) will be given as a function of VC2. Eqs. (1)–(7) show that the gain characteristics of the different voltage gain controlling schemes have dependence of fractional, square root, or polynomial form on the control voltages. 2.2. Exponential functions as a general solution The ideal solution to a dB-linear function or gain characteristic is exponential function 20 log ðkeαðx þ βÞ Þ ¼ 20 log k þ 20αðx þ β Þlog e
ð8Þ
The right side of (8) is an ideal distortion-less linear function. Thus, if we have a voltage gain characteristic with exponential form dependence on the control voltage as the independent variable, the gain characteristic will be exactly dB-linear versus the control voltage. Accordingly, we must find a mathematical function with similarity to MOS amplifier voltage gain characteristic that can estimate an exponential function. The Taylor expansion of a general exponential function suggests the solution
αðx þ βÞ
keαðx þ βÞ ¼ k þ k
1!
α2 ðx þ βÞ2
þk
2!
α3 ðx þ βÞ2
þk
3!
þUUU
ð9Þ
where k, α and β are constants and x is the independent variable (in this study it will be the control voltage). We can neglect the series terms that have degree greater than any arbitrary positive integer in order that we will have a polynomial as an estimation of the exponential function. To increase the accuracy of the estimation we must include more terms in the right side of (9). Fig. 3, which shows the three different estimations of the polynomial series expansion, illustrates that increase in the degree of the polynomials at right side of (9) increases the accuracy of the estimation and the interval that our estimation is acceptable. Even a first-degree polynomial can estimate the exponential function although the acceptable interval and accuracy are very limited. However, from Fig. 3, it is clear that, aside from the accuracy of the estimation and the number of terms included in the right side of (9), for a given polynomial the bounds and position of the acceptable interval on the independent variable axis are specified. This interval often is outside of the usable range of control voltage, where amplifier operates correctly, or just has a short overlap with this range. Thus, we must somehow adjust the position of the acceptable interval to usable range of the control voltage and extend it as much as possible. For this reason, we replace the control voltages of (1)–(7) by a polynomial of a degree more than one. Our calculations will include three first terms at the right side
Fig. 3. The exponential function (in black), and the sum of the first 2–4 terms of its Taylor series.
of (9) to have a rational tradeoff between complexity and accuracy of the calculations. Based upon what is discussed, if the voltage gain can be converted to a polynomial versus control voltage, the logarithm of it will be linear versus control voltage (in a certain interval of control voltage). It must be noted that because of neglecting the terms with exponent higher than two in (9), this equation is valid only for specific interval of x as calculated below: αðx þ βÞ⪡1 ) 1 β ⪡x⪡ 1 β
α
ð10Þ
α
In (1)–(7) the control voltage appears in these positions: as the independent variable of a polynomial that is in numerator or denominator of a fraction or under radical. It is obvious that while we assume a polynomial is equal to its exponential counterpart, if the polynomial is placed under radical, the exponent of its counterpart exponential will be halved and the new coefficient will be square root of old one (k). Besides, if the radical is added to a constant, like (7), we can convert the new exponential, which is obtained from radical, to a new polynomial, add the constant to it, and convert the result to final exponential. On the other hand, if the polynomial is placed in the denominator of a fraction, the sign exponent of exponential will be inverted and its coefficient will be reversed. Finally, it can be concluded that the operands of (1)–(7) cannot change the type of the resulting function, which is exponential, and they only change the coefficients. Therefore, by adjusting the coefficients of the polynomial we can obtain the desired exponential for the voltage gain. Neglecting the series terms that have degree greater than 2, we arrive at keαðx þ βÞ C k
α2 2
x2 þ ðkα þkα2 β Þx þ ðk þ kαβ þ
kα 2 β Þ 2 2
ð11Þ
And after comparing (11) with y ¼ K 1 x2 þ K 2 x þ K 3 one can calculate k¼
4K 1 K 3 K 22 2K 1 2K
1 ffi α ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4K 1 K 3 K 22
α, β and k versus K1, K2 and K3
ð12Þ
1082
β¼
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
K2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K 1 K 3 K 22
ð13Þ
2K 1
It is apparent that the coefficients k, α and β could be simply calculated in terms of Ki. For desired coefficients in the left side of (11), coefficients in (12) will be available, and if we need a specific linear characteristic, we can modify α, β and k by changing Ki. Replacing the control voltage with a polynomial of the control voltage with appropriate coefficients increases the accuracy of the estimation and adjusts the boundaries of the acceptable interval to the usable range. 2.3. Characteristic improvement of VGAs Based upon what has been discussed so far, we produce a secondary control voltage, V0 C V 0C ¼ K 1 V C 2 þ K 2 V C þ K 3
ð14Þ
of degree two that can estimate an exponential acceptably. The obtained voltage gain characteristic will be dB-linear, and it will be adjustable by changing Ki in (14). By substituting VC1 by V0 C, (1) and (2) are transformed to the below equations rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Av1;RL ¼
μn C ox W 1 W 2 L
2
ð15Þ sffiffiffiffiffiffiffiffiffiffiffi W2 2λ 0 0 0 Av1;AL ¼ 2 ffi k1 e α1 ðV C þ β1 Þ W 1 ðK 1 V C 2 þ K 2 V C þ K 3 V TH Þ
Av2;RL ¼ μn cox
ðK 01 V 2 C þ K 02 V C þ K 03 V TH Þ
W2
0
0
ð18Þ
The coefficients of exponentials in (15) and (18) can be simply calculated from (13), and they are adjustable by changing V0 C coefficients. As it can be seen, applying an appropriate second order control voltage (V0 C) instead of VC transforms the logarithm of amplifier voltage gain to a dB-linear function of control voltage (VC). As it can be seen, according to the voltage gain function versus VC, by using simple functions such as polynomials, an appropriate approximation of an exponential function is achieved in the both of categories of VGAs. Complete analysis of the proposed technique including second order effects is given in Appendix.
R
R1
ðK 01 V 2 C þ K 02 V C þ K 03 V TH ÞRL C k2 eα2 ðV C þ β2 Þ 2
λW 1 ðV GS1 V TH Þ2 ð1 þ λððK 1 K 01 ÞV C 2 þ ðK 2 K 02 ÞV C þ ðK 3 K 03 ÞÞÞ
0
+
VIN2
Ck2 e α2 ðV C þ β2 Þ VOUT
R VIN1
W L
ð17Þ
Av2;AL ¼ 2
R
ð16Þ
For (3) and (4), if we use polynomial–exponential and exponential–polynomial conversions, we finally will have new coefficients K0 i for the final polynomial of VGS2
This function is used to implement the predistorter, which eliminates nonlinearity of VGAs. If V0 C is applied to the gate of M1 instead of VC1, the voltage gain Eqs. (1) and (2) will be polynomials
R
ðK 1 V C 2 þ K 2 V C þ K 3 V TH ÞRL ffi k1 eα1 ðV C þ β1 Þ
R2
VOUT
+
VIN
VC
Amp1 Av=K1
R4 R3
V'C= K1VC2+K2VC+K3
R3
R2 VIN1 VIN2
Amp3 Av=K2
R1 R3 R2
+
K3
R5
+
VOUT
VC R4 R3
K1
R2
V'C= K1VC2+K2VC+K3
R1 R3
+
R3
R5
K2 Fig. 4. (a) Non-inverting voltage adder, (b) non-inverting amplifier and (c) voltage multiplier.
