# Two-Port Feedback Analysis On Miller-Compensated Amplifiers

Myungjun Kim Myungjun Kim is with Samsung Electronics, Hwaseong, Korea e-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, various Miller-compensated amplifiers are analyzed by using the
two-port feedback analysis together with the root-locus diagram. The proposed
analysis solves problems of Miller theorem/approximation that fail to predict
a pole-splitting and that require an impractical assumption that an initial
lower frequency pole before connecting a Miller capacitor in a two-stage
amplifier should be associated with the input of the amplifier. Since the
proposed analysis sheds light on how the closed-loop poles originate from the
open-loop poles in the $s$-plane, it allows the association of the closed-loop
poles with the circuit components and thus provides a design insight for
frequency compensation. The circuits analyzed are two-stage Miller-compensated
amplifiers with and without a current buffer and a three-stage nested Miller-
compensated amplifier.

###### Index Terms:

Operational amplifier, Miller theorem, stability, frequency compensation,
poles and zeros, two-port feedback analysis, root-locus diagram.

##  I Introduction

Miller theorem converts a feedback circuit into an open-loop circuit, allowing
the association of the poles with the circuit nodes, which thus provides a
design insight for frequency compensation. However, Miller theorem requires an
impractical condition that the voltage-gain transfer function
$a(s)=V_{2}/V_{1}$ in a given two-port network to be independent of $Y(s)$
which is an admittance connected across the ports [1]. For example, consider a
two-stage amplifier in Fig. 1(a) where $R_{1}$ and $C_{1}$ ($R_{2}$ and
$C_{2}$) model the impedance at the input (output) and $C_{c}$ is a
compensation capacitor. Since $Y(s)=sC_{c}$ and $a(s)=-K<0$ which is
independent of $Y(s)$, the Miller theorem can be applied to this circuit.
However, because a transistor is used to achieve the voltage-gain from the
input to output in practice, the circuit should be modeled by using a
controlled current source with a transconductance $g_{m}$ as shown in Fig.
1(b). Then, $a(s)=-g_{m}R_{2}(1-sC_{c}/g_{m})/[1+s(C_{2}+C_{c})R_{2}]$ is a
function of $Y(s)$; thus, Miller theorem cannot be applied to this circuit.

Alternatively, Miller approximation using a dc value of $a(s)$,
$\absolutevalue{a(0)}=g_{m}R_{2}$, is often used to achieve the open-loop
circuit as shown in Fig. 1(c). Assuming an initial lower frequency pole before
connecting $C_{c}$ is associated with the input of the amplifier (i.e.,
$\absolutevalue{p_{o1}}=1/R_{1}C_{1}<\absolutevalue{p_{o2}}=1/R_{2}C_{2}$),
the Miller approximation predicts that the input pole moves to a lower
frequency and is located such that $p_{c1}\simeq-1/g_{m}R_{1}R_{2}C_{c}$ which
is true. However, it also predicts that a pole associated with the output
becomes $p_{c2}$ with a lower frequency as well, which is in variance with a
pole-splitting [See Fig. 1(d)]. Moreover, if the initial low-frequency pole is
at the output (i.e.,
$\absolutevalue{p_{o1}}=1/R_{1}C_{1}>\absolutevalue{p_{o2}}=1/R_{2}C_{2}$),
which is a more general case because amplifiers are often required to drive a
large load-capacitance, the Miller approximation predicts that the input pole
moves to a lower frequency, which is completely false [See Fig. 1(e)]; the
pole-splitting still occurs, implying the input pole moves to a higher
frequency [1], which is very counter-intuitive.

Figure 1: Small-signal model of two-stage Miller-compensated amplifier using
(a) controlled voltage source or (b) controlled current source. (c) Circuit
obtained from applying Miller approximation to circuit in Fig. 1(b). Wrong
estimation of the poles of circuit in Fig. 1(c) when the initial lower
frequency pole before connecting $C_{c}$ is at (d) $V_{1}$ node or (e) $V_{2}$
node. Figure 2: (a) Two-stage Miller-compensated amplifier analyzed using two-
port feedback analysis. (b) $a$-circuit. (c) $\beta$-circuit.

On the other hand, the direct analysis (i.e., solving node equations to
achieve the desired transfer function) provides accurate the pole/zero
locations [2, 3]. However, it is so complex that the design insight for
frequency compensation cannot be readily obtained.

In this paper, we use a two-port feedback analysis [4, 5] together with the
root-locus diagram[6] to analyze two- and three-stage Miller-compensated
amplifiers. This method solves the problems of Miller theorem/approximation
mentioned above. Also, because the proposed analysis sheds light on how the
closed-loop poles of the amplifier originated from the open-loop poles which
can be readily found by inspection in the $s$-plane, it provides the design
insight for frequency compensation.

This paper is organized as follows. Section II analyzes a simple two-stage
Miller-compensated amplifier using the proposed method. Specifically, we will
see that a non-dominant pole location in a classical textbook [4] should be
modified to have a smaller magnitude when the right-half-plane (RHP) zero is
neglected in the amplifier transfer function. Section III analyzes a two-stage
Miller-compensated amplifier using a current buffer and shows that the
stability can be improved due to the additional third-pole from the current
buffer compared to the case without the current buffer. Section IV analyzes a
three-stage nested Miller-compensated (NMC) amplifier [7]. Section V concludes
the paper. Finally, appendices are included to present a pole-splitting
theorem and perform the direct analysis to investigate the exact locations of
the non-dominant poles in the NMC amplifier.

Figure 3: Positive root-locus diagram for $a(s)\beta(s)$ given by (5).
$p_{o1}$, $p_{o2}$ are the poles of $a(s)\beta(s)$ or equivalently the poles
of $A(s)$ when the feedback loop is opened. $p_{cd}$, $p_{cnd}$ are
respectively the possible dominant and non-dominant poles of $A(s)$ when the
feedback loop is closed.

