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									C MO S O PE R ATI O NAL
       PRO J E C T

                          EECS 413
                  December 15th, 2000

                     Joseph A. Potkay

         The CMOS operational amplifier shown below was designed to achieve the
specifications given in the “Performance” section and, in general, is a low power,
moderate gain, and fast settling time operational amplifier consisting of three stages. The
first stage provides of the biasing circuitry for the amplifier. Transistors M10 and M11
provide the gate bias voltage of transistor M9 and, in doing so, sets its “on” resistance.
Transistors M12, M13 and M14 are simply used to decrease the voltage drop across
resistor Rref, which sets the current for this stage. This current sets the gate voltage of
transistor M8, and this gate voltage is used as a gate bias voltage for the transistor current
sources, M5 and M7, which bias the second and third stages of the amplifier.

            M8                      M5                                        M7

                                            18u/0.8u                               54u/0.8u


          Rref                 M1                   M2
                         (-)                                   (+)

                 M12     66u/0.8u                   66u/0.8u





                 M10                                                 0.443p                   CL
           8u/0.8u                                                            M6

                 M11                  M3    M4
                         14u/0.8u                   14u/0.8u

                 Figure 1 The complete CMOS operational amplifier circuit.

        The second stage of the amplifier is the first gain stage and provides the
differential input for the op-amp. Transistors M1 and M2 are the drivers for this stage,
with M3 and M4 acting as the active current mirror load for this first stage. The high
output resistances of these transistors provide a high gain for this stage, and its single
ended output feeds the second gain stage.
        The last stage is the second gain stage and consists of transistors M6 and M7.
The NMOS transistor M6 is the driver with M7 acting as the load. Again, the high output
resistances of these two transistors equate to a relatively large gain for this stage and an
overall moderate gain for the complete amplifier.
        The large gain of the last stage is further utilized in the compensation of the
amplifier via the capacitance Cc. Without compensation, the op-amp will oscillate in
feedback circuits with a high loop gain. By taking advantage of the Miller Effect and the
high resistance at the drain of M2, a smaller value for the capacitance Cc is used than
would be needed otherwise. However, due to the low transconductance of MOSFETS,
the transistor M9 is needed to provide a nulling resistance to reduce the effects of that
right hand plane zero in the transfer function, and, in fact, can be used to improve the
frequency response of the amplifier.
        The three stages of the amplifier and its compensation circuitry provide a stable,
moderate gain, low power and fast settling time monolithic CMOS operational amplifier.
The following sections discuss the design of this amplifier, the constraints for the design,
the simulation and performance results, and a discussion of the overall amplifier.
         The design and design constraints were approached by first deciding on an
appropriate topology. Due to its well-studied and simple nature, the standard CMOS
operational topology was chosen. The folded cascode seemed to have also been a viable
solution with many advantages, but was passed up due to design time constraints. Due to
the fast settling time requirements, a PMOS input stage was chosen to increase the slew
rate of the amplifier and, a NMOS driver in the second gain stage would provide the
needed drive strength at the output. Then, after deciding upon the topology, the circuit
was analyzed and the design specifications were reduced to simpler design constraints.
Each specification led to different design constraints, as shown below.

DC Gain:
                                                                                 W        W
                                                            k p  2k n          ( ) 2  ( )6
       vo             g m 2  g m6                                               L         L
                                                                            
       vi ( g ds 2  g ds 4 )  ( g ds 6  g ds 7 ) ( 2   4)  ( 6   7)     I D5  I D7

       Given the design specification of 85dB and using the minimum L=0.8um to
       minimize capacitance, this reduces to the following:
        W        W
       ( )2  ( )6
        L         L  15.44 X 1012 A  2
         I D5  I D7

Common-Mode Input Range:

                                                 I D5                       I D5
       CMR   VGS 3  VDSAT1  VGS1                  0.75V        or,          32.4  10 6
                                                  W                         W
                                              kn ( )3                      ( )3
                                                   L                        L

                                         2I D5        I D5
       CMR   VDSAT 5  VGS 2                            Vto  0.75 , or
                                            W          W
                                       k p ( )5   k p ( )2
                                            L           L
          2I D5    I D5
                        380  10 6 A1 / 2
          W        W
         ( )5     ( )2
           L       L
Output Swing:

                                   2I D7            I
       Vout   V DSAT 7                  0.3 or, D 7  2.6  10 6 A
                                      W             W
                                 k p ( )7          ( )7
                                      L             L

