GYRATOR WITH FEEDBACK RESISTORS

Information

  • Patent Application
  • 20060197632
  • Publication Number
    20060197632
  • Date Filed
    May 11, 2005
    19 years ago
  • Date Published
    September 07, 2006
    18 years ago
Abstract
A gyrator includes a gyrator core and at least one common mode feedback circuit. The gyrator core includes four inverters mutually connected in a loop configuration between a pair of input ends and a pair of output ends. The common mode feedback circuit is provided between the pair of input ends and/or the pair of output ends and includes a forward-reverse connection inverter set and a backward-reverse connection inverter set. The forward-reverse connection inverter set has a first inverter, a second inverter connected in reverse series with the first inverter, and a first feedback resistor connected in parallel with the second inverter. The backward-reverse connection inverter set has a third inverter, a fourth inverter connected in reverse series with the third inverter, and a second feedback resistor connected in parallel with the fourth inverter.
Description
BACKGROUND OF INVENTION

1. Field of the Invention


The present invention relates to a gyrator, and more particularly, to a gyrator with feedback resistors.


2. Description of the Prior Art


Gyrators are one of many kinds of electronic circuits for converting impedance. For example, a gyrator has the capability to make an inductance circuit behave like a capacitance circuit. In the design of a continuous time filter, an integrator or a gyrator is frequently used for converting impedance. In fact, two integrators can be electrically connected in a loop to form a gyrator.


Please refer to FIG. 1 and FIG. 2. FIG. 1 is a circuit diagram of a Nauta cell gyrator NCG having a gyrator core GCi disclosed in U.S. Pat. No. 6,490,706. FIG. 2 is an equivalent circuit diagram of the gyrator NCG. In the design of a filter of the prior art, at least a filter section FSTi comprises the gyrator NCG. In FIG. 2, a relation between an input current I1 and an output voltage V2 is I1=−gm*V2, and another relation between an input voltage V1 and an output current is I2=gm*V1, where −gm is a negative trans-conductance of the gyrator NCG and gm is a positive trans-conductance.


In addition to the gyrator core GCi, the gyrator NCG shown in FIG. 1 further comprises two common mode feedback sections CMIi and CMOi. The gyrator core GCi comprises four inverters GI1i, GI2i, GI3i, and GI4i electrically connected between two input ends i_1 and i_2 and two output ends o_1 and o_2. The common mode feedback section CMIi comprises two non-reverse series connection inverter sets, each of which comprises an inverter CMI1 and a short-circuited inverter CMI2. The common mode feedback section CMOi also comprises two non-reverse series connection inverter sets, each of which comprises an inverter CMO1 and a short-circuited inverter CMO2.


Please refer to FIG. 3, which is an equivalent circuit diagram of an inverter (such as the inverter CMI1, or the inverter CMO1) of the gyrator NCG shown in FIG. 1. The inverter has an admittance matrix equal to
Yinv=[yi+yf-yfym-yfyo+yf],(eq.1)


where yi=sci is an input admittance, yo=go+sco is an output admittance, yf=scf is a trans-admittance from an output end to an input end, and ym is a trans-admittance from the input end to the output end.


Based on eq.1, an admittance matrix of the gyrator core GCi can be derived as
Ycore=[yl+yf-yf0ym-yfym-yfyl+yf-yf00ym-yfyl+yf-yf-yf0ym-yfyl+yf]


(without losing the generality, the gyrator NCG is assumed to comprise identical inverters GI1i, GI2i, GI3i, GI4i, CMI1, CMI2, CMO1, and CMO2), and an admittance matrix of the common mode feedback section can be derived as
YCM=[2yl+ym0ym-2yf002yl+ym0ym-2yfym-2yf02yl+ym00ym-2yf02yl+ym],


where yi is equal to yi+yf+yo. Accordingly, an admittance matrix of the gyrator NCG can be derived as
Ygyr=[3yl+ym+-yfym-2yfym-yfyfym-yf3yl+ym+-yfym-2yf`yfym-2yfym-yf3yl+ym+-yfyf-yfym-2yfym-yf3yl+ym+yf].(eq.2)


Under an assumption that the applied signal of the gyrator NCG is differential, that is Vi1=−Vi2, Vo1=−Vo2, Ii1=−Ii2, and Io1=−Io2, where Vi1, Vi2, Vo1 and Vo2 are four voltages on two input ends i_1 and i_2 and two output ends o_1 and o_2 respectively, and Ii1, Ii2, Io1, Io2 four currents flowing through the input ends i_1 and i_2 and the output ends o_1 and o_2 respectively, eq.2 can be simplified as
Ygyr=[3(yi+2yf+yo)+Δym-ymym3(yi+2yf+yo)+Δym].(eq.3)


If YI is defined to be equal to 3(yi+2yf+yo), eq.3 can be further simplified as
Ygyr=[Yl+Δym-ymymYl+Δym],(eq.4)


where Δym is a difference between trans-admittances ym.


