Analog Integrated Circuit Design
Nagendra Krishnapura
Address: Dept. of Electrical Engg., IIT Madras, Chennai, 600036, India.
Phone/Fax: +91-44-2257-4444/+91-44-2257-4402
e-mail: nagendra@iitm.ac.in
A video course under the NPTEL
Assignment problem set
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Contents
1 Negative feedback systems
3
2
Opamps using controlled sources
5
3
Opamp circuits
7
4
Components and their models
10
5
Noise and mismatch
12
6
Miscellaneous
15
7
Opamp design at the transistor level
17
8
Oscillators
19
9
Bandgap reference
20
10 Low dropout regulator
21
11 Continuous-time filters
22
12 Switched-capacitor filters
24
13 Appreciating approximations
26
2
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 1
Negative feedback systems
vi
vo
ωu
s
+ Σ
-
Vi
Ve
f(Ve)
+ Σ
-
f()
e-sTd
Vo
β
(a)
Vi
Figure 1.1: Problem 1
1
f()
1+f’(0)β
1.
(a) Setup the differential equation for the system
above.
Vnl
Vi
(b) Vi is 1 V for a long time and changes to 0 V at
+
+ Σ
t = 0. What is the equation for t > 0?
Vo
(b)
A
Vo
(c)
(c) Assume that the solution is of the form
Vp exp(σt). Obtain the equation from which
Figure 1.2: Problem 2
you will determine σ (You are not required to
solve it).
(a) In each case, denote the transfer characteristic
of the overall system by g, i.e. Vo = g(Vi ) and
calculate the first three terms of the Taylor se-
(d) Express the above equation as f (σ) = 0.
Sketch f (σ). Determine the extremum of f (σ)
in terms of Td . For what value of Td does the
ries of g about the operating point of the circuit
in terms of f and its derivatives. Assume that
extremum become equal to zero?
f (0) = 0.
(e) Assume that the solution is of the form
Vp exp((σ + jω)t). Obtain the equations from
(b) Fig. 1.2(c) shows the linear small signal equivalent circuit from Vi to Vo with an additional
which you will determine σ and ω (You are not
required to solve them).
input Vnl . For the systems in Fig. 1.2(a) and
(c) Fig. 1.2(b), compute the small signal equivalent
gain A and the additional input Vnl . What do
(f) Reduce the above to a single equation in ω.
2. Fig. 1.2(a) shows a nonlinearity f enclosed in a
negative feedback loop with a feedback fraction β.
VFig. 1.2(b) shows a nonlinearity f preceded by an
you infer from the results?
3. Fig. 1.3(a) shows the amplifier studied in class.
Fig. 1.3(b) shows the same system with the input
attenuation factor.
applied at a different place. Calculate the dc gain,
3
Nagendra Krishnapura, Dept. of EE
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4
Vi
+ Σ
-
Ve
Vo
ωu dt
(k-1)R
Vf
R
(a)
+ Σ
-
Ve
Vo
ωu dt
(k-1)R
Vf
R
(b)
Vi
Figure 1.3: Problem 3
the -3dB bandwidth, and the gain bandwidth product of the system and compare them to the corresponding quantities in Fig. 1.3(b). Also compare
the loop gains. Remark on conventional wisdom
such as “constant gain bandwidth product”, “closed
loop bandwidth = unity gain frequency/closed loop
dc gain”. What is the reason for the discrepancy?
Draw an equivalent block diagram of Fig. 1.3(b) such
that the classical form of feedback (sensed error integrated to drive the output) is clearly obvious (Hint:
compute the error voltage Ve ).