##  II Two-Stage Miller-Compensated Amplifier

A two-stage Miller-compensated amplifier is shown in Fig. 2(a). This circuit
is a feedback transimpedance amplifier that has a shunt-shunt feedback
topology where the basic amplifier and the feedback network are shown in the
dashed boxes. Thus, it can be decomposed into $a$\- and $\beta$-circuit as
shown Fig. 2(b) and (c), respectively. Note that the $a$-circuit includes the
loading effect of the feedback network. Also, note that there are two open-
loop transimpedances with [$a^{\prime}(s)$] or without [$a(s)$] a non-
inverting current source $sC_{c}V_{i}$ at the output that models the
feedforward current through $C_{c}$.

Let us first investigate $a(s)$. By inspection of Fig. 2(b),

$\displaystyle a(s)$ $\displaystyle=\frac{V_{o}}{I_{i}}(s)$
$\displaystyle=-\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)g_{m}\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right)$
$\displaystyle=-\frac{g_{m}R_{1}R_{2}}{[1+sR_{1}(C_{1}+C_{c})][1+sR_{2}(C_{2}+C_{c})]}$
(1)

where $X\parallel Y=XY/(X+Y)$. Thus, $a(s)$ has two left-half-plane (LHP) real
poles given by

$\displaystyle p_{o1}$ $\displaystyle=-\frac{1}{R_{1}(C_{1}+C_{c})}$ (2)
$\displaystyle p_{o2}$ $\displaystyle=-\frac{1}{R_{2}(C_{2}+C_{c})}$ (3)

where $p_{o1}$ and $p_{o2}$ are the open-loop poles associated with the input
and output of the amplifier, respectively.

Next, $\beta(s)$ from Fig. 2(c) is given by

$\displaystyle\beta(s)=\frac{I_{x}}{V_{o}}(s)=-sC_{c}.$ (4)

Thus, $\beta(s)$ has a zero $z_{\beta}$ at the origin (i.e., $z_{\beta}=0$).

$\displaystyle A_{exact}(s)$
$\displaystyle=\frac{-(g_{m}-sC_{c})R_{1}R_{2}}{1+s[R_{1}C_{1}+R_{2}C_{2}+(g_{m}R_{1}R_{2}+R_{1}+R_{2})C_{c}]+s^{2}R_{1}R_{2}[C_{1}C_{2}+C_{c}(C_{1}+C_{2})]}$
(10a)
$\displaystyle=\frac{-\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)(g_{m}-sC_{c})\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right)}{1+\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)(g_{m}-sC_{c})\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right)sC_{c}}=\frac{a^{\prime}(s)}{1+a^{\prime}(s)\beta(s)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(10b)

$\displaystyle A(s)$
$\displaystyle=\frac{-\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)g_{m}\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right)}{1+\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)g_{m}\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right)sC_{c}}=\frac{a(s)}{1+a(s)\beta(s)}$
(11a)
$\displaystyle=\frac{-g_{m}R_{1}R_{2}}{1+s[R_{1}C_{1}+R_{2}C_{2}+(g_{m}R_{1}R_{2}+R_{1}+R_{2})C_{c}]+s^{2}R_{1}R_{2}[C_{1}C_{2}+C_{c}(C_{1}+C_{2})+\boldsymbol{C_{c}^{2}}]}$
(11b)

Combining (1) and (4), the $a(s)\beta(s)$ is given by

$\displaystyle
a(s)\beta(s)=\frac{g_{m}R_{1}R_{2}C_{c}s}{[1+sR_{1}(C_{1}+C_{c})][1+sR_{2}(C_{2}+C_{c})]}.$
(5)

Using (5), we can draw the positive root-locus diagram for increasing $g_{m}$
as shown in Fig. 3 to find the poles of the closed-loop transimpedance
$A(s)=V_{2}/I_{1}$ in Fig. 2(a). Note that $p_{o1}$, $p_{o2}$ are the poles of
$a(s)\beta(s)$ or equivalently the poles of $A(s)$ when the feedback loop via
$C_{c}$ is opened, and $p_{cd}$, $p_{cnd}$ are the possible dominant and non-
dominant poles of $A(s)$ when the feedback loop via $C_{c}$ is closed. And
$z_{\beta}$ is not a zero of $A(s)$ because it is from $\beta(s)$. Also note
that the root-locus diagram includes the two cases where the initial lower
frequency pole is either $p_{o1}$ or $p_{o2}$. As can be readily seen, the
root-locus diagram reveals the fact that $p_{o1}$ and $p_{o2}$ always split
apart for increasing $g_{m}$ irrespective of the condition about the initial
low-frequency pole locations, which the Miller approximation fails to predict.

Let us now find the exact locations of the closed-loop pole $p_{cd}$,
$p_{cnd}$. Since $a(s)\beta(s)$ has two LHP real and distinct poles, one zero
at the origin and a large midband value111The closed-loop poles are found from
the midband value of $\absolutevalue{a(j\omega)\beta(j\omega)}$ if
$a(s)\beta(s)$ has zero(s) at the origin [8]. $a_{0}\beta_{0}$ assuming
$g_{m}R_{1}$, $g_{m}R_{2}$ and $C_{c}$ are large, we can apply the pole-
splitting theorem presented in Appendix A. Depending on whether the initial
lower frequency pole of $a(s)\beta(s)$ is $p_{o1}$ or $p_{o2}$,
$a_{0}\beta_{0}$ is expressed differently as

$\displaystyle a_{0}\beta_{0}=$
$\displaystyle\frac{g_{m}R_{2}C_{c}}{C_{1}+C_{c}}$ if
$\absolutevalue{p_{o1}}<\absolutevalue{p_{o2}}$ (6a) $\displaystyle
a_{0}\beta_{0}=$ $\displaystyle\frac{g_{m}R_{1}C_{c}}{C_{2}+C_{c}}$ if
$\absolutevalue{p_{o1}}>\absolutevalue{p_{o2}}$. (6b)

Applying the pole-splitting relation (39) yields

$\displaystyle p_{cd}$ $\displaystyle\simeq-\frac{1}{g_{m}R_{1}R_{2}C_{c}}$
(7) $\displaystyle p_{cnd}$
$\displaystyle\simeq-\frac{g_{m}C_{c}}{(C_{1}+C_{c})(C_{2}+C_{c})}.$ (8)

Figure 4: Negative root-locus diagram for $a^{\prime}(s)\beta(s)$.