                                   2I D7            I
       Vout   V DSAT 6                  0.3 or, D 7  5.7  10 6 A
                                      W             W
                                 k n ( )6          ( )6
                                      L             L
Power Dissipation:

      (ID8 + ID5 + ID7)*(VDD-VSS)  250uW, or, (ID8 + ID5 + ID7)  50uA

Unity Gain Frequency:
                                        W                          W
                           I D5 k p (     )2              I D5 (     )2
            g                           L  50MHz , or,            L  41.4  109 A1 / 2 F 1
       fu  m 2 
           2CC               2CC                          CC

Settling Time:
            C    C
        SR  C  L
            I D5 I D7
                     1                     (  4 )(6  7 ) I D 5 I D 7
       p1                      u  2
             g m 6 Ro1 Ro 2 CC        avo                   W
                                                 2CC 2k n ( ) 6
                            2I D7 k n ( ) 6
             g                         L
       p2   m6  
              CC                CC
        n  p1 p2 avo
                p1  p2
        
             2 p1 p2 avo
                            
       VOVERSHOOT  Vo exp         
                           1 2 
                                   
                          1     0.001 Vo 
       TS                  ln            
             n 1   2  n VOVERSHOOT 
Power Supply Rejection Ratio:
      GB  m 2
                                                               W W
                                                  k p kn   (      )2 ( )6
                         g m 2 g m6                            L       L
        Avo                              
                 ( g ds 2  g ds 4 ) g ds 6 (2  4 )6        I D5 I D7
                GB    (   4 )  6 I D 5 I D 7
        p       
                      2
                Avo                   W
                        2CC k p ( ) 6
                                                               W W
                                                  k p kn   (      )2 ( )6
                         g m2 g m6                             L       L
        Avo                              
                 ( g ds 2  g ds 4 ) g ds 7 (2  4 )7        I D5 I D7
                ( g ds 2  g ds 4 ) ( 2   4 ) I D 5
        p                       
                        CC               2C C

Using the above equations, an initial design was estimated and entered into spice and
simulated. To begin, a rough estimate of CC was determined as 0.2*CL. Then (W/L)2
and (W/L)5 were set to meet what appeared to be the major constraint on the design – the
CMR. ID5 was then set to meet the unity gain frequency and the CMR, and, initially ID7
was set about equal to ID5 to keep the gain high. As mentioned before, the CMR was the
major constraint for the design according to hand calculations, requiring a small ID5 and a
large (W/L)2 and (W/L)5. In spice, however, the CMR was not a constrain at all and, in
fact, the area of the transistors needed to be decreased significantly to improve the
frequency response and settling time of the amplifier. Throughout the design process, the
PSRR- and the settling time were the major constraints to the design. The slew rate of the
amplifier initially limited its settling time. To improve this, ID7 was increased to
improve the current available to charge the capacitance at the output node. The PSRR-
was improved by increasing the magnitude of its pole as shown in the equations above.
The whole design process was a set of tradeoffs, which finally converged to a working
design. When all parameters were extremely close to meeting the specifications, the
nulling resistor as implemented by M9 was tweaked to obtain the optimal frequency
response and settling time. The transistor and bias summary for this final design is
shown in the next section. The hand calculations for the final design are included in the
table in the “Design Performance” section. Additional design information and the
differences between the hand calculations and the spice simulations are further addressed
in the “Discussion” section.

Transistor    W/L (um/um)    ID (uA)       VGS (mV)      gm (uA/V)      gds (nA/V)
M1            66/0.8               4.107       738.861        211.485         482.505
M2            66/0.8               4.108       738.858        211.489         482.527
M3            14/0.8               4.107       761.959        132.705         643.662
M4            14/0.8               4.108       761.959        132.729         643.898
M5            18/0.8               8.215       809.476        150.287         890.241
M6            64/0.8             26.790        763.131        849.558        5779.200
M7            54/0.8             26.790        809.476        490.102        3155.900
M8            2.4/0.8              0.993       809.476         18.165          97.542
M9            40/0.8               0.000       717.635          0.000     111629.300
M10           8/0.8                0.993       740.383         49.208         155.086
M11           8/0.8                0.993       740.383         49.208         155.086
M12           1.2/2.4              0.993       890.014         10.479           3.686
M13           1.2/2.4              0.993       890.014         10.479           3.686
M14           1.2/2.4              0.993       890.014         10.479           3.686
             Table 1 A summary of the transistors and biasing.
                              DESIGN PERFORMANCE