K3
Fig. 5. Two different implementations of the controller: (a) constant coefficients and (b) adjustable coefficients.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
3. Controller implementation and specifications As mentioned previously, the conventional controllers [6,8], which are presented in the traditional VGAs, are unique for their amplifiers and they cannot be modified for other types of VGAs by simple modifications. Furthermore, the presented circuits for implementing controller consume considerable DC power [6,8]. Moreover, in the circuits which use structure and topology modification to control the voltage gain, the characteristic is not linear and do not offer a useful performance [9]. Accordingly, there is an essential need for a comprehensive and adjustable controller, which can modify the transfer function of diverse VGAs to adjust the voltage gain versus VC for achieving a dB-linear characteristic. 3.1. Controller implementation The general function that must be implemented for controller is (14). Due to fact that the control voltage is a low frequency signal, Op-Amp-based circuits can be used to implement it.
30 a=0,b=1,c=0 a=0.1,b=-0.6,c=0.65 a=0.1,b=0.45,c=0.3 a=0.1,b=0.45,c=0.2
20
To implement (7), voltage adder, amplifier and multiplier blocks must be implemented, and this can be easily done by diverse Op-Amp circuits, and one instance of each is depicted in Fig. 4(a)–(c). There are two ways for implementation of (14): constant coefficients and adjustable coefficients. Fig. 5 shows these structures. In the first case, after obtaining the optimum coefficients in (14) by circuit simulation, we implement the equation based on Fig. 5(a). It is apparent that this implementation requires less OpAmps than that in Fig. 5(b), and the controller has smaller power consumption. Nevertheless, after fabrication there is no opportunity to compensate technology-caused errors and mismatches. In the second case, we use Fig. 5(b) to implement the equation. Here, the coefficients, which can be produced by the output of a Digital to Analog Converter (DAC), are adjustable even after fabrication. As a result, we can compensate technology-caused errors and mismatches after fabrication. This type of controller implementation has larger power consumption while both of these implementations have smaller power consumption in comparison to conventional controllers since today different Op-Amps with power consumption below 50 mW are available [17–19]. As it has been discussed, any arbitrary value of k, α and β can be obtained by adjusting the polynomial coefficients, or on the other words, by modifying polynomial coefficients. This means that the characteristic, which is a straight line, can be moved forward or backward by varying β. To control the dynamic range, we can adjust the slope of the characteristic varying α. Adjusting the slope
0 30
-10
Voltage Gain (dB)
Voltage Gain (dB)
10
1083
-20 -30 -40 -50 -60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
20 10 0 -10 -20 -30
a=0.2,b=1.2,c=0 a=0,b=1,c=0 a=0.1,b=0.4,c=0.2 =0.1,b=0.4,c=0.1
-40
VC (V)
-50 -60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
VC (V)
0.7
0.6
1 a=0.2,b=1.2,c=0 a=0,b=1,c=0 a=0.1,b=0.4,c=0.2 a=0.1,b=0.4,c=0.3
0.5
V'C (V)
V'C (V)
0.8 0.4
0.3
0.2
a=0,b=1,c=0 a=0.1,b=-0.6,c=0.65 a=0.1,b=0.45,c=0.3 a=0.1,b=0.45,c=0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.6 0.4 0.2
0.7
VC (V) Fig. 6. (a) Voltage gain characteristic of Fig. 2; the control node is gate of M1 and (b) V0 C versus VC.
0
0
0.1
0.2
0.3
0.4
VC (V) Fig. 7. (a) Voltage gain characteristic of Fig. 2; the control node is gate of M1 and (b) V0 C versus VC.
1084
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
of characteristic and moving it forward or backward make it possible to adjust the variation range of VC. Finally, with respect to (8), the coefficient k will move the characteristic vertically. These capabilities make the proposed approach useful in conjunction with AGC loop and alleviate AGC design difficulties. Finally, it is noteworthy that the interval, which these coefficients can vary, is limited because the coefficients variation out of the interval that circuit operates correctly changes the operation region of the transistors, and the dB-linear characteristic will be degraded.
3.2. Sensitivity and mismatch effects All integrated circuits suffer from mismatch effects since there are some mismatch between the geometry of layout of the circuit and geometry of the fabricated circuit. Although resistors have mismatches, their mismatch effects are small and the most important mismatch errors are due to transistor's geometry mismatch [16]. In the conventional gain controllers, this kind of mismatch leads to deviation from optimum status. This deviation itself, in turn, often changes the VGA gain characteristic by changing its dynamic range, slope, or linearity. In the proposed
100
controller, OP-Amp is utilized to implement controller. Although errors and mismatch may vary the gain or input impedance of the Op-Amps to some extent, it is straightforward to prove that these changes have negligible effects on the performance of the circuits shown in Fig. 4. By proper selection of resistance value and implementation, we can minimize the mismatch effects of the fabrication on the controller. On the other hand, in the adjustablecoefficient controller we can simply adjust the coefficients to eliminate the effects of the errors and mismatches. As a result, this technique has more resilience against mismatches. The power–supply rejection ratio has become increasingly an important parameter in MOS technology as the level of integration increases. Integrating analog and digital on the same chip increases the coupling from the digital circuits to the analog supplies. A common solution to this issue is using a capacitor in parallel to the power supply. In the proposed technique, the amplifier is implemented differentially and it cancels small perturbations in supply voltage. Moreover, the functionality of the proposed controller, which uses differential Op-Amps, has a weak dependence on supply voltage variations because a small perturbation in the gain and input impedance of the Op-Amps has a negligible effect of the transfer functions of the circuits shown in Fig. 4. Accordingly, the proposed technique is stable and rejects supply voltage variations.
80
40
60
20
40
0
Voltage Gain (dB)
Voltage Gain (dB)
60
-20 -40 a=0,b=1,c=0 a=0.15,b=0,c=0 a=0.15,b=-0.36,c=0.22
-60 -80
0
0.2
0.4
0.6
0.8
1
20 0 -20 -40
1.2 -60
VC (V)
V : a=0.12,b=0.17,c=0.25
V : a=-0.21,b=-0.13,c=0.81
V : a=0.12,b=-0.55,c=0.81
-80
0.6
0.7
V : a=-0.21,b=0.8,c=0.1
0.8
0.9
1
1.1
0.8
0.8
0.75
0.6
0.65
0.8
0.7
0.6
0.6
VC2
0.4
0.55 0.5
0.5
0.45
0.2
0.4
0.4
0.35
0
1.2
a=-0.21,b=0.8,c=0.1
a=-0.21,b=-0.13,c=0.81
0.7
VC1
V'C (V)
0.5
VC (V)
a=0,b=1,c=0 a=0.15,b=0,c=0 a=0.15,b=-0.36,c=0.22
1
0.4
0
0.2
0.4
0.6 VC (V)
0.8
1
1.2
Fig. 8. (a) Voltage gain characteristic of Cherry–Hooper before and after applying the proposed approach and (b) V0 C versus VC.