It should be mentioned that while the dominant pole $p_{cd}$ given by (7) is
the same as in [4], the non-dominant pole $p_{cnd}$ given by (8) is in
variance because

$p_{cnd}\,\,\textrm{in~{}\cite[cite]{[\@@bibref{}{Sedra}{}{}]}}\simeq-\frac{g_{m}C_{c}}{(C_{1}+C_{c})(C_{2}+C_{c})-C_{c}^{2}},$
(9)

implying (8) is located at a lower frequency than (9). This variance is due to
the assumption that the feedforward current through $C_{c}$ is neglected. This
is verified as follow. The exact transimpedance $A_{exact}(s)=V_{2}/I_{1}$ of
the two-stage Miller-compensated amplifier in Fig. 2(a) is given by (10a).
With some algebra, (10a) can be written in the form of the feedback equation
as (10b). Comparing (10b) with (11a) obtained from the two-port feedback
analysis shows that the only difference is that ($g_{m}-sC_{c}$) has been
replaced by $g_{m}$. Thus, such a deviation vanishes if we include the effect
of $sC_{c}V_{i}$ current source in Fig. 2(b), yielding

$\displaystyle a^{\prime}(s)=\frac{V_{o}^{\prime}}{I_{i}}(s)$
$\displaystyle=-\left(R_{1}\parallel\frac{1}{s(C_{1}+C_{c})}\right)(g_{m}-sC_{c})\left(R_{2}\parallel\frac{1}{s(C_{2}+C_{c})}\right).$
(12)

Therefore, $A_{exact}(s)=a^{\prime}(s)/[1+a^{\prime}(s)\beta(s)]$ as shown in
(10b), validating the two-port feedback analysis.

Figure 5: Pole-zero diagram of (a) transimpedance $V_{2}/I_{1}$ and (b) input-
impedance $V_{1}/I_{1}$ when $\absolutevalue{p_{o1}}>\absolutevalue{p_{o2}}$.
Figure 6: (a) Two-stage Miller-compensated amplifier using current buffer
analyzed using two-port feedback analysis. (b) $a$-circuit. (c)
$\beta$-circuit. Figure 7: Positive root-locus diagram for
$a_{CB}(s)\beta_{CB}(s)$ given by (20). $p_{o1}$, $p_{o2}$ and $p_{o3}$ are
the poles of $a_{CB}(s)\beta_{CB}(s)$ or equivalently the poles of $A_{CB}(s)$
when the feedback loop is opened. Also, $p_{cd}$, $p_{cnd1}$ and $p_{cnd2}$
are respectively the possible dominant, first and second non-dominant poles of
$A_{CB}(s)$ when the feedback loop is closed.

Indeed, the non-dominant pole given by (8) is more accurate than (9) if the
RHP zero ($z_{rhp}=g_{m}/C_{c}$) of the two-stage Miller-compensated amplifier
is assumed to be at high frequency so as to be neglected. $z_{rhp}$ can be
ignored by assuming a large $g_{m}$ so that $(g_{m}-sC_{c})\simeq g_{m}$ at
the frequencies of interest. However, because the term $(g_{m}-sC_{c})$ also
exists in the denominator of (10b), such condition also alter the non-dominant
pole location by modifying the coefficient of $s^{2}R_{1}R_{2}$ to include an
additional term of $\boldsymbol{C_{c}^{2}}$ as shown in the denominator of
(11b), which results in (8) to have a smaller magnitude than (9).

Such increase in the magnitude of the non-dominant pole can also be explained
using the root-locus diagram for $a^{\prime}(s)\beta(s)$ as shown in Fig. 4.
Note that the negative locus rule [6] is applied here because of the low-
frequency sign reversal associated with the RHP zero [8]. Similar to Fig. 3,
$p_{cnd}$ moves toward high frequency in the LHP as $g_{m}$ increases.
However, it eventually moves into the RHP, manifesting itself at
$s=z_{rhp}^{+}$ if $g_{m}\rightarrow\infty$; thus, for a given finite $g_{m}$,
$p_{cnd}$ is located at nearer to $s=-\infty$ when the RHP zero is included as
illustrated in Fig. 4.

So far, we have investigated the pole/zero locations of the transimpedance
$V_{2}/I_{1}$ of the circuit in Fig. 2(a). It is worth mentioning of the
pole/zero locations of the input-impedance $V_{1}/I_{1}$. Since the natural
structure of the circuit is unchanged, $p_{cd}$, $p_{cnd}$ are also unchanged.
However, the voltage-amplifier $V_{o}/V_{i}$ or $V^{\prime}_{o}/V_{i}$ in Fig.
2(b) that has a pole $p_{o2}$ is now included not in the basic amplifier but
in the feedback network, and thus $p_{o2}$ becomes a zero in the input-
impedance. For example, the pole-zero diagram of the transimpedance
$V_{2}/I_{1}$, input-impedance $V_{1}/I_{1}$ when
$\absolutevalue{p_{o1}}>\absolutevalue{p_{o2}}$ are shown in Fig. 5(a), (b),
respectively. The pole/zero pattern of the input-impedance can be employed as
a lag compensating network to control a damping factor of the complex poles in
a three-stage amplifier in [9].

##  III Two-Stage Miller-Compensated Amplifier Using Current Buffer

The RHP zero in the Miller-compensated two-stage amplifier in the previous
section was ignored by assuming a large $g_{m}$, which requires a large power
consumption in practice. Instead, the RHP zero can be removed by employing a
unilateral current buffer [10, 11] in the feedback network, allowing only a
current through $C_{c}$ to flow backward from the output to the input of the
transconductor ($g_{m}$) as shown in Fig. 6(a). Note that the feedback network
now has a current buffer with an input impedance of $1/g_{mc}$ and the
transconductance $g_{mc}$. Since this circuit also has a shunt-shunt feedback
topology, it can be decomposed into $a$\- and $\beta$-circuit as shown Fig.
6(b) and (c), respectively. Note that unlike the amplifier without the current
buffer, the feedback loading only occurs at the output of the amplifier in the
$a$-circuit.