In U.S. Pat. No. 6,490,706, a channel delay effect is taken into consideration, that is ym=gme−Sτ, and τ=cm/gm, where τ is an effective channel delay of the gyrator core GCi, gm is an effective gyrating constant, and cm is an effective trans-capacitance. The gyrator NCG has to function in a stable condition: g*c is not smaller than gm*cm, where g is an effective conductive loading of the gyrator GCi, and c is an effective capacitive loading.


However, the above stable condition is applied to a specific case only.


Moreover, since an integrator of the gyrator NCG has a DC gain equal to A0=gm/g, and g is proportional to I/L, and gm is proportional to I/Vod, where I is a bias current of the inverter, L is a channel length, and Vod is an overdrive voltage, if gm is kept constant, A0 can be increased through a decrease in I, or another increase in L so as to decrease g. However, in the meantime of the decreasing 1, Vod is decreased accordingly. Therefore, the gyrator has a poor linearity. On the other hand, cm is increased as L is increased, and the stability of the gyrator NCG will be reduced further.


SUMMARY OF INVENTION

It is therefore a primary objective of the claimed invention to provide a gyrator with a feedback resistor, so as to solve the problems of deteriorating linearity and decreasing stability resulting from the efforts to increase DC gain.


According to the claimed invention, the gyrator includes a gyrator core and at least a common mode feedback section. The gyrator core includes four inverters mutually connected in a loop configuration between a pair of input ends and a pair of output ends. The common mode feedback section is electrically connected between the pair of the input ends or the pair of the output ends. The common mode feedback section includes a forward-reverse series connection inverter set and a backward-reverse series connection inverter set electrically connected in anti-parallel with the forward-reverse series connection inverter set. The forward-reverse series connection inverter set includes a first inverter, a second inverter electrically connected in reverse series with the first inverter, and a first feedback resistor electrically connected in parallel with the second inverter. The backward-reverse series connection inverter set includes a third inverter, a fourth inverter electrically connected in reverse series with the third inverter, and a second feedback resistor electrically connected in parallel with the fourth inverter.


Taking a channel delay effect into consideration, the gyrator has a stable condition |yLE2(jw0)|>gm2, where w0 is an existing smallest positive number making 2w0τ+∠yLE2(jw0)=π, YLE2(s) is an effective loading product of the gyrator core, τ is an effective channel delay, and gm is an effective gyrating constant.


These and other objectives of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment that is illustrated in the various figures and drawings.




BRIEF DESCRIPTION OF DRAWINGS


FIG. 1 is a circuit diagram of a gyrator according to the prior art.



FIG. 2 is an equivalent circuit diagram of the gyrator shown in FIG. 1.



FIG. 3 is an equivalent circuit diagram of an inverter of the gyrator shown in FIG. 1.



FIG. 4 is a circuit diagram of a gyrator of the preferred embodiment according to the present invention.



FIG. 5 is an equivalent circuit diagram of a forward-reverse series connection inverter set of an input common mode feedback section of the gyrator shown in FIG. 4.



FIG. 6 and FIG. 7 are two equivalent circuit diagrams of an input loading admittance or an output loading admittance of the gyrator shown in FIG. 4.



FIG. 8 is a plot diagram of zeros and poles of the input loading admittance or the output loading admittance shown in FIG. 7.




DETAILED DESCRIPTION

Please refer to FIG. 4, which is a circuit diagram of a gyrator 10 having a gyrator core GCi of the preferred embodiment according to the present invention. In addition to the gyrator core GCi, the gyrator 10 further comprises an input common mode feedback section 12, and an output common mode feedback section 14


Different from the common mode feedback section CMIi of the gyrator NCG shown in FIG. 1, which comprises two non-reverse series connection inverter sets, the input common mode feedback section 12 comprises two reverse series connection inverter sets, a forward-reverse series connection inverter set 16 and a backward-reverse series connection inverter set 18, each of which comprises an inverter CMI1, an inverter CMI2, and an input feedback resister rfi1 electrically connected in parallel with the inverter CMI2. Similarly, the output common mode feedback section 14 comprises two reverse series connection inverter sets, each of which comprises an inverter CMO1, an inverter CMO2, and an output feedback resister rf01 electrically connected in parallel with the inverter CMO2.


