BANDWIDTH ENHANCEMENT OF MULTI-STAGE AMPLIFIERS USING ACTIVE FEEDBACK M. Reza Samadi, AydÕn ø. KarúÕlayan and Jose Silva-Martinez Texas A&M University Department of Electrical Engineering College Station, TX, 77843-3128, Email:
[email protected]
ABSTRACT
1
A new topology for wideband multistage amplifiers (MA) is introduced. The proposed method uses active negative feedback in a chain of amplifiers to extend the bandwidth and improve gain-bandwidth product. The topology has several advantages such as having capability of widening bandwidth as the number of stage increases and enhancing bandwidth by several times that of the dominant pole of each stage. To verify the performance of topology, an 8-stage amplifier in 0.35µm CMOS was designed, where more than 2.8GHz bandwidth and 40dB gain were obtained from simulations.
1. INTRODUCTION For both data amplification and clock distribution, multistage amplifiers (MA) must have high gain and wide bandwidth with frequency response ranging from DC to multi-gigahertz-band frequencies. A conventional MA is composed of n cascaded amplifier stages such that each stage is presented as a transfer function of gj(s) with a DC gain of G. To simplify, let us assume
ωG g j (s) = p s + ωp
ω bw ≈ ω p 2 1 / n − 1
GBP1 = Gωp 21/ n − 1
(4)
GBPT = Gnω p 21/ n −1
(5)
If the number of stages (n) is increased, GBP1 in Equation (4) is decreased. Consider an n-stage conventional MA with passive negative feedback within each stage as illustrated in Figure 1. Assume each stage has a feedback of F (frequency independent) and each forward gain stage can be presented as Equation (1). Using feedback, the dominant pole of each stage is ideally shifted to (1+GF)ωp. The overall bandwidth of an n-stage MA with passive feedback can be written as:
ω bw ≈ ω p (1 + GF ) 21 / n − 1 -f1(s) Vi
-fj(s)
(6)
-fn(s)
+
+ g1(s)
Vo
+ gj(s)
gn(s)
Figure 1. n-stage conventional MA with passive local feedback
(2)
Increasing n decreases ωbw, whereas enlarging the bandwidth of each stage increases the overall bandwidth of MA. Several techniques have been used to increase the speed of amplifiers [2]-[7]. One of the methods to improve bandwidth is using local feedback [2]-[4]. Some techniques, such as capacitance and inductance peaking, enhance the bandwidth by placing a peak in the transfer function at high frequencies (these will be referred as peaking techniques). Active feedback [2]-[3] uses peaking technique to improve the bandwidth of the amplifier by reducing feedback at high frequencies. In previously reported methods [2][7], extending the bandwidth of one stage broadens the overall bandwidth of MA. However, the combination of the poles of all stages degrades the overall bandwidth such that ωbw is always less than the bandwidth of each stage. A meaningful definition for performance of an n-stage MA is the gain-bandwidth product of a single stage [8] (GBP1), which can be written as:
;,(((
The GBP1 and total GBP (GBPT) for an n-stage conventional MA, when all stages are designed as in Eq. (1), are obtained by
(1)
Then the overall DC gain and the bandwidth of the MA are obtained by [1]
GT = G n
GBP1 = (Overall Gain ) n × ( 3 − dB bandwidth ) (3)
,
The GBP1 of Figure 1 is obtained by Equation (4) and GBPT can be written as: GBPT =
Gn ωp 21/ n −1 (1 + GF)n−1
(7)
Equations (6) and (7) show that increasing feedback for having a wider bandwidth decreases GBPT proportional to 1/(1+G×F)n-1. This paper introduces a new topology to build a multi-stage wideband amplifier. It uses a chain of amplifiers with active feedback to expand the bandwidth, and offers several advantages such as: • Improved bandwidth by several times that of the dominant pole of each stage. • Being capable of increasing bandwidth as n increases. • Increased GBP1 by several times that of the conventional MA.
,6&$6
•
Being capable of increasing overall gain-bandwidth product in comparison with the MA with local passive feedback for the same bandwidth.
To validate the proposed topology, an 8-stage MA in 0.35µm CMOS process was designed and simulated. Section II presents the new topology or chained-feedback multistage amplifier (CMA). Circuit simulation results are presented in Section III. Finally, summary of the results and outline of the work are given.