Assuming $g_{mc}R_{2}\gg 1$, the open-loop transimpedance $a_{CB}(s)$ is

$\displaystyle a_{CB}(s)=\frac{V_{o}}{I_{i}}(s)$
$\displaystyle=-\frac{g_{m}R_{1}R_{2}(1+sC_{c}/g_{mc})}{[1+sR_{1}C_{1}][1+sR_{2}(C_{2}+C_{c})]\left[1+s\frac{C_{2}\parallel
C_{c}}{g_{mc}}\right]}.$ (13)

Thus, $a_{CB}(s)$ has three LHP real poles as

$\displaystyle p_{o1}$ $\displaystyle=-\frac{1}{R_{1}C_{1}}$ (14)
$\displaystyle p_{o2}$ $\displaystyle=-\frac{1}{R_{2}(C_{2}+C_{c})}$ (15)
$\displaystyle p_{o3}$ $\displaystyle=-\frac{g_{mc}}{C_{2}\parallel C_{c}}$
(16)

and one LHP zero as

$\displaystyle z_{a}$ $\displaystyle=-\frac{g_{mc}}{C_{c}}$ (17)

From Fig. 6(c), $\beta_{CB}(s)$ is

$\displaystyle\beta_{CB}(s)=\frac{I_{x}}{V_{o}}(s)=-\frac{sC_{c}}{1+sC_{c}/g_{mc}}.$
(18)

Thus, $\beta_{CB}(s)$ has one LHP real pole as

$\displaystyle p_{\beta}=-\frac{g_{mc}}{C_{c}}$ (19)

and a zero $z_{\beta}=0$.

$\displaystyle
A_{CB}(s)=\frac{a_{CB}(s)}{1+a_{CB}(s)\beta_{CB}(s)}\simeq\frac{-g_{m}R_{1}R_{2}\left(1+s\dfrac{C_{c}}{g_{mc}}\right)}{(1+sg_{m}R_{1}R_{2}C_{c})\left(1+s\dfrac{C_{c}C_{2}+g_{mc}R_{1}C_{1}(C_{c}+C_{2})}{g_{m}R_{1}g_{mc}C_{c}}+s^{2}\dfrac{C_{1}C_{2}}{g_{m}g_{mc}}\right)}$
(24) Figure 8: Pole-zero diagram of the closed-loop transimpedance
$V_{2}/I_{1}$ of the two-stage amplifier (Not to scale): (a) without current
buffer, (b) with ideal current buffer that has infinite $g_{mc}$, (c) with
non-ideal current buffer that has large $g_{mc}$, (d) with non-ideal current
buffer that has small $g_{mc}$, (e) with non-ideal current buffer that has
optimum $g_{mc}$.

Since (19) coincides with (17), the $a_{CB}(s)\beta_{CB}(s)$ has the three LHP
poles, a zero $z_{\beta}$ at the origin, and is expressed as

$\displaystyle a_{CB}(s)\beta_{CB}(s)$
$\displaystyle=\frac{g_{m}R_{1}R_{2}C_{c}s}{[1+sR_{1}C_{1}][1+sR_{2}(C_{2}+C_{c})]\left[1+s\frac{C_{2}\parallel
C_{c}}{g_{mc}}\right]}.$ (20)

The positive root-locus diagram of $a_{CB}(s)\beta_{CB}(s)$ for increasing
$g_{m}$ is shown in Fig. 7. The resulting loci indicate the locations of the
poles of the closed-loop transimpedance $A_{CB}(s)=V_{2}/I_{1}$ in Fig. 6(a).
As $g_{m}$ increases, $p_{o1}$ and $p_{o2}$ split apart and become the
dominant pole $p_{cd}$ and the first non-dominant pole $p_{cnd1}$,
respectively. Also, $p_{o3}$ moves toward lower frequency and becomes the
second non-dominant pole $p_{cnd2}$ for increasing $g_{m}$, implying that
$p_{cnd1}$ and $p_{cnd2}$ can form a complex pole pair beyond some value of
$g_{m}$. Thus, we should analyze how effective this amplifier is compared to
the previous amplifier in Section II and also investigate the effect of the
third pole $p_{o3}$ on the stability of the amplifier.

Figure 9: (a) Three-stage NMC amplifier analyzed using two-port feedback
analysis. (b) $a$-circuit. (c) $\beta$-circuit.

First, let us compare the Miller compensated amplifier using the current
buffer with the previous amplifier by assuming that $p_{o3}\rightarrow-\infty$
so as to be neglected. It is achieved by assuming the ideal current buffer
with zero input impednace (i.e., $g_{mc}\rightarrow\infty$). Then, (20)
reduces to have the two LHP poles, a zero at the origin and the midband value
$a_{0}\beta_{0}|_{CB}$ as

$\displaystyle a_{0}\beta_{0}|_{CB}=$
$\displaystyle\frac{g_{m}R_{2}C_{c}}{C_{1}}$ if
$\absolutevalue{p_{o1}}<\absolutevalue{p_{o2}}$ (21a) $\displaystyle
a_{0}\beta_{0}|_{CB}=$ $\displaystyle\frac{g_{m}R_{1}C_{c}}{C_{2}+C_{c}}$ if
$\absolutevalue{p_{o1}}>\absolutevalue{p_{o2}}$. (21b)

Since the midband value is typically very large, we apply the pole-splitting
theorem in Appendix A to find the splitted closed-loop poles as

$\displaystyle p_{cd(p_{o3}\rightarrow-\infty)}$
$\displaystyle\simeq-\frac{1}{g_{m}R_{1}R_{2}C_{c}}$ (22) $\displaystyle
p_{cnd1(p_{o3}\rightarrow-\infty)}$
$\displaystyle\simeq-\frac{g_{m}}{C_{2}+C_{c}}\frac{C_{c}}{C_{1}}.$ (23)

While the dominant pole (22) is the same with (7), the non-dominant pole given
by (23) is located at higher frequency than (8) by a multiplication factor of
$1+C_{c}/C_{1}$ which is typically large because $C_{1}$ is a small parasitic
capacitance. Thus, the amplifier with the current buffer has the better
relative stability than the amplifier without the current buffer if their
gain-bandwidth products (GBWs) are the same.222GBWs of the voltage-gain
$V_{2}/V_{s}$ of the amplifier in Fig. 2(a) and Fig. 6(a) are expressed as
$g_{m0}/C_{c}$ by replacing the input current source $I_{1}$ with
$g_{m0}V_{s}$ where $g_{m0}$ is an additional input transconductance applied
to the input voltage $V_{s}$. The pole-zero diagrams of the transimpedance
$V_{2}/I_{1}$ of the two-stage amplifier without the current buffer and the
two-stage amplifier with the ideal current buffer is shown in Fig. 8(a), (b),
respectively.