According to the preferred embodiment, any one of the inverters of the gyrator 10 can comprise a transistor, and the transistor can be a MOS, a COMS, or a bi-polar transistor.


The gyrator 10 has a larger DC gain A0 due to the installation of the feedback resistor rfi1(rfo1), and this is described as follows.


Please refer to FIG. 5, which is an equivalent circuit diagram of the forward-reverse connection inverter set 16 (the backward-reverse connection inverter set 18 is similar) of the input common mode feedback section 12 (the output common mode feedback section 14 is similar) of the gyrator 10 shown in FIG. 4. In FIG. 5, current flowing into an intermediate point Vx is equal to current flowing out from the intermediate point Vx, that is yf1(Vi1−Vx)+yf2(Vi2−Vx)=ym1Vi1+ym2Vi2+(yo1+y02)Vx (eq.5). Eq.5 can be derived as
-Vx=(ym1-yf1yx)Vi1+(ym2-yf2yx)Vi2,(eq.6)


where yx=yo1+yo2+yf1+yf2. Based on eq.6, a relation between Ii1, Ii2, Vi1, and Vi2 can be derived as:
Ii1=yi1Vi1+yf1(Vi1-Vx)=(yi1+yf1+yf1ym1-yf1yx)Vi1+yf1ym2-yf2yxVi2and(eq.7)Ii2=yi2Vi2+yf2(Vi2-Vx)=yf2ym1-yf1yxVi1+(yi2+yf2+yf2ym2-yf2yx)Vi2.(eq.8)


Eq.7 and eq.8 can be combined to form
[Ii1Ii2]=[yi1+yf1+yf1ym1-yf1yxyf1ym2-yf2yxyf2ym1-yf1yxyi2+yf2+yf2ym2-yf2yx][Vi1Vi2]=[y11y12y21y22][Vi1Vi2](eq.9)


Without losing the generality, all of the inverters of the gyrator 10 are assumed to be identical and have a feedback resistance of rf=1/gf, that is yi1=yi2=yi, yo1=yo2=yo, yf1=yf, yf2=yf+gf, and ym1=ym2=ym. Therefore, an admittance matrix of the input common mode feedback section 12 can be derived as
YCM=[y11+y220y12+y2100y11+y220y12+y21y12+y210y11+y2200y12+y210y11+y22].


Similarly, assume differential signal is applied to the gyrator 10. Therefore, the admittance matrix can be simplified as
Ygyr=[YL-ymymYL],whereYL=(y11+y22)-(y12+y21)+(yi+2yf+yo),andYL=3yi+4yf+y0+gf(2yo+2yf)2yo+2yf+gf(eq.10)


In accordance with eq.10, an equivalent circuit diagram of YL is shown in FIG. 6. It is apparent that the installation of the feedback resistor rfi1(rfo1) increases the impedance of YL, and the gyrator 10 has therefore a higher DC gain A0.


The stable condition of gyrator 10 is derived as follows. Under an assumption that differential signal is applied to the gyrator 10, the admittance matrix Ygyr of the gyrator 10 can be represented as
Ygyr=[yLI-ymOymIyLO],


where yLI is an input loading admittance, yLO is an output loading admittance, ymI is a trans-admittance from an input end to an output end, and ymO is another trans-admittance from the output end to the input end. The channel delay effect is taken into consideration, that is ymIymO=gm2e−2Sτ, where τis an effective channel delay of the gyrator core GCi, and gm is an effective gyrating constant. Accordingly, the characteristic function of gyrator 10 can be derived as
Δ(s)=1+ymIymOyLIyLO=1+gm2-2sτyLE2(s)=1+Fd(s)=1+F(s)-2Sτ,(eq.11)


where YLE2(s) is an effective loading product of the gyrator core GCi, Fd(s) is a loop transfer function, and F(s) is another loop transfer function without taking the channel delay effect into consideration. The stable condition of the gyrator 10 is sustained if all of zeros of Δ(s) are located in the left half region of the S-plane. Whether or not a zero of Δ(s) is located in the left half region of the S-plane can be determined through an application of a Nyquist plot.