4. The loop gain L(s) of a system with N extra poles is
given by
L(s) =
ωu,loop
1
PN
m
s
m=0 am s
a0 = 1. What does the loop gain step response (inverse laplace transform of L(s)/s) look
like after an initial transient period? Give your answer in terms of the poles of the additional factor (Hint: Split L(s) into a sum of two parts, one of
which is ωu,loop /s)
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 2
Opamps using controlled sources
Gf
Vi
Gi
case). Comment on the results.
voltage buffer
−
Vo
+
+
GL
Gm2vo1 Go2
1
+
vo1
-
vd
CL
Cc
Gm1vd Go1 Co1
-
out
(a)
current controlled current source
+
Ci1
Gm1 Go1 Co1 Cc
Rc
Gm2 Go2
ic
out
+
Gm1vd Go1 Co1
Gm2vo1 Go2
+
vo1
-
vd
-
Cc
ic
-
out
(b)
+
Figure 2.1: Problem 1
Gm2vo1 Go2
+
vo1
-
vd
-
1. Gm1 = 20µS, Go1 = 0.25µS, Gm2 = 80µS, Go2 =
Cc
Gm1vd Go1 Co1
out
(c)
2µS, GL = Gf = Gi = 4µS, Ci1 = 10 fF, Co1 =
40 fF, Cc = 250 fF, CL = 1 pF, Rc = 12.5 kΩ.
Figure 2.2: Problem 2
Determine the poles and zeros of the loop gain—
2. The circuits in Fig. 2.2(a, b) are modified
versions of the two stage miller compensated
Calculate them based on approximations discussed
in the class, and by calculating the Loop gain func-
opamp (Fig. 2.2(c)). Calculate their transfer functions and compare them to that of the conventional
tion symbolically and extracting the roots numerically. Comment on the accuracy of approximations.
structure. What is the difference? Explain the results.
Determine the closed loop transfer function and calculate its poles and zeros. How do these relate to
poles and zeros of the loop gain function.
3. Design a three stage opamp (Fig. 2.3(a)) using the
opamp in Fig. 2.1 as the “inner” opamp (Fig. 2.3(b)).
Plot the unit step response and the loop gain magnitude and phase response.
Change each (one at a time) of Cc , Rc , GL = Gf =
Exclude Rc , Ci1 , Gi , Gf , and GL from Fig. 2.1. Use
C3 = 1 pF. For the first stage of Fig. 2.3, use the
same values as in the first stage of Fig. 2.1. Deter-
Gi to 0.5× and 2× their nominal values. Plot the
unit step response and the loop gain magnitude and
mine the value of Cm1 to obtain a phase margin of
60◦ . What is the unity gain frequency of the three
phase response (overlaid on the same plot for each
stage opamp?
5
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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6
Cm1
components have their nominal values). Comment
on the results.
Cm2
Gm1
Vi
Ro1
C1
Gm2
Ro2
C2
Gm3
Ro3
Vf
5. Calculate the poles and zeros for each case in the
C3
+
Ve
-
+
Vo
-
above problem. How do they compare to the approximate expressions?
(a)
Cm1
Gm1
Vi
Vf
Ro1
C1
−
A(s)
+
+
Ve
-
Vo
(b)
Figure 2.3: Problem 3
Where are the poles and zeros? (Derive the expression assuming G1 = G2 = G3 = 0 and find the
roots exactly. Calculate the dc gain and unity gain
frequency separately). Comment on the location of
the zeros.
Connect a zero cancelling resistor in series with Cm2
such that the corresponding zero moves to infinity.
What is the phase margin?
With the zero cancelling resistor in place, adjust Cm1
such that the phase margin is 60◦ . What is the new
unity gain frequency?