$\displaystyle
a_{NMC}(s)\beta_{NMC}(s)=\frac{g_{m1}g_{m2}R_{0}R_{1}R_{2}C_{c0}s}{[1+sR_{0}(C_{0}+C_{c0})][1+sg_{m2}R_{1}R_{2}C_{c1}]\left[1+s\dfrac{(C_{1}+C_{c1})(C_{2}+C_{c0}+C_{c1})}{g_{m2}C_{c1}}\right]}$
(27)

Such increase of magnitude of the non-dominant pole is explained as follows.
When $\absolutevalue{p_{o1}}<\absolutevalue{p_{o2}}$, (21a) is larger than
(6a) by the multiplication factor of $1+C_{c}/C_{1}$; thus, such increased
midband value help move $p_{o2}$ given by (15) to become $p_{cnd1}$ that is
located at higher frequency than (8). When
$\absolutevalue{p_{o1}}>\absolutevalue{p_{o2}}$, though the midband value
given by (21b) is unchanged compared to (6b), the initial higher frequency
pole $p_{o1}$ given by (14) is located at more higher frequency than (2) by
the multiplication factor of $1+C_{c}/C_{1}$ because the loading effect does
not occur at the input of the amplifier; thus, $p_{cnd1}$ is located at higher
frequency than (8) as well.

Next, we consider a non-ideal current buffer that has a finite $g_{mc}$ so
that $p_{o3}$ and thus $p_{cnd2}$ are not neglected. Assuming $g_{m},g_{mc}\gg
1/R_{1},1/R_{2}$; $C_{c},C_{2}\gg C_{1}$; $g_{m}R_{1}\gg C_{2}/C_{c}$,
$A_{CB}(s)$ is given by (24) at the bottom of the page. Note that the dominant
pole $p_{cd}$ is the same as (22); this is because $p_{o1}$ is typically
located at much lower frequency than $p_{o3}$ and thus
$\absolutevalue{a_{CB}(j\omega)\beta_{CB}(j\omega)}$ at
$\omega=\absolutevalue{p_{o1}}$ is little affected by $p_{o3}$. Therefore,
$p_{o1}$ moves toward the origin for the same amount and becomes $p_{cd}$ as
before when feedback-loop is closed for a given $g_{m}$.

The other poles $p_{cnd1}$, $p_{cnd2}$ can be found by factoring the second-
order polynomial in the denominator of $A_{CB}(s)$. Though simple expressions
for $p_{cnd1}$, $p_{cnd2}$ cannot be readily generated, they have the
following simple relation:

$\displaystyle p_{cnd1}p_{cnd2}=\frac{g_{m}g_{mc}}{C_{1}C_{2}}.$ (25)

Since the right-hand side of (25) can be obtained by multiplying (16) with
(23), we write

$\displaystyle
p_{cnd1}=\frac{p_{o3}}{p_{cnd2}}p_{cnd1(p_{o3}\rightarrow-\infty)}.$ (26)

Thus, $p_{cnd1}$ is modified with a multiplication factor of $p_{o3}/p_{cnd2}$
compared to $p_{cnd1(p_{o3}\rightarrow-\infty)}$. Let us consider the
following two cases where $g_{mc}$ and thus $p_{o3}$ (16) are different but
the midband values of $a_{CB}(s)\beta_{CB}(s)$ are preserved since (21a) and
(21b) are not functions of $g_{mc}$: 1) $g_{mc}$ is large and thus $p_{o3}$ is
located at high frequency so that $p_{cnd1}$ and $p_{cnd2}$ are real and 2)
$g_{mc}$ is small and thus $p_{o3}$ is located at low frequency so that
$p_{cnd1}$ and $p_{cnd2}$ are complex.

1) When $g_{mc}$ is large so that $p_{cnd1}$ and $p_{cnd2}$ are real,
$p_{o3}/p_{cnd2}$ is obviously larger than ”1” as shown in the root-locus
diagram in Fig. 7; thus, $p_{cnd1}$ is located at higher frequency than
$p_{cnd1(p_{o3}\rightarrow-\infty)}$ as if attracted by $p_{cnd2}$ as shown in
the pole-zero diagram of $V_{2}/I_{1}$ in Fig. 8(c). Note that because $z_{a}$
(17) is the zero of $a_{CB}(s)$ and $p_{\beta}$ (19) is not the pole of
$a_{CB}(s)\beta_{CB}(s)$, a zero $z_{a}$ exists in $A_{CB}(s)$. Though
$p_{cnd2}$ is at the lower frequency than in the previous ideal current buffer
case, the stability is not degraded if $z_{a}$ and $p_{cnd2}$ are closely
spaced so as to be canceled as shown in Fig. 8(c). Thus, this case is better
than the Miller compensation with the ideal current buffer since it achieves
both the improved stability margin and the lower power consumption with the
finite $g_{mc}$.