It can be seen from eq.10 that YL is a network comprising nothing but resistors and capacitors, so YL can be further simplified as
YI(s)=g1+sc1+gf(g2+sc2)gf+(g2+sc2),(eq.12)


where g1=go, g2=2go, c2=2cf+2co, and c1=3ci+4cf+co+cext, where cext is an external capacitance of the gyrator 10. Please refer to FIG. 7, which is an equivalent circuit diagram of the simplified YL. Eq.12 can be rewritten as
YI(s)=(gf+g1+sc1)-gf2gf+g2+sc2=c1(s+z1)(s+z2)(s+p2),(eq.13)


where −p2=−(gf+g2)/c2<0 is a pole of yL(s), and −z1 and −z2 are two zeros. A curve yL(s) shown in FIG. 8 is obtained if s in eq.13 changes along a negative real axis (s=0→−∞), wherein when s=0, yL(s)>0, when s=−p2+, yL(s)→−∞, when s=−p2, yL(s)→∞, and when s→−∞, yL(s)→−∞. Because yL(s) is a continuous function besides the pole−p2, the two zeros −z1 and −z2 are easily determined to be two negative real numbers with the pole −p2 located between the two zeros −z1 and −z2. Without losing the generality, the pole −p2 and the two zeros −z1 and −z2 are assumed to be located in an order of z2>p2>z1.


Therefore, it is concluded that:


1. Neither Fd(s) nor F(s) has any pole or zero located in the right half region of the S-plane;


2. when s→∞, Fd(s)→0; and


3. Fd(−jw)=Fd(jw)*.


Hence, whether or not the gyrator is stable now can be determined through observing whether the Nyquist plot of Fd(s) when s=j∞→j0 encircles a specific point (−1,j0).


When s changes along a positive imaginary axis (s=j0→j∞), each of |(s+z1)/(s+p2)| and |(s+z2)| is a monotonic increasing function, so when w=0−∞, |YL(jw)| is also a monotonic increasing function, while |Fd(jw)|=|F(jw)| is a monotonic decreasing function. Moreover, a deduction of ∠yL(s)=∠(s+z1)+∠(s+z2)−∠(s+p2) can obtain ∠YL(jw)≧∠(jw+z2)≧0 and ∠YL(jw)<∠(jw+z1)<π/2, so−π<∠F(jw)≧0. Since a phase delay −2wτ due to the channel delay effect is also a monotonic decreasing function, ∠Fd(jw)≧0.


In summary, the stable condition of the gyrator 10 can be represented through the introduction of a concept of gain margin, that is |Fd(jw0)|<1, where w0 is an existing smallest positive real number making ∠Fd(jw0)=−π or |yLE2(jw0)|>gm2, where w0 is a smallest positive real number making 2w0τ+∠yLE20(jw0)=π.


However, the solution to this is not easy to obtain. Since |Fd(jw)| is a monotonic decreasing function when w=0→∞, if a positive real number w0 makes |Fd(jw0)|=1 and ∠Fd(jw) is restricted to be within a specific function larger than −π when w≧w0, the system is stable. In short, through the introduction of a bounding function and the concept of a phase margin, a sufficient condition for stability can be obtained.


For example, assume yp(s) has the following characteristics: when w=0→∞, if ∠yp(jw) is a monotonic increasing function and ∠yp(jw)≧∠yL(jw), yp(s) is a phase upper bound function of yL(s). Based on the above, if a positive real number w0 makes |Fd(jw0)|=1 and 2w0τ+∠yPI(jw0)+∠yPO(jw0)≧π, where yPI(s) and yPO(s) are two phase upper bound functions of yLI(s) and yLO(s) respectively, 0≧∠Fd(jw)>−π when w≧w0, and the system is determined to be stable.


Similarly, assume yA(s) has the following characteristics: when w=0→∞, if |yA(jw)| is a monotonic increasing function and |yA(jw)|≧|yL(jw)|, yA(s) is an amplitude lower bound function of yL(s). Based on the above, if a positive real number w0 makes |yAI(jw0)|*|yAO(jw0)|=gm2 and 2w0τ+∠yPI(jw0)+∠yPO(jw0)<π, where yAI(s) and yAO(s) are two amplitude lower bound functions of yLI(s) and yLO(s) respectively, when w>w0|Fd(jw)|<1 and when w≧w00≧∠Fd(jw)>−π, and the system is determined to be stable.


Inspecting eq.12 again,
yI(jw)=g1+jωc1+gf(g2+jωc2)gf+(g2+jωc2)=g1+g3+gf2gf+g2*k1+k+jω[c1+c3(1-k1+k)](eq.14)

is obtained, where k=(wc2)2/(gf+g2)2>0, g3=(gfg2)/(gf+g2), and c3=c2(gf/(gf+g2))2.