0.3
a=0.12,b=0.17,c=0.25
0.4
0.5
0.6
0.7
a=0.12,b=-0.55,c=0.81
0.8
0.9
1
1.1
1.2
VC (V) Fig. 9. (a) Voltage gain characteristic of Gm-Ratioed VGA before and after applying the proposed approach and (b) V0 C versus VC.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
1085
Table 1 Comparison of previously reported VGAs and this work. Ref.
Linear gain range (dB)
VDD (V)
Power (mW)
Number of stages
3-dB bandwidth (MHz)
VC range (V)
Frequency Input (MHz) P1dB
[6] [8] [9] Cherry–Hooper [12] [13] Gm-Ratioed [14] This workb Cherry–Hooper Gm-Ratioed
39 55 0 60a 10 50a 42 42 52 43 18.5 21.5 –61 70 74 47
1.8 2.5 1 1.8 1.8 1.2 1.2 1.2
20.5 22.7 2.5 5.4 6.5 21.6 6 6.3 7.4
3 3 4 3 2 1 3 3
4 900 2.78 2200 350 32 1050 800 3000 230 1060 220 1020
0.7 1.52 0 2.5 0 0.9 0 1.8 0.4 1.4 0.6 0.8 0.1 1.2 0.35 1.15
100 NA NA NA NA NA 600 500
a b
59 11 NA 55 13 NA 48 17 52 27 73.5 28.7 46.4 11.9
NF (dB)
Tech (nm)
6.8 53 5.2 24.1 17 30 NA NA NA 8.6 75.5 8.16 85.7
180 350 90 180 180 130 130 130
Nonlinear characteristic. Simulation results.
4. Simulations and results The capability of the proposed approach is proved by simulation of some circuits using 0.13 mm CMOS process model. This approach has the capability of linearization of different VGAs; however, for the sake of brevity it is applied to four circuits, two of which are basic differential amplifiers and the others are two traditional VGAs. Although some polynomial coefficients are of centesimal scale, it is easy to create them either by DAC or by voltage amplifier. To make the voltage of node B stable and welldefined, we tied the gates of M4 and M5 together and connected this node to the drain of these devices by a large resistor; this topology serves a simple implementation of common-mode feedback (CMFB). The VGA and voltage controller use 1.2 V supply voltage. First, the capability of the proposed approach to linearly control the voltage gain of basic amplifiers is demonstrated. Fig. 6 shows the simulation results of voltage gain of Fig. 2, while VC is directly applied to the gate of M1 and also for the case that V0 C is applied to it. When VC is directly applied to M1, it varies from 0.3 V to 0.6 V for a linear characteristic in decibel scale; however, VGA is roughly sensitive to VC. To extend and control the dynamic range of VC, a second order polynomial (aVc2 þbVc þc) is used as V0 C. The values of coefficients a, b and c are given in figures. The slope of the characteristic can be decreased and the variation range of VC can be increased by applying a second order polynomial controller. Also, it is possible to achieve a negative slope, a shifted characteristic or even a higher slope if the coefficients of polynomial vary. In other words, the characteristic is adjustable. Fig. 6 shows the gain characteristics and variation range of V0 C. The linear dynamic range of Av is about 60 dB in all cases. Fig. 7 shows the simulation result for the case that the control nodes are the gates of M2 and M3. Second, Cherry–Hooper based and Gm-Ratioed VGAs are simulated using the proposed technique. Therefore, in Fig. 1(b) VC is replaced by V0 C where it is a second order polynomial versus VC. Fig. 8 depicts the ability of the approach to linearize the characteristic and to invert the sign of slope. Finally, the gain of GmRatioed VGA is controlled by two different first order polynomials. For this, the two control voltages VC1 and VC2, which are two function of VC, are applied to the gates of M1 and M6 respectively. As depicted in Fig. 9 when two polynomials are applied, the dB-linear dynamic range of VGA is about 121 dB which is 26 dB more than that of the traditional main VGA. It depicts that the proposed approach not also accomplishes the function of conventional predistorter but also gives broader dynamic range. Even more linear dynamic range could be achieved at the cost of accepting more values for maximum noise figure. The complete simulation results are given and compared in Table 1. It must be noted that, in Table 1, large noise figures occur because of large losses (negative gains in dB scale).
To investigate the stability of the gain characteristic against supply voltage variations, in all circuits, the supply voltage was changed in a range between 1.1 V and 1.3 V, and the change in the voltage gain characteristic was negligible and smaller than 0.2 dB.
5. Conclusion In this study, a comprehensive dB-linear approach is presented to linearize and broaden gain characteristic of various VGA. By analytical calculation, the general function for linear gain control of a generic amplifier is obtained. Conventional Op-Amp circuits are used to implement controller mathematical function. The proposed technique has lower sensitivity than conventional techniques to mismatches and supply voltage variations. To demonstrate the validity of calculations, gains of a conventional differential amplifier and two traditional VGAs are improved. Due to the low frequency of control signal, narrow band-very low power Op-Amps can be used to implement the controller so that its power consumption is negligible.
Appendix A To obtain the overall performance of the amplifier and its dependence on control voltages taking into the account second order effects we can utilize polynomial–exponential and exponential–polynomial conversions. The current if transistors give us the basic equations. In the case that we change the gate voltage of M1, M1 operates in saturation region, and we replace VC1 with V0 C. Using KCL in nodes A and B in Fig. 2, we arrive at 1 W 1 ðK 1 V C 2 þ K 2 V C þ K 3 V TH0N Þ2 ð1 þ λV A Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi ¼ W 2 ðV C2 V TH0N γ ð 2φ þ V A 2φ ÞÞ2 ð1 þ λðV B V A ÞÞ F
F
1 μ W 1 ðK 1 V C 2 þ K 2 V C þK 3 V TH0N Þ2 ð1 þ λV A Þ 2 n ¼ μp W 3 ðV DD V B V TH0P Þ2 ð1 þ λðV DD V B ÞÞ
ðA1Þ
where I is the drain current of M1. In these equations, VC2 is fixed, and our objective is to calculate coefficients of VB either in polynomial or exponential form as functions of constants in (1), VC2, VC, and coefficients of V0 C in (14). Using (1), one can calculate VB as a function VC1, VC2, and VDD are given. Because the operands in (1) cannot change the form of exponential functions, as it has been discussed, the total form of VA and VB will be exponential too V A ¼ kA expðaA V C þ βA Þ ¼ K 1A V C 2 þ K 2A V C þ K 3A V B ¼ kB expðaB V C þ β B Þ ¼ K 1B V C 2 þ K 2B V C þ K 3B
ðA2Þ
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Here coefficients in both forms are functions of Ki for i¼1, 2, 3, VC2, VC, and other constants in (1). To calculate the coefficients of (2), we must use polynomial–exponential and exponential–polynomial conversions, (12) and (13), to eliminate operands, radicals and exponents, and achieve a unified form, exponential or polynomial, in the both sides of equations in (2). Then we will have three pair of equations making it possible to calculate the six coefficients, exponential or polynomial, in (2). It is straightforward to calculate coefficients of VB by equating corresponding coefficients. Still, these calculations are lengthy and the results will be large functions, and we forbear to accomplish them for the sake of brevity. After calculating coefficients, one can calculate the voltage gain simply by AV 1 ¼
∂V B ∂V C2
ðA3Þ
In the second case, when VC2 controls the voltage gain, M1 operates in Triode region 1 1 W 1 ½ðV C1 V TH0N ÞV A V A 2 Þ 2 2 ¼ W 2 ðK 1 V C 2 þ K 2 V C þ K 3 V TH0N γ ð qffiffiffiffiffiffiffiffiffiffiffiffi 2φF ÞÞ2 ð1 þ λðV B V A ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2φ þ V A F
1 μ W 1 ½ðV C1 V TH0N ÞV A V A 2 Þ 2 n ¼ μp C ox W 3 ðV DD V B V TH0P Þ2 ð1 þ λðV DD V B ÞÞ
ðA4Þ
The voltage gain in this case will be AV 2 ¼
∂V B ∂V B ∂V C ∂V B 1 ¼ ¼ ∂V C2 ∂V C ∂V C2 ∂V C 2K 1 V C þ K 2
ðA5Þ
References [1] S.-H. Yahng, C.-C. Wang, A low power 48-dB/stage linear-in-dB variable gain amplifier for direct-conversion receivers, Microelectron. J. 43 (4) (2012) 274–279.
