模拟集成电路中的频率补偿(3)

function

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IEEE CIRCUITS AND SYSTEMS MAGAZINE

31

Ti1s2 is calculated by cutting the loop at the input of 2gmb1, and it would be approximately given byTi1s2<2

gmb1

11sgob1a

go2

Vgob1

,1

RbCb

CbCLCb

111sRbCpb12gob1

(3)Apparently, gob1/Cb is the dominant pole while 1/1RbCpb12 is the non-dominant pole. To achieve a PM $ 45°, the non-dominant pole must be higher than the loop’s GBW, which According to (6), the size of Cp is required to be

nearly as large as Cb, given that gm and gmb have the same order of magnitude, increasing the silicon area. Moreover, the closed-loop bandwidth is not improved in comparison with a simple source follower. Thus, this case will not be effective in the reduction of the size of CM’s physical capacitor.

In a second possibility, the loop is characterized with P3 as the non-dominant pole in Fig. 7(e) and the above phase-margin requirement changes to,means that the following inequality should hold,

1

R$gmbC3Cb$gmbRb. (4)

bCpb1bCpb1

This requirement is easily satisfied because gmbRb is

typically in the order of ten and Cb is much larger than Cpb1. In contrast, the conditions on compensating local feedback loops in former enhanced CMs, illustrated in Figs. 6(c) and (d), are quite stringent. In practice, the local feedback amplifier is commonly realized by one transistor or a simple differential-to-single-ended ampli-fier. Its frequency response can be well described by a single-pole system, i.e.,

AAv102v1s25

5gmro11sP11

s3roCp

where gm, ro and Cp are the local amplifier’s transconduc-tance, output resistance, and output capacitance, respec-tively. There are two alternatives to stabilize the local

loop with the assumption of unity-gain in the source fol-lower stage. First, as depicted in Fig. 7(d) the dominant pole of the local loop is P3 located at the output of the local amplifier. The non-dominant pole occurring at the source follower must be beyond the loop’s GBW to obtain

a PM of at least 45°, which is mathematically expressed as

gmbC$Agmgmv102#P353Cp$gb. (6)bCpmb

3

2 IEEE CIRCUITS AND SYSTEMS MAGAZINE

1r$A02#gmbv13Cb.1gmgmbro

2

2Cp. (7)oCpCb

Obviously, this case is also not suitable for capaci-tance multiplication with very small Cb. For instance, assuming Cp is 10 fF, and gmro and gmbro are both equal to ten, roughly the same as gmbRb, the resulting con-dition of Cb$1 pF would still be required, while (4) for the proposed CM will be sufficiently fulfilled in this case.

Therefore, the proposed CM is the most desirable for reducing the dimension of the physical capacitor and it is more powerful in terms of building an enhanced CM, as shown in Fig. 7(f).

In order to compare the high-frequency perfor-mance between the proposed CM and other types of CMs and choose a proper mirroring factor k, the small-signal equivalent model of the proposed CM is given in Fig. 8, with gob2 and Cpb2 denoting the para-sitic capacitance and output conductance at the drain of Mb2, respectively. Under the assumptions that CbWCgs1, Cpb1 and 11/gob12WRb.11/gmb12, the trans-fer function of the CM can be approximately given by,

Y5iosMCb

iv<iss2

11a10a0a1

where M5k1gmb1Rb212 is the multiplication factor,

a0 and a1 are gmb1/Cb and 1/1RbCpb12, respectively. The

effective bandwidth of a CM, BCM, is defined at the fre-quency where the phase magnitude drops by 45° (for a positive CM, its phase magnitude decreases from 90° to 45° while an inverting CM has 45° reduction from –90° to –135°). Hence, the BCM is determined by,

BCM arctan≥

a0

2

1B22¥545°. (9)1CMa0a1Solving (9), BCM is obtained as,

FIRST QUARTER 2011

B5"a2114a0a12a1

CM2

It can be observed that BCM is not affected by the type of poles, no matter they are two real poles or a com-plex pair (the damping factor must be no less than 1/2 according to (4)). Providing that the effective capaci-tance is a given constraint, the BCM can be extended by increasing the value of a0 (i.e. reducing the value of Cb while increasing Rb to fully utilize the characteristic of small parasitic Cpb). For example, if a1$2a0 is the re-quirement for a stable local loop, BCM is $ 0.732a0 (i.e. 0.732 (gmb1/Cb)).In terms of the power budget, the bias current can be accurately measured by the transconductance of all transistors [17]. The total transconductance of each CM is 2gmb1. The current-mirror CM’s bandwidth BCM21 is given by, B2gmb1CM215M11CbFIRST QUARTER 2011

From (11), it would be possible to demonstrate that

the frequency performance of the proposed CM is su-perior, when compared with the current-mirror CM

because M must be set to be greater than one to per-form capacitance amplification. As for other complex CMs, due to the existence of parasitic low-frequency poles, their bandwidth is even smaller than that of a current-mirror CM.To prove the forgoing assertions, different designs aiming to obtain 9-pF effective capacitance with 10-mA quiescent current dissipation are carried out. Figure 9(a) and (b) shows the frequency characterization of the proposed CMs with different values of Rb and Cb. As shown in Fig. 9(b), BCM of the proposed CM’s increases with a larger Rb that corresponds to a smaller Cb. How-ever, the magnitude peaking also grows fast as shown in Fig. 9(a); the upper boundary of BCM is limited by the stability imposed by the local resistive feedback.Figure 10(a) and (b) shows the magnitude and phase

responses of different CMs, respectively. Notice that the

phase responses of the basic current mirror and current-mirror CM are intentionally inverted from drop, beginning

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