It can be seen from eq.14 that yPI(s)=(g1+g3)+s(c1+c3) is a phase upper bound function of yL(s). Naturally, yP2(s)=(g1+g3)+s(c1+c2) is also a phase upper bound function of yL(s). On the other hand, yA1(s)=(g1+g3)+sc1 is an amplitude lower bound function of yL(s). Naturally, yA2(s)=g1+sc1 and yA3(s)=sc1 are another amplitude lower bound functions of yL(s). Through the use of the above bounding functions, a stable condition can be obtained.


For example, when all of the inverters of the gyrator 10 are identical and all the feedback resistors are also identical, the above yA3(s) and yPI(s) are selected to be the bounding functions of yL(s), and one sufficient condition for the stability of the gyrator 10 is w0=gm/c1 and W0τ+∠yPI(jw0)<π/2, or tan(w0τ)<(g1+g3)/(w0(c1+C3))=c1(g1+g3)/(gm(c1+c3)), or tan(cm/c1)<c1/(A0(c1+c3)).


In particular, when rf=0, yL(s) can be simplified as yL(s)=(g1+g2)+s(c1+c2)=g+sc. Since both |yL(s)| and ∠yL(s) are themselves monotonic increasing functions when w=0→∞, without any additional auxiliary of a bounding function, they can make use of the gain margin (and the phase margin as well) of Fd(s) to be the stable condition of the gyrator 10 directly. For example, when all of the inverters of the gyrator 10 are identical, the necessary and sufficient condition for the stability of the gyrator 10 is w0=
1c(gm2-g2)=gmc(1-1A02)andw0τ+∠yL(jw0)<π/2ortan(w0τ)<gω0c=1(A02-1)ortan(cmc1-1A02<1(A02-1)).


In particular,
(A02-1)tan(cmc1-1A02)<A0cmc=gmcmgc.


Therefore, gmcm<gc is a sufficient condition for the stability of the gyrator 10.


In contrast to the prior art, the present invention can provide a gyrator comprising a gyrator core and at least a common mode feedback section. The common mode feedback section comprises two reverse series connection inverter sets, each of which comprises an inverter CMI1, an inverter CMI2, and a feedback resistor electrically connected in parallel with the inverter CMI2. Therefore, a DC gain of the gyrator is increased due to the installation of the feedback resistor. Moreover, when the feedback resistor has a resistance equal to zero, the gyrator has a stable condition equal to
tan(cmc1-1A02)<1(A02-1).(eq.15)


In contrast to eq.15, the stable condition g*c is not smaller than gm*cm of the gyrator of the prior art is only a specific sufficient condition.


Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above disclosure should be construed as limited only by the metes and bounds of the appended claims.

Claims
  • 1. A gyrator with feedback resistors, the gyrator comprising: a gyrator core comprising four inverters mutually connected in a loop configuration between a pair of input ends and a pair of output ends; and at least a common mode feedback section electrically connected between the pair of the input ends or the pair of the output ends, the common mode feedback section comprising: a forward-reverse series connection inverter set comprising: a first inverter; a second inverter electrically connected in reverse series with the first inverter; and a first feedback resistor electrically connected in parallel with the second inverter; a backward-reverse series connection inverter set electrically connected in anti-parallel with the forward-reverse series connection inverter set, the backward-reverse series connection inverter set comprising: a third inverter; a fourth inverter electrically connected in reverse series with the third inverter; and a second feedback resistor electrically connected in parallel with the fourth inverter.
  • 2. The gyrator of claim 1, wherein the inverter comprises a transistor.
  • 3. The gyrator of claim 2, wherein the transistor is a MOS transistor.
  • 4. The gyrator of claim 2, wherein the transistor is a CMOS transistor.
  • 5. The gyrator of claim 2, wherein the transistor is a bi-polar transistor.
  • 6. The gyrator of claim 2, wherein the transistor is selected to satisfy a stable condition |yLE2(jw0)|>gm2, where w0 is an existing smallest positive number making 2w0τ+∠yLE2(jw0)=π, yLE2(s) is an effective loading product of the gyrator core, πis an effective channel delay, and gm is an effective gyrating constant.
  • 7. The gyrator of claim 2, wherein the inverters are all identical, the first feedback resistor has a zero first resistance, the second feedback resistor has a zero second resistance, and the transistor is selected to satisfy a stable condition
Priority Claims (1)
Number Date Country Kind
094106103 Mar 2005 TW national