H dc ω n21ω n22 (s + 2ζ 1ω n1 s + ω n21 )(s 2 + 2ζ 2ω n 2 s + ω n22 ) where ωn1, ωn2, ζ1, ζ2 and an overall DC gain are
f j (s) =
ωp F
ζ 1 = (1 + 0.38GF )
-fj(s)
-f2(s)
Vi
+
+ g1(s)
g2(s)
-fn(s)
+
+
+ g3(s)
gj(s)
0
-fn-1(s)
-fj-1(s)
-f1(s)
gj+1(s)
Vo
+ gn(s)
(13)
ζ 2 = (1 + 2.62GF )−0.5
−0.5
H dc =
(8)
s + ωp
ωn2 = ω p 1 + 2.62GF
ω n1 = ω p 1 + 0.38GF
2. CHAINED FEEDBACK TOPOLOGY Figure 2 shows the proposed n-stage CMA topology, where active feedback is used between stages. The overall structure is composed of n amplifier stages g1(s) ,…, gn(s) with active feedback gains f1(s) ,…, fn(s). The outputs of forward gain stages, gj(s), and feedback stages, -fj(s), are added together. For simplicity, assume that the amplifier blocks in Figure 2 have a single dominant pole and can be represented as in Equation (1) and feedback gains can be given as
(12)
2
(14)
(G )4 [1 + 3GF + (GF )2 ]
(15)
Since ωns and ζs are different for both sections, each 2nd-order transfer function has a peak at different frequencies. Matlab simulation of the transfer function of 4-stage CMA and two 2ndorder functions (@ F=1) for Gs of 2.3 and 6.1, respectively, are illustrated in Figure 3. The peak of one of the 2nd-order functions is placed where the other 2nd-order function is decreasing. For small Gs, the –3dB frequency of CMA is determined by the –3dB frequency of the first 2nd-order function. Increasing DC gain of forward stages (G) increases the ripple of the overall function and pushes the –3dB frequency to higher frequencies and extends the bandwidth. Increasing DC loop gain widens the bandwidth up to the point where the first 2nd-order function produces a peak of more than 1.5dB. AC Response G=6.1
Figure 2. Scheme of an n-stage CMA To explain how CMA uses the peaking technique to widen the bandwidth, let us consider n=2, so that there is only feedback from the second amplifier to the first one. This two-stage CMA has a 2nd-order transfer function given by
H dc ω n2 s 2 + 2ζω n s + ω n2
G=2.3
(9)
Overall transfer function First 2nd-order Second 2nd-order .-.-
where the natural frequency, damping factor, DC gain and bandwidth are given by
ζ = (1 + GF )
−0 . 5
ωn = ω p 1 + GF H dc =
G2 1 + GF
(10)
ωbw = ω n 1 − 2ζ 2 + 4ζ 4 − 4ζ 2 + 2 (11)
For the underdamped case (ζ1) ωbw can be improved up to 2.69ωp, while the peak gain is less than 1.5dB (for GF≤ 3.34). In fact, using feedback mostly decreases Hdc rather than increasing the bandwidth, i.e., GBPT decreases more as the bandwidth is widened.
2.1 Bandwidth of CMA
Frequency/ ωp
Figure 3. Matlab plot of magnitude of two 2nd-order transfer functions and the overall function of 4-stage CMA -f2(s) -f1(s) Vi g1(s)
+ g1(s)
,
+ g3(s)
g2(s)
-f2(s)
-f1(s) Vi
Transfer function of a 4-stage CMA is a 4th-order function. It can be presented as a product of two 2nd-order transfer functions as:
+
+
+ g2(s)
-f3(s)
0
0 Vo
+ g4(s) -f3(s)
+ g3(s)
-f4(s)
+ g4(s)
(a)
-f4(s)
0 Vo
(b)
Figure 4. The schemes of a) cascaded two 2-stage CMAs, b) 4-stage CMA
To clarify how much the feedback between stages improves the bandwidth of a 4-stage CMA, consider two 2-stage CMAs in cascade form (see Figure 4). It can be intuitively seen that cascade combination of two 2-stage CMAs has a bandwidth less than the single 2-stage CMA. Also Figure 5 shows the Matlab plots of the magnitude of transfer functions of a 4-stage CMA and two cascaded 2-stage CMAs for different Gs and F=1. Indeed, a 4-stage CMA has one extra feedback path from the output of the third stage to the second stage. For G>2.3 in two cascaded 2-stage CMAs, there is a peak (>1.5 dB). The maximum bandwidth in two cascaded 2-stage CMAs is 1.96ωp that is about 71% of the bandwidth of one 2-stage CMA. Not only did not the maximum achievable bandwidth of 4-stage CMA decrease, but also it can reach up to 2.9ωp without incurring a significant peak in transfer function. In this case the maximum bandwidth is 6.7 times of the bandwidth of a 4-stage conventional MA.