Parameter                      Design Objective             Hand Calculated*            Simulated Performance
DC Gain                             85 dB or more                 92.03 dB                     85.02 dB
Common Mode Input Range
   Positive                          1.75 V or more                  1.69 V                       1.79 V
   Negative                          -1.75 V or less                 -2.44 V                      -2.43 V
Output Swing
   Positive                          2.2 V or more                   2.413 V                     2.499 V
   Negative                           -2.2 V or less                 -2.439 V                   -2.490 V
Power Dissipation                    250uW or less                   180 uW                      180 uW
Unity Gain Frequency                     50 MHz                       95 MHz                    65.9 MHz
Settling Time
   0 to +1 V Output Step             250 ns to 0.1%                   210 ns                     158 ns
   0 to -1 V Output Step             250 ns to 0.1%                   210 ns                     132 ns
CMRR at DC                           85 dB or more                    inifinite                  95.2 dB
  PSRR+ at DC                        85 dB or more                   97.1 dB                     94.0 dB
  PSRR+ at 800kHz                    40 dB or more                   86.4 dB                     60.6 dB
  PSRR- at DC                        85 dB or more                   87.4 dB                     94.5 dB
  PSRR- at 800kHz                    40 dB or more                   32.6 dB                     40.0 dB
       Table 2 Comparison of the design objective and the calculated and simulated results.
       * Using the drain current values from the spice simulation and the equations in the “Design” section

         Overall, the final design performed above and beyond the design specification. In
particular, the unity gain frequency, settling time, and power particularly excelled. The
low power biasing, the moderate to small device size and hence small device
capacitances, and the moderate current available to charge the capacitances helped to
meet and improve on the design specifications. Each specification is further examined
below, but a few issues affected the whole circuit and are described here.
         In the design of the amplifier, the bulks were tied to the sources for simplicity in
design and this is a valid simplification for several reasons. First of all, by looking at the
circuit, it can be seen that most of the transistors naturally have their bulks and sources
connected. Second, a current dual well process could implement this type of
configuration. Finally, if the bulks were tied to their respective power supply, it would
have little impact on the performance of the circuit. The only impact on the circuit would
be to increase Vt1 and Vt2 by p[(2f+VSB)1/2 – (2f)1/2)=60 mV. The gain of the circuit
would be unaffected, because, gm1 and gm2 are fixed by the value of ID5. The CMR+
would decrease by 60 mV and this change would have to be compensated by a slightly
larger (W/L)1 and (W/L)2, but would otherwise affect the circuit very little. In addition,
the bias circuitry would have to change slightly to produce the same current in its branch,
but would otherwise not affect the performance of the amplifier. However, CMOS
processes have at least one well and thus either the M1 and M2 would have to be adjusted
slightly or the biasing circuitry would, but not both.
         Another issue that hampered the initial design stages was the fact that the
calculated values differed somewhat from the simulated values, as can be seen in table 2.
It appears that short channel effects caused the differences in gain and in the ouput swing
and CMR values. With an Leff of 0.44 um for the PMOS transistors and 0.56 um for
NMOS, short channel effects become significant, reducing the gain and VDSAT of the
devices, especially for the smaller PMOS transistors. This helped to meet the output
swing and CMR specifications, but hurt the gain slightly. The difference in the frequency
response and hence unity gain frequency was due to assumption that the first pole was
dominant and the frequency response was not affected by the others. In reality, a second
pole that was below the unity gain frequency existed at about 35 MHz and thus affected
the frequency response.
         Each specification is further examined and validated in the following subsections.
                                        DC GAIN

       Throughout the design, the dc gain was not a factor and was reduced in trading off
other performance characteristics of the design. The gain was measure using the
configuration shown in figure 2, below, because it allows the direct measurement of vo/vin
at dc.
                                 VOS         VDD


                                               OUT            Vo

                                         -               2p


           Figure 2 Configuration for the measurement of the open loop gain.

                 Figure 3 DC Gain and –3dB frequency of the amplifier.