[2] S. D’Amico, M.D. Matteis, A. Baschirotto, A 6.4 mW, 4.9 nV/√Hz, 24 dBm IIP3 VGA for a multi-standard (WLAN, UMTS, GSM, and Bluetooth) receiver, J . Analog Integr. Circuits Signal Process. 61 (2009) 1–7. [3] B. Sewiolo, G. Fischer, R. Weigel, A 30 GHz variable gain amplifier with high output voltage swing for ultra-wideband radar, IEEE Microw. Wireless Compon. Lett. 19 (9) (2009) 590–592. [4] A.A. El-Adawy, A.M. Soliman, H.O. Elwan, Low voltage fully differential CMOS voltage mode digitally controlled variable gain amplifier, Microelectron. J. 31 (2) (2000) 139–146. [5] Y. Zheng, J. Yan, Y.P. Xu, A CMOS dB-linear VGA with predistortion compensation for wireless communication applications, in: Proceedings of the IEEE International Circuits System Symposium, 1 (5) 2004, pp. 23–26. [6] H.D. Lee, K.A. Lee, S. Hong, A wideband CMOS variable gain amplifier with an exponential gain control, IEEE Trans. Microw. Theory Tech. 55 (6) (2007) 1363–1373. [7] W.C. Song, C.J. Oh, G.H. Cho, H.B. Juiig, High frequency/high dynamic range CMOS VGA, Electron. Lett. 36 (6) (2000) 1096–1098. [8] Y. Zheng, J. Yan, Y.P. Xu, A CMOS VGA with DC offset cancellation for directconversion receivers, IEEE Trans. Circuits Syst.-I 56 (1) (2009) 103–113. [9] Y. Wang, B. Afshar, L. Ye, V.C. Gaudet, A.M. Niknejad, Design of a low power, inductorless wideband variable-gain amplifier for high-speed receiver systems, IEEE J. Solid-State Circuits 59 (4) (2012) 696–707. [10] C. Wu, C. Liu, S. Liu, A 2 GHz CMOS variable-gain amplifier with 50 db linear-inmagnitude controlled gain range for 10GBase-LX4 Ethernet, in: Proceedings of the International Solid-State Circuits Conference (ISSCC), 1 (2) 2004, pp. 484–541. [11] X. Huang, X., Qin, Y. Qin, H. Fang, Z. Hong, A 0.8–3 GHz 40 dB dynamic range CMOS variable-gain amplifier, in: Proceedings of the International Conference on ASIC, 9 (10) 2011 pp. 1030–1033. [12] J.K. Kwon, K.D. Kim, W.C. Song, G.H. Cho, Wideband high dynamic range CMOS variable gain amplifier for low voltage and low power for wireless applications, Electron. Lett. 39 (10) (2003) 759–760. [13] Q.-H. Duong, Q. Le, C.-W. Kim, S.-G. Lee, A 95-dB linear low-power variable gain amplifier, IEEE Trans. Circuits Syst.-I 53 (8) (2006) 1648–1657. [14] Z. Chen, Y. Zheng, F.C. Choong, M. Je, A low-power variable-gain amplifier with improved linearity: analysis and design, IEEE Trans. Circuits Syst.-I 59 (10) (2012) 2176–2185. [15] E.M. Cherry, D.E. Hooper, The design of wide-band transistor feedback amplifier, IEEE Proc. 110 (2) (1963) 375–389. [16] D. Manstretta, M. Brandolini, F. Svelto, Second-order inter-modulation mechanisms in CMOS downconverters, IEEE J. Solid-State Circuits 38 (3) (2003) 394–406. [17] D. Chen, H. Lin, An 1 V rail–rail low-power CMOS op-amp, in: Proceedings of the IEEE ISCAS Symposium, 1 (5) 2002, pp. 309–312. [18] E. Kargaran, H. Khosrowjerdi, K. Ghaffarzadegan, A 1.5 v high swing ultra-lowpower two stage CMOS op-amp in 0.18 mm technology, in: Proceedings of the ICMEE International Conference, 1 (8) 2010, pp. 68–71. [19] D.J. Comer, D.T. Comer, R.P. Singh, A high-gain, low-power CMOS op amp using composite cascode stages, in: Proceedings of the MWSCAS IEEE Symposium, 53 (8) 2010, pp. 600–603.
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
A comprehensive, adjustable approach for linearizing and broadening the gain characteristic of variable gain amplifiers H. Bameri a,n, H. Abdollahi b, A. Hakimi a a b
Electrical Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran Electrical and Computer Engineering Department, Tarbiat Modares University, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 16 October 2013 Received in revised form 26 April 2014 Accepted 28 April 2014
In this paper a comprehensive approach is presented to linearize and adjust gain characteristic of variable gain amplifiers (VGAs). It is also capable of increasing the output linear dynamic range of VGAs and modifying variation range of control voltage. The approach is able to change the voltage gain characteristic of an amplifier, even after fabrication, to a desired one by means of a digital control signal and a digital to analog converter. Using this approach, the gain of basic differential amplifier is controlled by two different predistorters, and adjustable dB-linear characteristics in range of greater than 60 dB are achieved. The approach, also, is applied to two conventional VGAs, the gain characteristic of first VGA is linearized, and in the second VGA, the linear dynamic range is expanded about 26 dB. The controller uses 1.2 V voltage supply, and simulations are done using 0.13 mm CMOS process model. The other characteristics of each mode of control are reported completely. & 2014 Elsevier Ltd. All rights reserved.