2) When $g_{mc}$ is small so that $p_{cnd1}$ and $p_{cnd2}$ are complex, the
magnitude of the complex poles is still larger than that of
$p_{cnd1(p_{o3}\rightarrow-\infty)}$ [13]. Thus, power consumption can be
further reduced by decreasing $g_{mc}$ compared to the previous real non-
dominant poles case without deteriorating the stability. However, since a real
part of the complex pole-pair is fixed at $s\simeq p_{o3}/2$, too small
$g_{mc}$ can result in a small damping ratio of the pole-pair, degrading
stability margins shown in Fig. 8(d). Therefore, for a given $g_{m}$, an
optimum design that achieves both good stability and low power consumption can
be achieved by locating $p_{o3}$ with the optimum $g_{mc}$, allowing
$p_{cnd1}$ and $p_{cnd2}$ to be located at near $z_{a}$ so that they can be
considered a LHP real pole as shown in Fig. 8(e) (i.e., $p_{c2}\simeq
p_{cM}\simeq z_{a}$).

$A_{NMC}(s)=\frac{V_{2}}{I_{0}}(s)=\frac{a_{NMC}(s)}{1+a_{NMC}(s)\beta_{NMC}(s)}\simeq\frac{-g_{m1}g_{m2}R_{0}R_{1}R_{2}}{(1+sg_{m1}g_{m2}R_{0}R_{1}R_{2}C_{c0})\left[1+s\dfrac{C_{c1}}{g_{m1}}+s^{2}\dfrac{C_{c1}C_{2}}{g_{m1}g_{m2}}\right]}$
(35)

##  IV Three-Stage NMC Amplifier

In this section, we analyze three-stage NMC amplifiers [7] by using two-port
feedback analysis.

A three-stage NMC amplifier is shown in Fig. 9(a). Note that two Miller
capacitors $C_{c0}$, $C_{c1}$ are employed and the input current source
$I_{0}$ is expressed as $g_{m0}V_{s}$ where $V_{s}$ is the input voltage of
the NMC amplifier. Also, note that a two-stage Miller-compensated amplifier in
Section II is included in this NMC amplifier. This circuit is also a feedback
transimpedance amplifier that has a shunt-shunt feedback topology where the
basic amplifier and the feedback network are shown in the dashed boxes in Fig.
9(a). Thus, it can be decomposed into $a$\- and $\beta$-circuit as shown Fig.
9(b) and (c), respectively. It should be noted that the $a$-circuit is drawn
by neglecting the signal feedforward through $C_{c0}$ for simplicity. Also,
the $a$-circuit includes the two-stage amplifier in Fig. 2(a) where the output
impedance includes the loading effect by $C_{c0}$.

Using the same approches and the results in Section II,
$a_{NMC}(s)\beta_{NMC}(s)$ which is given by (27) shown at the bottom of the
page has three LHP poles given by

$\displaystyle p_{o0}$
$\displaystyle=-\frac{1}{R_{0}(C_{0}+C_{c0})}\simeq-\frac{1}{R_{0}C_{c0}}$
(28) $\displaystyle p_{o1}$ $\displaystyle=-\frac{1}{g_{m2}R_{1}R_{2}C_{c1}}$
(29) $\displaystyle p_{o2}$
$\displaystyle=-\frac{g_{m2}C_{c1}}{(C_{1}+C_{c1})(C_{2}+C_{c0}+C_{c1})}\simeq-\frac{g_{m2}}{C_{2}}$
(30)

where the approximations in (28) and (30) follow if $C_{2}\gg C_{c0},C_{c1}\gg
C_{0},C_{1}$ since typically $C_{2}$ is a large load capacitance and
$C_{0},C_{1}$ are small parasitics. Also, $a_{NMC}(s)\beta_{NMC}(s)$ has a
$z_{\beta}$ at the origin from the $\beta_{NMC}(s)=I_{f0}/V_{o}$ in Fig. 9(c).
Note that $p_{o1}$ and $p_{o2}$ are expressed by neglecting the signal
feedforward through $C_{c1}$.

Typically,
$\absolutevalue{p_{o1}}<\absolutevalue{p_{o0}}<\absolutevalue{p_{o2}}$ because
of a large $g_{m2}$ in the design of the NMC amplifier [2]. Thus, the positive
root-locus diagram for increasing $g_{m1}$ can be drawn as shown in Fig. 10.
As can be seen, the two low frequency poles $p_{o1}$ and $p_{o0}$ split apart
as $g_{m1}$ increases and become a dominant pole $p_{cd}$ and a first non-
dominant pole $p_{cnd1}$, respectively. It should also be noted that $p_{o2}$
moves toward low frequency and becomes a second non-dominant pole $p_{cnd2}$
as $g_{m1}$ increases. Also, note that $p_{cnd1}$ and $p_{cnd2}$ can form a
complex-conjugate pole-pair beyond some value of $g_{m1}$.

Figure 10: Positive root-locus diagram for $a_{NMC}(s)\beta_{NMC}(s)$ given by
(27). $p_{o0}$, $p_{o1}$ and $p_{o2}$ are the poles of
$a_{NMC}(s)\beta_{NMC}(s)$ or equivalently the poles of
$A_{NMC}(s)=V_{2}/I_{0}$ when the feedback loop via $C_{c0}$ is opened. Also,
$p_{cd}$, $p_{cnd1}$ and $p_{cnd2}$ are respectively the possible dominant,
first and second non-dominant poles of $A_{NMC}(s)$ when the feedback loop via
$C_{c0}$ is closed.