The graph in Figure 3, above, shows the result of the simulation. The DC gain is just
slightly over 85 dB, 85.02 dB to be exact, and the gain starts to roll off around the –3dB
frequency of 4.06 kHz.
                         COMMON-MODE INPUT RANGE

        In hand calculations, the CMR appeared to be the most limiting constraint for the
design, requiring a small value for ID5 and large values for (W/L)5 and (W/L)2. In
simulations, however, the CMR posed no limit on the design and stayed relatively
independent to design changes. The CMR was measured using the circuit in figure 4 by
sweeping the input voltage from VSS to VDD. This configuration was chosen because, in
a high gain configuration, the output swing of the amplifier limits the linearity of the
circuit. In the unity gain configuration, however, the linear portion of the curve
represents the CMR of the amplifier. The results from this simulation are shown in
figures 5, 6 and 7 and show a CMR+ of 1.79 V and a CMR- or –2.42 V. The difference
between hand calculations and simulation was due to the short channel nature of the
devices, which caused the VDSAT of the devices to be less than the (VGS-Vth) that long
channel theory predicts. This effect was more pronounced in the PMOS devices, which
had a smaller Leff. In addition, this effect was also seen when simulating the output
swing of the device, which is discussed in the next section.

                              Vin     +

                                            OUT         Vout

                                      -                 CL

              Figure 4 The configuration for the measurement of the CMR.

                Figure 5 The common mode input range of the amplifier.
Figure 6 A close-up of the CMR+.                           Figure 7 A close-up of the CMR-.

                                       OUTPUT SWING

             The output swing of the device was not a major constraint on the design, but was
    even less so in simulations than in hand calculations. The reason for this short channel
    effects, as described above in the CMR section. The circuit in figure 8 was used to
    measure the output swing of the amplifier. In the unity gain configuration, the linearity
    of the transfer curve is limited by the CMR. Using this configuration of higher gain, the
    linearity region of a plot of vo vs. vin corresponds to the output swing of the amplifier.
    The results of the simulation are displayed in figures 9 through 11 and plot vo vs. vin.



                                                    OUT         Vout

                               Vin           -                 CL


              Figure 8 The configuration for the measurement of the output swing.
                            Figure 6 The output voltage swing of the amplifier.

Figure 10 A close-up of the positive output swing.      Figure 11 A close-up of the negative output swing.
                                               POWER DISSIPATION

                         The circuit was initially designed for a maximum open loop quiescent power or
                250 uW. To save power, the current source biasing and biasing for the nulling resistor
                were combined in a single low current branch. This allowed a lot of flexibility in
                choosing and modifying the current in the two gain stages. In the final design, this
                biasing stage consumed only about 4.5 uW of quiescent power. Overall, the total open
                loop quiescent power was 180 uW, as given by the output of the spice simulation.
                Furthermore, figures 13 and 15 below show that, with a capacitive load, the maximum dc
                power that this amplifier will consume under any conditions is 248 uW. This occurs
                when the amplifier is connected in unity gain feedback and its output is at –2.5 V. As Vo
                varies, VDS7 varies and causes a corresponding change in ID7 due to the channel
                modulation effect. This causes the power to vary with output voltage. The additional
                increase in power and current in the unity gain configuration is due to the fact that the
                output voltage is connected to the input, and thus modulates VDS5 as well as VDS7. It
                should be noted that the distortion in the power curve for the unity gain feedback
                configuration is due the current source transistor, M5, exiting its saturation region.

      VOS        VDD


                    OUT            Vo

            -                 2p


Figure 12 Circuit for the                   Figure 13 The open loop power, in watts, as a function of vo.
measurement of the open loop power.

Vin     +

              OUT         Vout

        -                 CL

Figure 14 Circuit for the measurement     Figure 15 The unity gain power, in watts, as a function of vo.
of the unity gain feedback power.

                                           UNITY GAIN FREQUENCY

                       The unity gain frequency was not much of a limitation on the design until the
              final stages, at which point it was necessary to make small modifications to Cc to trade
              off frequency response for settling time. This is discussed more in the section on settling
              time. When the design was almost complete, (W/L)9 was modified to modulate the
              nulling resistance and determine the optimal pole-zero configuration. This resulted in the
              open-loop frequency response shown in figure 16 and a unity gain frequency of 65.9
              MHz. For reference, the frequency response of the amplifier in the unity gain
              configuration is shown in figure 17. This plot exhibits a slight peaking near the unity
              gain frequency due to the 39 phase margin that was achieved, but is consistent with the
              open-loop results.
     Figure 16 The open-loop gain of the amplifier, highlighting the unity-gain frequency.