Keywords: VGA dB-linear Dynamic range Adjustable gain characteristic
1. Introduction The variable gain amplifier (VGA) is a significant analog block in wireless communication receivers. Applications of this block in Bluetooth, UWB, WLAN, and WSN (Wireless Sensor Network) indicate its substantial role [1–3]. In such wireless links, as a result of channel fading, the amplitude of received signal varies greatly. In conjunction with an automatic gain control (AGC) feedback loop, VGA is implemented to compensate variations and provide fairly constant amplitude for ADC in order that the dynamic range of receiver will be increased [4]. It is desired to a have dB-linear gain characteristic for VGA to achieve a constant settling time for AGC loop [1]. It may be possible to achieve this gain characteristic by using the exponential relation between collector current and base–emitter voltage in BJT or BiCMOS technology [3]. Nevertheless, an all-CMOS integration of digital and analog circuits is easier and more economical. The other solution is a subthreshold-biased MOS device, which has an exponential transconductance versus gate–source voltage [5]. However, it suffers from undesired effects such as poor noise performance and limited bandwidth. The final solution is a parasitic BJT in deep n-well CMOS technology which can imitate
n
Corresponding author. Tel.: þ 98 9132968531; fax: þ98 3413235900. E-mail address: [email protected] (H. Bameri).
http://dx.doi.org/10.1016/j.mejo.2014.04.041 0026-2692/& 2014 Elsevier Ltd. All rights reserved.
the function of an inherent BJT, but imperfect effects of lateral NPN/PNP transistors degrade the gain characteristic [6]. Based on mentioned difficulties, designers commonly use the square or linear characteristics of MOS devices, which inherently are not dB-linear. Several approaches have been proposed to realize a linear-in-decibel gain characteristic. These approaches are divided into two categories: the first approach is predistortion [6–8], and the second is topology modification and the combination of VGA elements characteristics [9–11]. In some cases a combination of both approaches is used [12–14]. In the first category, the transfer function of a predistorter block converts VC to a secondary voltage or current in such a way that if it is applied to the VGA, a dB-linear gain characteristic will be achieved. In the second category, the circuit topology of VGA is changed and modified to provide the desired gain characteristic. Examples of such categories are a differential amplifier modified by diode-connected loads (Gm-Ratioed amplifier) and Cherry– Hooper amplifier [15] with tunable resistive feedback, which are shown in Fig. 1. Among described categories, in the former, the application of the proposed predistorter block is limited to the VGA circuit topology, so it is not feasible to use the predistorter as the controller of other VGAs, which have different circuit topologies. In addition, the utilized predistorters dissipate significant amount of DC power that increases the total power consumption of VGA [7,8]. On the other hand, in the latter, the proposed approach is unique for a specific VGA, and cannot be applied to others.
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H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
diverse MOS amplifier formulas must be obtained so that it will be possible to devise an exhaustive controller.
VDD
2.1. Gain analysis
M4
M5 CMFB
+ V OUT + VIN -
M2
VC1
M7
M3
M1
VC2
M8
M6
VDD Rb
Rd
Rd
Rb
VC
Mf1
+ V OUT M3
+
M1
Mf2
M4
ISS2
M2
VIN
A variable gain could be realized by changing the transfer function of the transconductor (Gm) or the load resistance by control voltage. Depending on the nodes where this voltage is applied (transconductor, active load, or tail current source), the operation and the characteristics of VGA will be different. According to the intrinsic properties of MOS devices, and regardless of how it operates in the circuit (saturation or triode), the relation between the logarithm of the voltage gain and control voltage will be nonlinear. As shown in Fig. 2, the control voltage can be applied to three different nodes of circuit. The analysis of each control voltage effect on the gain characteristic enables us to have a deep insight into how the circuit operates and how its gain depends on the control voltages, and as a result, a general form for predistorter operation will be obtainable. Moreover, more complex VGAs could be designed by applying a couple of the control voltages as shown in Fig. 2, and there will be several scenarios. It must be noted that active loads (M4 and M5) can be replaced by resistive loads occasionally. In Fig. 2, the input small signal is applied to the gate of transistors M2 and M3, and the output is obtained from drains of them. In the simplest scenario, each control signal is applied individually while others are fixed. According to the large signal relation between drain current and gate–source voltage in a MOS device, the amplifier gain can be calculated in terms of VCi (i¼1:3). First, when VC1 is applied to the gate of the current source of the amplifier, the gain is calculated by (1) and (2). RL and AL suffixes depict whether the load of the amplifier is a resistive load or an active load respectively rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ C ox W 1 W 2 Av1;RL ¼ n ð1Þ ðV C1 V TH ÞRL L 2
ISS1
Fig. 1. (a) Modified Cherry–Hooper gain cell and (b) Gm-Ratioed amplifier.
In this paper, a comprehensive approach is proposed to realize a dB-linear VGA. In addition to low power consumption, by some simple modifications, the proposed controller is capable to be applied to other VGAs. Using this approach, the slope of the linear gain characteristic as well as control voltage range can be adjusted arbitrarily even after fabrication to facilitate the AGC loop design. Moreover, this approach has the ability of linearization and linear dynamic-range extension of traditional VGAs. The remainder of this paper is organized as follows. In Section 2, the gain equations and general forms of fundamental amplifiers are calculated to find the optimum mathematical function for the predistorter based on the circuit of VGA. Section 3 describes advantages of this approach and how the predistorter of the proposed approach can be implemented. In Section 4, simulation results are presented, and Section 5 concludes the paper.
Av1;AL ¼
sffiffiffiffiffiffiffiffiffiffiffi W2 1 2 W 1 λðV C1 V TH Þ
ð2Þ
where Wi, L and VTH are the widths, length and threshold voltage of the MOS devices, i¼1, 2, and RL is the resistive load of the amplifier. Second, in the case of applying VC2, (3) and (4) are obtained for the voltage gain of the amplifier Av2;RL ¼ μn C ox
W2 ðV GS2 V TH ÞRL L
ð3Þ
VDD W3/L
VC3
W3/L
M4
B M2
VC2
To obtain a general solution for the controller that can convert the VGA to a dB-linear high dynamic range one, the total form of
M3
A
W2/L
VC2 W2/L
M1
VC1 2. Systematic analysis of control scheme
VC3
M5
W1/L Fig. 2. Differential amplifier with current-source and three nodes which control signals can be applied.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
Av2;AL ¼ 2
W2 ðV GS2 V TH Þ W 1 λðV GS1 V TH Þ2 ð1 þ λðV C2 V GS2 ÞÞ
1081
ð4Þ
Finally, the general form of the voltage gain equation after applying VC3 will be similar to the case of applying VC1. In (3) and (4), VC2 is not equal to VGS2, and VGS2 must be calculated as a function of it. Eqs. (13)–(17) show how VGS2 is dependent on VC2: I D1 ¼ 2I D2
ð5Þ
W W μn C ox ðV GS1 V TH ÞðV C2 V GS2 Þ ¼ μn C ox ðV GS2 V TH Þ2 L 1 L 2 ð6Þ Therefore, VGS2 will be V GS2 ¼ V TH
W1 1 ðV GS1 V TH Þ 7 2W 2 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W1 W1 ðV GS1 V TH Þ ðV GS1 V TH Þ þ 4ðV C2 V TH Þ W2 W2
ð7Þ The negative sign of the radical in (7) is not the solution because it results in a VGS2 less than VTH. Replacing VGS2 in (3) and (4) by (7), the voltage gains in (3) and (4) will be given as a function of VC2. Eqs. (1)–(7) show that the gain characteristics of the different voltage gain controlling schemes have dependence of fractional, square root, or polynomial form on the control voltages. 2.2. Exponential functions as a general solution The ideal solution to a dB-linear function or gain characteristic is exponential function 20 log ðkeαðx þ βÞ Þ ¼ 20 log k þ 20αðx þ β Þlog e
ð8Þ
The right side of (8) is an ideal distortion-less linear function. Thus, if we have a voltage gain characteristic with exponential form dependence on the control voltage as the independent variable, the gain characteristic will be exactly dB-linear versus the control voltage. Accordingly, we must find a mathematical function with similarity to MOS amplifier voltage gain characteristic that can estimate an exponential function. The Taylor expansion of a general exponential function suggests the solution
αðx þ βÞ
keαðx þ βÞ ¼ k þ k
1!