Using a similar approach in the Section III, let us first assume $p_{o2}$ is
located at very high frequency so as to be neglected and investigate a
dominant ($p_{cd}$) and a first non-dominant ($p_{cnd1}$) pole locations; in
this case, (27) can be approximated to have the two LHP poles, a zero at the
origin, and a midband value $a_{0}\beta_{0}|_{NMC}$ as given by

$a_{0}\beta_{0}|_{NMC}=\frac{g_{m1}R_{0}C_{c0}}{C_{c1}}$ (31)

which is typically large. Thus, the splitted closed-loop poles can be found by
applying the pole-splitting relation (39), which results in

$\displaystyle p_{cd(p_{o2}\,\textrm{at}\,HF)}$
$\displaystyle\simeq-\frac{1}{g_{m1}g_{m2}R_{0}R_{1}R_{2}C_{c0}}$ (32)
$\displaystyle p_{cnd1(p_{o2}\,\textrm{at}\,HF)}$
$\displaystyle\simeq-\frac{g_{m1}}{C_{c1}}$ (33)

Thus, GBW of this NMC amplifier is $g_{m0}/C_{c0}$. Typically, a magnitude of
the first non-dominant pole given by (33) is set to be

$\displaystyle\frac{g_{m1}}{C_{c1}}=2\textrm{GBW}=2\frac{g_{m0}}{C_{c0}}$ (34)

so that the phase margin (PM) $\simeq$ 90°$-tan^{-1}(1/2)\simeq 63$° [2]; the
pole-zero diagram of the voltage-gain of this NMC amplifier is illustrated in
Fig. 11(a). It is worth mentioning that the proposed analysis shows that
$p_{o2}$ should be located at a high frequency such that
$\absolutevalue{p_{o2}}=\absolutevalue{p_{cnd2}}\gg$ GBW to satisfy (34),
implying a seperate pole approach in [2] that satisfies not only (34) but
$\absolutevalue{p_{cnd2}}$ = 2 GBW cannot be achieved.

Figure 11: Pole-zero diagram of the voltage-gain $V_{2}/V_{s}$ of the three-
stage NMC amplifier (Not to scale): (a) when $p_{o2}$ is located at a very
high frequency so as to be neglected, (b) when $p_{o2}$ is located at
$s=-8$GBW, and (c) when $p_{o2}$ is located at $s=-4$GBW.

Next, we consider the case when $p_{o2}$ is located at a lower frequency and
investigate its effect on the stability. Though this can be achieved by
decreasing $g_{m2}$, it will also affect $p_{o1}$ given by (29). Instead,
increasing $C_{2}$ only allows $p_{o2}$ to be located at a lower frequency.
Then, as mentioned eariler, $p_{o2}$ moves toward low frequency and becomes
the second-nondominant pole $p_{cnd2}$. Similar to the case in the Miller
compensated two-stage using the non-ideal current buffer in Section III, the
first non-dominant pole $p_{cnd1}$ here is at a higher frequency than (33) as
if attracted by $p_{cnd2}$. This is verified as follows.

Assuming $g_{m1},g_{m2}\gg 1/R_{0},1/R_{1},1/R_{2}$; $C_{2}\gg
C_{c0},C_{c1}\gg C_{0},C_{1}$; $A_{NMC}(s)=V_{2}/I_{0}$ in Fig. 9(a) is
approximately expressed as (35) shown at the bottom of the page. From a
quadratic function in the denominator of (35), $p_{cnd1}$ and $p_{cnd2}$ have
the following relation:

$\displaystyle p_{cnd1}p_{cnd2}\simeq\frac{g_{m1}g_{m2}}{C_{c1}C_{2}}.$ (36)

Because the right-hand side of (36) can also be obtained by multiplying (30)
with (33), we can write

$\displaystyle
p_{cnd1}\simeq\frac{p_{o2}}{p_{cnd2}}p_{cnd1(p_{o2}\,\textrm{at}\,HF)}.$ (37)

Thus, $p_{cnd1}$ is located at higher frequency than
$p_{cnd1(p_{o2}\,\textrm{at}\,HF)}$ by a multiplication factor of
$p_{o2}/p_{cnd2}$ that is obviously larger than unity as shown in the root-
locus diagram in Fig. 10. This implies that even if $p_{o2}$ is not located at
very high frequency, the stability margin may be little affected by $p_{cnd2}$
because the magnitude of $p_{cnd1}$ is also increased.

For example, we consider two cases that have different values of $C_{2}$ so
that 1) $p_{o2}=-8$GBW, and 2) $p_{o2}=-4$GBW and the condition (34) holds in
the both cases. The exact analysis is carried out for the both cases in
Appendix B.

1) When $p_{o2}=-8$GBW, the NMC amplifier has the pole-zero diagram as
illustrated in Fig. 11(b). Note that the dominant pole $p_{cd}$ is at the same
location as in the previous case because $p_{o1}$ is typically located at much
lower frequency than $p_{o2}$ and thus
$\absolutevalue{a_{NMC}(j\omega)\beta_{NMC}(j\omega)}$ at
$\omega=\absolutevalue{p_{o2}}$ is little affected by $p_{o2}$. However,
$p_{o0}$ and $p_{o2}$ attract each other, becoming the first ($p_{cnd1}$) and
second non-dominant ($p_{cnd2}$) pole, and form a double pole-pair at
$s=-4$GBW, which results in PM $\simeq$ 90°$-2tan^{-1}(1/4)\simeq 62$°. Thus,
the stability of the NMC amplifier is little affected compared to the previous
case.

2) When $p_{o2}=-4$GBW, the NMC amplifier has the pole-zero diagram as
illustrated in Fig. 11(c). The location of $p_{cd}$ is still the same as the
previous cases. However, $p_{cnd1}$ and $p_{cnd2}$ form a complex-conjugate
pole-pair that has the same magnitude of the real- and imaginary-part as 2GBW
(i.e., $\absolutevalue{p_{cnd1}}=\absolutevalue{p_{cnd2}}=2\sqrt{2}$GBW) with
a damping ratio ($\xi$) of $1/\sqrt{2}\simeq 0.707$, which results in

$\displaystyle PM\simeq
90\textrm{\textdegree{}}-tan^{-1}\left[\frac{2\xi\frac{GBW}{2\sqrt{2}GBW}}{1-(\frac{GBW}{2\sqrt{2}GBW})^{2}}\right]\simeq
60\textrm{\textdegree{}}.$ (38)

Thus, the stability of the NMC amplifier is also little affected compared to
the previous cases.

##  V Conclusion

In this paper, various Miller-compensated amplifiers are analyzed by using the
two-port feedback analysis together with the root-locus diagram. This method
solves the problems of Miller theorem/approximation that fail to predict the
pole-splitting and that requires an an impractical assumption where an initial
low-frequency pole before connecting a Miller capacitor in a multi-stage
amplifier should be associated with the input of the amplifier. Also, because
the proposed analysis sheds light on how the closed-loop poles of the
amplifier originated from the open-loop poles which can be readily found by
inspection in the $s$-plane, it provides the design insight for frequency
compensation, helping allow the association of the closed-loop poles with the
circuit parameters.