Gs
Cgs Cgd g g
ds
m
CL
GL
+
Vo
-
+
Vs
-
Figure 2.4: Problem 4
4. Fig. 2.4 shows the small signal equivalent circuit of a
common source amplifier. gm = 100µS, gds = 1µS,
GL = 2µS, Gs = 1µS, Cgs = 0.1 pF, Cgd =
0.05 pF, CL = 0.5 pF. Plot the magnitude and phase
response of the circuit (overlaid) for the following
cases: a) Cgd = {0, 0.05, 0.1, 0.2, 0.4, 0.8, 1.6} pF,
b) CL = {0, 0.05, 0.1, 0.2, 0.4, 0.8, 1.6} pF, c)
GL = {0, 1, 2, 4, 8, 16} µS. (In each case all other
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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Topic 3
Opamp circuits
(k-1)R
Vi
kR
Rin
+
Rin
Rout
−
R
+
Rout
Vo
Vo
−
(k-1)R
−
R
Vo
+
Vi
Vi
(a)
−
R
Ii
Vo
R
+
(a)
Vi
(b)
Rin
+
(b)
vopa
−
Figure 3.1: Problem 1
Rout
1. Fig. 3.1(a) and Fig. 3.1(b) shows amplifiers which
Rin
+
vopa
−
+
− vopa
R
(c)
Io
Io
+
− vopa
Ii
load
Rout
(k-1)R
R
load
(d)
realize gains of k and −k respectively with ideal
opamps. Compare the following parameters of the Figure 3.2: Problem 2. (a) VCVS, (b) CCVS, (c) VCCS,
two circuits. Model the opamp as an integrator ωu /s. (d) CCCS
(a) Input impedance
3. Due to some parasitic effects, an opamp has a trans-
(b) Bandwidth
fer function with an extra pole p2 (ωu /s(1 + s/p2 )
instead of ωu /s). This is used to realize an amplifier with a closed loop dc gain k. Instead of
(c) Differential (V+ (s) − V− (s)) and common
mode ((V+ (s) + V− (s))/2) input voltages of
the opamps
the step response, the criterion here is the bandwidth. Find the conditions to maximize the band-
Assuming that the sign of the gain is unimportant
width without the closed loop gain increasing above
k for any frequency (This condition is known as max-
in your application, what would make you choose
one over the other? Is there any reason to choose
Fig. 3.1(b) at all?
imal flatness, and the mathematical condition is to
have dn /dω n |H(jω)|2 = 0, n = 1, 2, . . . for as large
2. Fig. 3.2 shows the four types of controlled sources
an n as possible). To avoid mess, assume a general
form of the second order transfer function, evaluate
the damping factor for maximal flatness, and substi-
using an opamp. Model the opamp as an integrator ωu /s. For each of these, calculate the transfer ratio (output/input), input impedance, and output
impedance at (a) dc, and (b) an arbitrary frequency
ω. For (b), set Rout = 0 when calculating the input
tute the values from the transfer function of the amplifier. How does it compare to a critically damped
system?
impedance and Rin = ∞ while calculating the output impedance. What happens to these three quanti-
4. Fig. 3.3 shows a transimpedance amplifier driven by
a photodiode. The photodiode can be modelled as
ties at high frequencies in each case?
7
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Nagendra Krishnapura (nagendra@iitm.ac.in)
8
Llarge
Cf
Clarge
Rf
−
Vi(jω)
−
+
Vi(jω)
+
Vo
−
Vo(jω)
Vo1(jω)
+
(a)
(b)
3
Ao=10
fu=100MHz
Figure 3.4: Problem 5: Measuring an opamp’s frequency
p2=400MHz
response
1pF
in Fig. 3.4(a), the opamp may not be biased cor-
Is
rectly. As you know, the opamp is biased correctly
only when there is dc negative feedback around the
Figure 3.3: Problem 4
a current source in parallel with a capacitor. The
opamp. A trick to maintain dc negative feedback,
but break the feedback loop for higher frequencies is
shown in Fig. 3.4(b). For frequencies where the volt-
opamp has Ao = 103 , ωu /2π = 100 MHz and
p2 /2π = 400 MHz (Make a model of the opamp us-
age drop across the capacitor and the current through
the inductor are negligible, the input voltage appears
ing controlled sources and passive elements. A parameterized macromodel of the opamp is very useful
directly across the opamp and there is no feedback.
Since this is a simulator, use comfortably large val-
for future circuit designs).
ues like Clarge = 1 F and Llarge = 1 kH.