α2 ðx þ βÞ2
þk
2!
α3 ðx þ βÞ2
þk
3!
þUUU
ð9Þ
where k, α and β are constants and x is the independent variable (in this study it will be the control voltage). We can neglect the series terms that have degree greater than any arbitrary positive integer in order that we will have a polynomial as an estimation of the exponential function. To increase the accuracy of the estimation we must include more terms in the right side of (9). Fig. 3, which shows the three different estimations of the polynomial series expansion, illustrates that increase in the degree of the polynomials at right side of (9) increases the accuracy of the estimation and the interval that our estimation is acceptable. Even a first-degree polynomial can estimate the exponential function although the acceptable interval and accuracy are very limited. However, from Fig. 3, it is clear that, aside from the accuracy of the estimation and the number of terms included in the right side of (9), for a given polynomial the bounds and position of the acceptable interval on the independent variable axis are specified. This interval often is outside of the usable range of control voltage, where amplifier operates correctly, or just has a short overlap with this range. Thus, we must somehow adjust the position of the acceptable interval to usable range of the control voltage and extend it as much as possible. For this reason, we replace the control voltages of (1)–(7) by a polynomial of a degree more than one. Our calculations will include three first terms at the right side
Fig. 3. The exponential function (in black), and the sum of the first 2–4 terms of its Taylor series.
of (9) to have a rational tradeoff between complexity and accuracy of the calculations. Based upon what is discussed, if the voltage gain can be converted to a polynomial versus control voltage, the logarithm of it will be linear versus control voltage (in a certain interval of control voltage). It must be noted that because of neglecting the terms with exponent higher than two in (9), this equation is valid only for specific interval of x as calculated below: αðx þ βÞ⪡1 ) 1 β ⪡x⪡ 1 β
α
ð10Þ
α
In (1)–(7) the control voltage appears in these positions: as the independent variable of a polynomial that is in numerator or denominator of a fraction or under radical. It is obvious that while we assume a polynomial is equal to its exponential counterpart, if the polynomial is placed under radical, the exponent of its counterpart exponential will be halved and the new coefficient will be square root of old one (k). Besides, if the radical is added to a constant, like (7), we can convert the new exponential, which is obtained from radical, to a new polynomial, add the constant to it, and convert the result to final exponential. On the other hand, if the polynomial is placed in the denominator of a fraction, the sign exponent of exponential will be inverted and its coefficient will be reversed. Finally, it can be concluded that the operands of (1)–(7) cannot change the type of the resulting function, which is exponential, and they only change the coefficients. Therefore, by adjusting the coefficients of the polynomial we can obtain the desired exponential for the voltage gain. Neglecting the series terms that have degree greater than 2, we arrive at keαðx þ βÞ C k
α2 2
x2 þ ðkα þkα2 β Þx þ ðk þ kαβ þ
kα 2 β Þ 2 2
ð11Þ
And after comparing (11) with y ¼ K 1 x2 þ K 2 x þ K 3 one can calculate k¼
4K 1 K 3 K 22 2K 1 2K
1 ffi α ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4K 1 K 3 K 22
α, β and k versus K1, K2 and K3
ð12Þ
1082
β¼
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
K2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K 1 K 3 K 22
ð13Þ
2K 1
It is apparent that the coefficients k, α and β could be simply calculated in terms of Ki. For desired coefficients in the left side of (11), coefficients in (12) will be available, and if we need a specific linear characteristic, we can modify α, β and k by changing Ki. Replacing the control voltage with a polynomial of the control voltage with appropriate coefficients increases the accuracy of the estimation and adjusts the boundaries of the acceptable interval to the usable range. 2.3. Characteristic improvement of VGAs Based upon what has been discussed so far, we produce a secondary control voltage, V0 C V 0C ¼ K 1 V C 2 þ K 2 V C þ K 3
ð14Þ
of degree two that can estimate an exponential acceptably. The obtained voltage gain characteristic will be dB-linear, and it will be adjustable by changing Ki in (14). By substituting VC1 by V0 C, (1) and (2) are transformed to the below equations rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Av1;RL ¼
μn C ox W 1 W 2 L
2
ð15Þ sffiffiffiffiffiffiffiffiffiffiffi W2 2λ 0 0 0 Av1;AL ¼ 2 ffi k1 e α1 ðV C þ β1 Þ W 1 ðK 1 V C 2 þ K 2 V C þ K 3 V TH Þ
Av2;RL ¼ μn cox
ðK 01 V 2 C þ K 02 V C þ K 03 V TH Þ
W2
0
0
ð18Þ
The coefficients of exponentials in (15) and (18) can be simply calculated from (13), and they are adjustable by changing V0 C coefficients. As it can be seen, applying an appropriate second order control voltage (V0 C) instead of VC transforms the logarithm of amplifier voltage gain to a dB-linear function of control voltage (VC). As it can be seen, according to the voltage gain function versus VC, by using simple functions such as polynomials, an appropriate approximation of an exponential function is achieved in the both of categories of VGAs. Complete analysis of the proposed technique including second order effects is given in Appendix.
R
R1
ðK 01 V 2 C þ K 02 V C þ K 03 V TH ÞRL C k2 eα2 ðV C þ β2 Þ 2
λW 1 ðV GS1 V TH Þ2 ð1 þ λððK 1 K 01 ÞV C 2 þ ðK 2 K 02 ÞV C þ ðK 3 K 03 ÞÞÞ
0
+
VIN2
Ck2 e α2 ðV C þ β2 Þ VOUT
R VIN1
W L
ð17Þ
Av2;AL ¼ 2
R
ð16Þ
For (3) and (4), if we use polynomial–exponential and exponential–polynomial conversions, we finally will have new coefficients K0 i for the final polynomial of VGS2
This function is used to implement the predistorter, which eliminates nonlinearity of VGAs. If V0 C is applied to the gate of M1 instead of VC1, the voltage gain Eqs. (1) and (2) will be polynomials
R
ðK 1 V C 2 þ K 2 V C þ K 3 V TH ÞRL ffi k1 eα1 ðV C þ β1 Þ
R2
VOUT
+
VIN
VC
Amp1 Av=K1
R4 R3
V'C= K1VC2+K2VC+K3
R3
R2 VIN1 VIN2
Amp3 Av=K2
R1 R3 R2
+
K3
R5
+
VOUT
VC R4 R3
K1
R2
V'C= K1VC2+K2VC+K3
R1 R3
+
R3
R5
K2 Fig. 4. (a) Non-inverting voltage adder, (b) non-inverting amplifier and (c) voltage multiplier.