have high loop gains due to passive feedback, which limits the expansion of the bandwidth. The ratio of GBPT of CMA and conventional MA shows how much GBPT is decreased. Unfortunately, GBPT of CMA in comparison with that of the conventional MA is decreased (as n and DC loop gain are increased this ratio decreases further). However, GBPT of CMA in comparison with that of other structures (such as Figure 1) is much better. A simulation of the ratio of GBPT of an n-stage CMA and an n-stage conventional MA with passive feedback for n=2, 4, 6 and 8 for different DC gain loops is shown in Figure 7. As it shows, increasing DC loop gain increases CMA’s GBPT. As n is increased, this ratio also increases. Another parameter is the ratio of GBP1 of CMA and a similar conventional MA that shows how much the GBP1 is improved. This ratio is simulated in Figure 8. It shows that GBP1 of an n-stage CMA can be several times of GBP1 of an n-stage conventional MA and the structure shown in Figure 1.
AC Response Increasing G
Bandwidth
Bandwidth(×ωp)
ω-3dB=2.91ωp G=6.1 ω-3dB=1.96ωp G=2.3
Cascaded two 2-stage CMAs 4-stage CMA
-----
8-Stage 6-Stage 4-Stage 2-Stage
x o * +
Frequency/ωp
DC Loop Gain (G× F)
The transfer function of the CMA for n=6 and 8 can also be written as a product of 2nd-order transfer functions. Figure 6 shows the bandwidth of n-stage CMA extracted from the magnitude response simulation result for different DC loop gains (GF) for n=2, 4, 6, and 8. The n-stage CMA for odd ns has a real pole (@ ωp) which limits the expansion of bandwidth to some extent. Figure 6 shows that the CMA has two advantages. First, its bandwidth can be several times of ωp (the bandwidth of one stage); whereas for n-stage conventional MA, ωbw is always less than ωp. Second, CMA can have more bandwidth as n increases. As shown above, 4-stage CMA has more bandwidth than 2-stage CMA. On the contrary, 4-stage conventional MA has less bandwidth than 2-stage conventional MA. The maximum bandwidth that can be obtained for n=8 is 4.51ωp. As F decreases, a higher G is needed to have the same bandwidth. Although as n increases CMA can have more bandwidth, it also needs more GF. Figure 6 shows that if GF is constant, smaller n gives more bandwidth. To evaluate the performance of a wideband MA topology, several parameters can be calculated. One of them is the bandwidth of MA. It can be proven that for the same number of stages and the same GF, both structures of CMA and Figure 1 have almost the same bandwidth and comparable group delay variation. In contrast to CMA, the topology in Figure 1 cannot
,
Figure 6. Matlab plot of the bandwidth of 2, 4, 6 and 8stage CMA in ωp for different DC loop gains (GF)
⎛ ⎞ GBPT of CMA 20 log⎜⎜ GBP of Convention al MA with Passive FeedBack T ⎠ ⎝
dB
Figure 5. Matlab plot of magnitude of transfer functions of schemes in Figure 4 for different Gs and F=1.
8-Stage 6-Stage 4-Stage 2-Stage
x o * +
DC Loop Gain (G×F)
Figure 7. Matlab plot of the ratio of GBPT of n-stage CMA and conventional MA with passive feedback for n=2, 4, 6 and 8.
3. TOPOLOGY VALIDATION The proposed topology was validated through simulation an 8stage CMA in 0.35µm CMOS. A simple circuit was used as forward and feedback stages to be easily modeled as Equations (1) and (8). CMA was combined with a buffer to drive 50Ω in series with 1pF capacitor at 3V single power supply. The circuit of two stages of CMA is shown in Figure 9. To increase the gain of forward and feedback stages, Rl=1.8kΩ was chosen, where Ml was used for gain boosting. The Mf and Mg paired transistors (feedback and forward transistors, respectively) are matched, so the Miller effect of Cgd is partially canceled. Because of low-DCgain stage (
