A Low-Power Wideband High Dynamic Range Single-Stage Variable Gain Amplifier - VLSI Design and Test - page 20

2 Approximated Exponential Functions

To achieve a wide dynamic range, it is necessary that the gain variation be an
exponential function of the control voltage. As MOSFET characteristics are not
exponential, different pseudo-exponential functions have been proposed [2,11,12] that
enable implementation in MOSFET circuits. The following are three such functions:
()

( )

a + ax 1

2 ax 2

f x = e ≅ (1) f ( x ) = [ + ( + 2 ax ) ] (2)

( )

a − ax 2

2

[ k + ( + ax ) ]

()

f x = (3)

2

[ k + ( − ax ) ]

Of these, the first two functions provide a linear range of about 15dB; the third,
obtains a range of more than 90 dB, for k = 0.12 [2]. We use the third function.

3 VGA Circuit Design

3.1 Control Circuit Design

Figure 1 shows a control circuit [2] which generates (3). The numerator and
denominator of (3) are proportional to the currents I C2 and I C1 in Fig. 1, respectively.
To minimize channel length modulation, the lengths of M1 and M2 must be kept
large. The current ratio (I C2 / I C1 ) as a function of the control voltage V C is given in (4)

2

 

I V

o

 


c

 


+ 1 +

( )

( )

2

V − V

K V − V

I

dd TH

dd TH

c 2

= (4)

2

I

 

c

I V

o

 


c

 


+ 1 −

( )

( )

2

V − V

K V − V

dd TH

dd TH

which is equivalent to (3) with

I

1 o

k =

and a =

( )

( )

2

V − V

K V − V

dd TH

dd TH

Equation (4) is derived by assuming V DD = V SS , with V C in the range (V SS + V thn ) to
(V DD -│V thp │). If V DD and V SS are V supply and 0 V, respectively, then V C stays in the
range V thn to (V DD -│V thp │). Circuit simulation results of the circuit of Fig. 1 are
shown in Fig. 2, which shows that a 90dB range of √(I C2 /I C1 ) is achieved.

3.2 Proposed VGA Design

Figure 3 shows the circuit of the proposed variable gain amplifier (VGA). The circuit
includes a common-mode feedback (CMFB) circuit [2]. In the figure, the core
amplifier consists of the differential pair (M 9 and M 12 ) and diode connected loads
(M 10 and M \11 ). The sizes of M 9 and M 12 are equal, and the sizes of M 10 and M 11 are
equal. The differential gain of this amplifier cell is (g m-M9 /g m-M10 ), with the assumption
that the R out of the current sources is high. Note that this configuration is different