Specifically, in Section II, the proposed analysis shows that when the RHP
zero is neglected the nondominant pole location of the two-stage Miller-
compensated amplifier should be modified to have a smaller magnitude than in a
classical textbook[4]. Also, in Section III, when the current buffer is used
in a Miller compensated two-stage amplifier to block the feedforward path
through the compensation capacitor, the stability can be improved because of
the absence of the RHP zero and the loading effect of compensation capacitor
at the input of the amplifier. Moreover, the additional third pole from the
current buffer and the zero of the $a$-circuit can be optimally located to
improve the stability and to lower the power consumption. Finally, in Section
IV, the stabiltiy of the three-stage NMC amplifier can be little affected even
if the second non-dominant pole is considered because the first non-dominant
pole is located at a higher frequency as if attracted by the second non-
dominant pole.

##  Appendix A Pole-splitting Theorem

Let $a(s)\beta(s)$ of a feedback circuit has two distinct real poles, $p_{od}$
and $p_{ond}$, in the left-half of the $s$-plane such that
$\absolutevalue{p_{od}}<\absolutevalue{p_{ond}}$, a zero at the origin, and a
midband value $a_{0}\beta_{0}\gg 2$. Then, the following Pole-splitting
relation holds:

$\boxed{a_{0}\beta_{0}\simeq\frac{p_{od}}{p_{cd}}\simeq\frac{p_{cnd}}{p_{ond}}}$
(39)

where $p_{cd}$ and $p_{cnd}$ are the closed-loop poles of the circuit.

$\blacksquare$

Proof: For a given condition, $a(s)\beta(s)$ can be written

$\displaystyle
a(s)\beta(s)=\frac{a_{0}\beta_{0}\frac{s}{\absolutevalue{p_{od}}}}{\left(1+\frac{s}{\absolutevalue{p_{od}}}\right)\left(1+\frac{s}{\absolutevalue{p_{ond}}}\right)}.$
(40)

The closed-loop poles of the circuit [or equivalently the zeros of the
characteristic equation $1+a(s)\beta(s)$] can be found as the zeros of

$\displaystyle
P(s)=1+s\frac{a_{0}\beta_{0}}{\absolutevalue{p_{od}}}+s^{2}\frac{1}{\absolutevalue{p_{od}p_{ond}}}.$
(41)

The two zeros of $P(s)$, $p_{cd}$ and $p_{cnd}$, are real and widely spaced
(i.e., $\absolutevalue{p_{cnd}}\gg\absolutevalue{p_{cd}}$). This can be
verified by comparing (41) with the standard form of the second-order
polynomial with a damping ratio $\xi$ and a natural frequency $\omega_{n}$
given by

$\displaystyle S(s)=1+s\frac{2\xi}{\omega_{n}}+s^{2}\frac{1}{\omega_{n}^{2}}.$
(42)

Equating (41) with (42) yields

$\displaystyle\xi=\frac{1}{2}\sqrt{\frac{p_{ond}}{p_{od}}}a_{0}\beta_{0}>\frac{1}{2}a_{0}\beta_{0}\gg
1.$ (43)

Thus, we can write

$\displaystyle P(s)\simeq
1+s\frac{1}{\absolutevalue{p_{cd}}}+s^{2}\frac{1}{\absolutevalue{p_{cd}p_{cnd}}}.$
(44)

Equating the coefficient of $s$ in (41) and (44) results in

$\displaystyle\absolutevalue{p_{cd}}\simeq\frac{\absolutevalue{p_{od}}}{a_{0}\beta_{0}}.$
(45)

Similarly, by equating coefficients of $s^{2}$ in (41) and (44) and using
(45), $p_{cnd}$ can be estimated as

$\displaystyle\absolutevalue{p_{cnd}}\simeq\absolutevalue{p_{ond}}a_{0}\beta_{0}.$
(46)

Using (45), (46) and eliminating the absolute functions, the pole-splitting
relation given by (39) is established and is illustrated in Fig. 12.

$\square$

Figure 12: Pole-splitting relation

## Appendix B

Applying the quadractic formula to the quadratic function in the denominator
of (35), $p_{cnd1}$ and $p_{cnd2}$ are

$\displaystyle
p_{cnd1},p_{cnd2}=-\frac{g_{m2}}{2C_{2}}\pm\frac{g_{m2}}{2C_{2}}\sqrt{1-4\frac{g_{m1}}{C_{c1}}\frac{C_{2}}{g_{m2}}}.$
(47)

Using (30) and (34), we can write

$\displaystyle
p_{cnd1},p_{cnd2}\simeq\frac{p_{o2}}{2}\mp\frac{p_{o2}}{2}\sqrt{1+\frac{8\textrm{GBW}}{p_{o2}}}.$
(48)

Thus, when $p_{o2}=-8$GBW, $p_{cnd1},p_{cnd2}\simeq p_{o2}/2=-4$GBW. And, when
$p_{o2}=-4$GBW, $p_{cnd1},p_{cnd2}\simeq p_{o2}/2\mp jp_{o2}/2=-2$GBW $\mp
j2$GBW.

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|  Myungjun Kim received the Ph.D. degree in electrical engineering from Korea
Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in Feb,
2022. His Ph.D. study has focused on the stability analysis and the
development of the methodology for frequency compensation of various linear
analog integrated circuits (ICs) such as Badngap reference and (output-
capacitorless) low-dropout regulator. His research interests also include the
exploration for (Hopf) bifurcation and chaos that can occur in modern analog
CMOS ICs from the nonlinear dynamical system theory standpoint. He is
currently working at the Memory Division, Samsung Electronics, Hwaseong, South
Korea, as an analog IC designer. Dr. Kim also serves as a reviewer of IEEE
Transactions on Circuits and Systems—I: Regular Papers and IEEE Transactions
on Circuits and Systems—II: Express Briefs.
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