K3
Fig. 5. Two different implementations of the controller: (a) constant coefficients and (b) adjustable coefficients.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
3. Controller implementation and specifications As mentioned previously, the conventional controllers [6,8], which are presented in the traditional VGAs, are unique for their amplifiers and they cannot be modified for other types of VGAs by simple modifications. Furthermore, the presented circuits for implementing controller consume considerable DC power [6,8]. Moreover, in the circuits which use structure and topology modification to control the voltage gain, the characteristic is not linear and do not offer a useful performance [9]. Accordingly, there is an essential need for a comprehensive and adjustable controller, which can modify the transfer function of diverse VGAs to adjust the voltage gain versus VC for achieving a dB-linear characteristic. 3.1. Controller implementation The general function that must be implemented for controller is (14). Due to fact that the control voltage is a low frequency signal, Op-Amp-based circuits can be used to implement it.
30 a=0,b=1,c=0 a=0.1,b=-0.6,c=0.65 a=0.1,b=0.45,c=0.3 a=0.1,b=0.45,c=0.2
20
To implement (7), voltage adder, amplifier and multiplier blocks must be implemented, and this can be easily done by diverse Op-Amp circuits, and one instance of each is depicted in Fig. 4(a)–(c). There are two ways for implementation of (14): constant coefficients and adjustable coefficients. Fig. 5 shows these structures. In the first case, after obtaining the optimum coefficients in (14) by circuit simulation, we implement the equation based on Fig. 5(a). It is apparent that this implementation requires less OpAmps than that in Fig. 5(b), and the controller has smaller power consumption. Nevertheless, after fabrication there is no opportunity to compensate technology-caused errors and mismatches. In the second case, we use Fig. 5(b) to implement the equation. Here, the coefficients, which can be produced by the output of a Digital to Analog Converter (DAC), are adjustable even after fabrication. As a result, we can compensate technology-caused errors and mismatches after fabrication. This type of controller implementation has larger power consumption while both of these implementations have smaller power consumption in comparison to conventional controllers since today different Op-Amps with power consumption below 50 mW are available [17–19]. As it has been discussed, any arbitrary value of k, α and β can be obtained by adjusting the polynomial coefficients, or on the other words, by modifying polynomial coefficients. This means that the characteristic, which is a straight line, can be moved forward or backward by varying β. To control the dynamic range, we can adjust the slope of the characteristic varying α. Adjusting the slope
0 30
-10
Voltage Gain (dB)
Voltage Gain (dB)
10
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-20 -30 -40 -50 -60
0
0.1
0.2
0.3
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0.7
20 10 0 -10 -20 -30
a=0.2,b=1.2,c=0 a=0,b=1,c=0 a=0.1,b=0.4,c=0.2 =0.1,b=0.4,c=0.1
-40
VC (V)
-50 -60
0
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VC (V)
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1 a=0.2,b=1.2,c=0 a=0,b=1,c=0 a=0.1,b=0.4,c=0.2 a=0.1,b=0.4,c=0.3
0.5
V'C (V)
V'C (V)
0.8 0.4
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a=0,b=1,c=0 a=0.1,b=-0.6,c=0.65 a=0.1,b=0.45,c=0.3 a=0.1,b=0.45,c=0.2
0.1
0
0
0.1
0.2
0.3
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0.6
0.6 0.4 0.2
0.7
VC (V) Fig. 6. (a) Voltage gain characteristic of Fig. 2; the control node is gate of M1 and (b) V0 C versus VC.
0
0
0.1
0.2
0.3
0.4
VC (V) Fig. 7. (a) Voltage gain characteristic of Fig. 2; the control node is gate of M1 and (b) V0 C versus VC.
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of characteristic and moving it forward or backward make it possible to adjust the variation range of VC. Finally, with respect to (8), the coefficient k will move the characteristic vertically. These capabilities make the proposed approach useful in conjunction with AGC loop and alleviate AGC design difficulties. Finally, it is noteworthy that the interval, which these coefficients can vary, is limited because the coefficients variation out of the interval that circuit operates correctly changes the operation region of the transistors, and the dB-linear characteristic will be degraded.
3.2. Sensitivity and mismatch effects All integrated circuits suffer from mismatch effects since there are some mismatch between the geometry of layout of the circuit and geometry of the fabricated circuit. Although resistors have mismatches, their mismatch effects are small and the most important mismatch errors are due to transistor's geometry mismatch [16]. In the conventional gain controllers, this kind of mismatch leads to deviation from optimum status. This deviation itself, in turn, often changes the VGA gain characteristic by changing its dynamic range, slope, or linearity. In the proposed
100
controller, OP-Amp is utilized to implement controller. Although errors and mismatch may vary the gain or input impedance of the Op-Amps to some extent, it is straightforward to prove that these changes have negligible effects on the performance of the circuits shown in Fig. 4. By proper selection of resistance value and implementation, we can minimize the mismatch effects of the fabrication on the controller. On the other hand, in the adjustablecoefficient controller we can simply adjust the coefficients to eliminate the effects of the errors and mismatches. As a result, this technique has more resilience against mismatches. The power–supply rejection ratio has become increasingly an important parameter in MOS technology as the level of integration increases. Integrating analog and digital on the same chip increases the coupling from the digital circuits to the analog supplies. A common solution to this issue is using a capacitor in parallel to the power supply. In the proposed technique, the amplifier is implemented differentially and it cancels small perturbations in supply voltage. Moreover, the functionality of the proposed controller, which uses differential Op-Amps, has a weak dependence on supply voltage variations because a small perturbation in the gain and input impedance of the Op-Amps has a negligible effect of the transfer functions of the circuits shown in Fig. 4. Accordingly, the proposed technique is stable and rejects supply voltage variations.
80
40
60
20
40
0
Voltage Gain (dB)
Voltage Gain (dB)
60
-20 -40 a=0,b=1,c=0 a=0.15,b=0,c=0 a=0.15,b=-0.36,c=0.22
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20 0 -20 -40
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VC (V)
V : a=0.12,b=0.17,c=0.25
V : a=-0.21,b=-0.13,c=0.81
V : a=0.12,b=-0.55,c=0.81
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VC1
V'C (V)
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1
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Fig. 8. (a) Voltage gain characteristic of Cherry–Hooper before and after applying the proposed approach and (b) V0 C versus VC.
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a=0.12,b=0.17,c=0.25
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a=0.12,b=-0.55,c=0.81
0.8
0.9
1
1.1
1.2
VC (V) Fig. 9. (a) Voltage gain characteristic of Gm-Ratioed VGA before and after applying the proposed approach and (b) V0 C versus VC.
H. Bameri et al. / Microelectronics Journal 45 (2014) 1079–1086
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Table 1 Comparison of previously reported VGAs and this work. Ref.