Figure 17 Frequency response of the amplifier in the unity gain configuration, displaying
           a peak near the unity gain frequency due to the 39 phase margin
                                    SETTLING TIME

        The settling time was the most demanding constraint in the design of the
amplifier. The initial design contained large (W/L) ratios and a small current in the
differential pair in order to meet the CMR specification. This, however, led to a
extremely poor settling time hampered by both a small charging current and large device
capacitances. After discovering the CMR was not much of a limiting constraint, the
device sizes were scaled aggressively to improve the settling time of the amplifier, at the
expense of the gain. At this point, the capacitance at the output node was the limiting
factor for the settling time and therefore the current in the second gain stage was
increased to improve the charging current for that capacitance. The ringing that was
observed was improved through the increase in Cc and through the modulation of the
(W/L)9. These were used to obtain the optimal pole-zero configuration and resulted in a
phase margin of 39 as shown in figure 18 below. This phase margin reduced the
overshoot and improved the settling time significantly. The configuration in figure 19
was used to measure the settling time of the final design and was used for accuracy in
setting the output voltage step range. The results of this simulation are shown in figures
20 through 22. The settling time to 0.1% for the amplifier was 158 ns for a 0 to +1 V
output step and 132 ns for a 0 to –1 V output step.

Figure 18 The phase response of the open-loop amplifier, displaying a phase margin of 39.


                              Vin      +

                                             OUT         Vout

                                       -                 CL

      Figure 19 The amplifier configuration for the measurement of the settling time
                Figure 20 The settling time of the amplifier for a 0 to +1 and –1 output step.

Figure 21 A close-up of the settling time for the            Figure 22 A close-up of the settling time for the
          0 to +1 V step.                                              0 to –1 V step

                 The CMRR specification was not much of a constraint during the design of the
        amplifier, and basically required a commode-gain of less than 1 V/V. The deviation from
        the ideal CMRR of infinity is due to the finite output resistance of the current-mirror load
        transistor M3. This causes the small signal current that is mirrored in M4 to vary slightly
        and thus, id3 and id4 to not cancel exactly. The CMRR of the circuit was measure using
        the circuit in figure 23. The output of this circuit is approximately equal to 1/CMRR.
        The graph in figure 24 plots –20log(1/CMRR) to give the CMRR of the amplifier, in dB.
        The CMRR at DC is highlighted and is equal to 95.2 dB.

      Vcm        VDD


      Vcm          OUT            Vout

             -               CL

Figure 23 The circuit for the             Figure 24 CMRR as a function of frequency for the amplifier.
  measurement of the CMRR.


                The PSRR turned out to be an important constraint near the end of the design
        process. After meeting all of the other constraints, the PSRR was measured and found to
        not meet the specification. In order to fix this, and not adversely affect the other
        specifications, the p- of the PSRR was increased by increasing ID5 and decreasing
        (W/L)6. This improved the p-, but had very little effect on the gain. These relationships
        can be seen in the equations in the “Design” section of this document. It was also
        interesting to note that as the offset voltage of the amplifier was reduced by changing
        (W/L)3 and (W/L)4, the DC value of the PSRR increased. The reason for this is unclear,
        but it appears that when the circuit is unbalanced, a small DC voltage exists across either
        the M1 and M2 or M3 and M4. This small DC voltage would tend to decrease the output
        resistance of that device as the device neared the linear region, and this decrease in output
        resistance leads to a decrease in the PSRR. Also, to reduce the PSRR over all, the Rref
        was kept as large as possible to minimize the change in bias currents due to a change in
        the supply voltage.
                The PSRR was measured using the circuit of figure 25, with vss active for PSRR-
        and vdd active for PSRR+. As was shown in class, the unity gain configuration produces
        an output that is approximately equal to 1/PSRR. Figure 25 and 26 plot the PSRR+ and
        PSRR-, respectively.
                                                 v dd


                                     OUT                Vout

                               -                        CL


                                         v ss

           Figure 25 A circuit for the measurement of the PSRR.

Figure 7 A plot of the PSRR+ of the amplifier as a function of frequency.
Figure 8 A plot of the PSRR- of the amplifier as a function of frequency.
         The design process that was followed resulted in a complete CMOS operational
amplifier that at least met and, in a few cases, exceeded the design objectives by a large
margin. The notable performance areas were the settling time of 158 ns, the unity gain
frequency of 67 MHz, and the power consumption of 180 uW.
         A great deal was learned in the design process, including how to approach a
design project, the tradeoffs involved in a CMOS op-amp design, patience, and how to
stay up late. I felt that not much knowledge was gained from doing the report, but that it
was still a necessary part of the project. I felt that the specification could have been a
little more explicit in some areas, such as the process type being used (N-well, dual well,
etc.), and maybe the maximum area or a maximum resistance value given from the start.
Some redesign was necessary to decrease Rref below 40 k. Overall, it was a
worthwhile and valuable experience.

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