Linear gain range (dB)
VDD (V)
Power (mW)
Number of stages
3-dB bandwidth (MHz)
VC range (V)
Frequency Input (MHz) P1dB
[6] [8] [9] Cherry–Hooper [12] [13] Gm-Ratioed [14] This workb Cherry–Hooper Gm-Ratioed
39 55 0 60a 10 50a 42 42 52 43 18.5 21.5 –61 70 74 47
1.8 2.5 1 1.8 1.8 1.2 1.2 1.2
20.5 22.7 2.5 5.4 6.5 21.6 6 6.3 7.4
3 3 4 3 2 1 3 3
4 900 2.78 2200 350 32 1050 800 3000 230 1060 220 1020
0.7 1.52 0 2.5 0 0.9 0 1.8 0.4 1.4 0.6 0.8 0.1 1.2 0.35 1.15
100 NA NA NA NA NA 600 500
a b
59 11 NA 55 13 NA 48 17 52 27 73.5 28.7 46.4 11.9
NF (dB)
Tech (nm)
6.8 53 5.2 24.1 17 30 NA NA NA 8.6 75.5 8.16 85.7
180 350 90 180 180 130 130 130
Nonlinear characteristic. Simulation results.
4. Simulations and results The capability of the proposed approach is proved by simulation of some circuits using 0.13 mm CMOS process model. This approach has the capability of linearization of different VGAs; however, for the sake of brevity it is applied to four circuits, two of which are basic differential amplifiers and the others are two traditional VGAs. Although some polynomial coefficients are of centesimal scale, it is easy to create them either by DAC or by voltage amplifier. To make the voltage of node B stable and welldefined, we tied the gates of M4 and M5 together and connected this node to the drain of these devices by a large resistor; this topology serves a simple implementation of common-mode feedback (CMFB). The VGA and voltage controller use 1.2 V supply voltage. First, the capability of the proposed approach to linearly control the voltage gain of basic amplifiers is demonstrated. Fig. 6 shows the simulation results of voltage gain of Fig. 2, while VC is directly applied to the gate of M1 and also for the case that V0 C is applied to it. When VC is directly applied to M1, it varies from 0.3 V to 0.6 V for a linear characteristic in decibel scale; however, VGA is roughly sensitive to VC. To extend and control the dynamic range of VC, a second order polynomial (aVc2 þbVc þc) is used as V0 C. The values of coefficients a, b and c are given in figures. The slope of the characteristic can be decreased and the variation range of VC can be increased by applying a second order polynomial controller. Also, it is possible to achieve a negative slope, a shifted characteristic or even a higher slope if the coefficients of polynomial vary. In other words, the characteristic is adjustable. Fig. 6 shows the gain characteristics and variation range of V0 C. The linear dynamic range of Av is about 60 dB in all cases. Fig. 7 shows the simulation result for the case that the control nodes are the gates of M2 and M3. Second, Cherry–Hooper based and Gm-Ratioed VGAs are simulated using the proposed technique. Therefore, in Fig. 1(b) VC is replaced by V0 C where it is a second order polynomial versus VC. Fig. 8 depicts the ability of the approach to linearize the characteristic and to invert the sign of slope. Finally, the gain of GmRatioed VGA is controlled by two different first order polynomials. For this, the two control voltages VC1 and VC2, which are two function of VC, are applied to the gates of M1 and M6 respectively. As depicted in Fig. 9 when two polynomials are applied, the dB-linear dynamic range of VGA is about 121 dB which is 26 dB more than that of the traditional main VGA. It depicts that the proposed approach not also accomplishes the function of conventional predistorter but also gives broader dynamic range. Even more linear dynamic range could be achieved at the cost of accepting more values for maximum noise figure. The complete simulation results are given and compared in Table 1. It must be noted that, in Table 1, large noise figures occur because of large losses (negative gains in dB scale).
To investigate the stability of the gain characteristic against supply voltage variations, in all circuits, the supply voltage was changed in a range between 1.1 V and 1.3 V, and the change in the voltage gain characteristic was negligible and smaller than 0.2 dB.
5. Conclusion In this study, a comprehensive dB-linear approach is presented to linearize and broaden gain characteristic of various VGA. By analytical calculation, the general function for linear gain control of a generic amplifier is obtained. Conventional Op-Amp circuits are used to implement controller mathematical function. The proposed technique has lower sensitivity than conventional techniques to mismatches and supply voltage variations. To demonstrate the validity of calculations, gains of a conventional differential amplifier and two traditional VGAs are improved. Due to the low frequency of control signal, narrow band-very low power Op-Amps can be used to implement the controller so that its power consumption is negligible.
Appendix A To obtain the overall performance of the amplifier and its dependence on control voltages taking into the account second order effects we can utilize polynomial–exponential and exponential–polynomial conversions. The current if transistors give us the basic equations. In the case that we change the gate voltage of M1, M1 operates in saturation region, and we replace VC1 with V0 C. Using KCL in nodes A and B in Fig. 2, we arrive at 1 W 1 ðK 1 V C 2 þ K 2 V C þ K 3 V TH0N Þ2 ð1 þ λV A Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi ¼ W 2 ðV C2 V TH0N γ ð 2φ þ V A 2φ ÞÞ2 ð1 þ λðV B V A ÞÞ F
F
1 μ W 1 ðK 1 V C 2 þ K 2 V C þK 3 V TH0N Þ2 ð1 þ λV A Þ 2 n ¼ μp W 3 ðV DD V B V TH0P Þ2 ð1 þ λðV DD V B ÞÞ
ðA1Þ
where I is the drain current of M1. In these equations, VC2 is fixed, and our objective is to calculate coefficients of VB either in polynomial or exponential form as functions of constants in (1), VC2, VC, and coefficients of V0 C in (14). Using (1), one can calculate VB as a function VC1, VC2, and VDD are given. Because the operands in (1) cannot change the form of exponential functions, as it has been discussed, the total form of VA and VB will be exponential too V A ¼ kA expðaA V C þ βA Þ ¼ K 1A V C 2 þ K 2A V C þ K 3A V B ¼ kB expðaB V C þ β B Þ ¼ K 1B V C 2 þ K 2B V C þ K 3B
ðA2Þ
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Here coefficients in both forms are functions of Ki for i¼1, 2, 3, VC2, VC, and other constants in (1). To calculate the coefficients of (2), we must use polynomial–exponential and exponential–polynomial conversions, (12) and (13), to eliminate operands, radicals and exponents, and achieve a unified form, exponential or polynomial, in the both sides of equations in (2). Then we will have three pair of equations making it possible to calculate the six coefficients, exponential or polynomial, in (2). It is straightforward to calculate coefficients of VB by equating corresponding coefficients. Still, these calculations are lengthy and the results will be large functions, and we forbear to accomplish them for the sake of brevity. After calculating coefficients, one can calculate the voltage gain simply by AV 1 ¼
∂V B ∂V C2
ðA3Þ
In the second case, when VC2 controls the voltage gain, M1 operates in Triode region 1 1 W 1 ½ðV C1 V TH0N ÞV A V A 2 Þ 2 2 ¼ W 2 ðK 1 V C 2 þ K 2 V C þ K 3 V TH0N γ ð qffiffiffiffiffiffiffiffiffiffiffiffi 2φF ÞÞ2 ð1 þ λðV B V A ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2φ þ V A F
1 μ W 1 ½ðV C1 V TH0N ÞV A V A 2 Þ 2 n ¼ μp C ox W 3 ðV DD V B V TH0P Þ2 ð1 þ λðV DD V B ÞÞ
ðA4Þ
The voltage gain in this case will be AV 2 ¼
∂V B ∂V B ∂V C ∂V B 1 ¼ ¼ ∂V C2 ∂V C ∂V C2 ∂V C 2K 1 V C þ K 2
ðA5Þ
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