(Don’t include Cf for this part) What is the largest
transimpedance Rf you can have without peaking in
What are the dc gain, unity gain frequency, and
the frequency response Vo /Is ? Show the ac magnitude response and the transient response to a current
nondominant pole(s)? Estimate these from magnitude/phase plots.
R2
step of 1/Rf Amperes with a 100 ps risetime?
Increase Rf by 20× and show the ac magnitude response step response (current step of 1/Rf Amperes.
Vi
R1
(a)
Vi=0 R1
R2
−
+
for maximal flatness. Calculate Cf for the increased
value of Rf and show the magnitude response and
Vtest(jω)
-L(jω)Vtest(jω)
the step response. What does the loop gain look like
for this circuit?
Calculate the expression for the “gain-bandwidth
Vo
+
Compare this to the earlier case and comment on the
results.
Calculate |Vo /Is | including Cf . Find the condition
−
(b)
R2
Vi=0 R1
−
product” with Cf (gain = Rf ).
+
(For analytical calculations of maximally flat magnitude response, it’ll be simpler to use an ideal integra-
-L(jω)Vtest(jω)
Llarge
Clarge
Vtest(jω)
(c)
tor model for the opamp, and then adjust the values
to account for the second pole).
Figure 3.5: Problem 6: Inverting amplifier
5. Simulate the open loop frequency response of the
opamp OPA656. If you try to measure it as given
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
6. Simulate the frequency response and the loop gain of
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an inverting amplifier (Fig. 3.5(a)) of gain 100. Loop
gain L(jω) can be determined by breaking the loop
as shown in Fig. 3.5(b). DC negative feedback has
to be maintained and the same trick as in the previous problem can be used (Fig. 3.5(c)). Do these
simulations for R1 = 100 Ω and R1 = 10k Ω. Do
the closed loop bandwidths match the unity loop gain
frequencies? Are the latter in turn consistent with the
opamp’s unity gain frequency evaluated in the previous experiment? Explain the results clearly.
7. Design inverting and non-inverting amplifers with
gains −5 and +5 respectively using the opamp
OPA656 and ±5 V supplies. Simulate these amplifiers with 10 MHz sinusoidal inputs of 400 mV peak.
Compute the distortion components upto the fifth
harmonic and compare the distortion performance of
the two amplifiers.
Plot the differential and common mode inputs of the
opamp in the two cases and explain the results using
the results from the previous problem.
When taking the DFT for distortion analysis, ensure that steady state is reached (wait
for a sufficiently long time before taking the
first point) and that you use an integer number of cycles to avoid spectral leakage (Refer to
http://www.ee.iitm.ac.in/∼nagendra/E6316/current/handouts.html
or the relevant lecture from EE658 at
http://www.ee.iitm.ac.in/∼nagendra/videolectures/)
OPA656
model
is
available
at
http://www.ee.iitm.ac.in/∼nagendra/cadinfo.html
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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Topic 4
Components and their models
Vbias
V0+vx/2
plate parasitic capacitance.
3. (Repeat this for nMOS and pMOS and compare the
results) Bias a transistor with VGS = VDS = 1.0 V
V0-vx/2
V0+vx/2
and determine W (with L = 0.18 µm) to get a current of 200 µA. Simulate SID the noise spectral density of drain current from 100 Hz to 100 MHz.
V0-vx/2
200kΩ
Double the length and resize W to get 200 µA, and
simulate SID . Repeat until L = 5.76 µm. Over-
Figure 4.1: Problem 1
1. (For this problem, The minimum usable dimension
is 0.3 µm.) A MOSFET is used as a 200 kΩ re-
lay the spectral density plots (log y axis) and identify the 1/f noise corners. Plot the 1/f noise corners vs. L. Briefly explain the results. Plot ID vs.
sistor (Fig. 4.1) V0 = 0.5 V and vx is restricted to
0.25 V. The nonlinear part of the current (Difference
VDS (0 to 1.8 V) for VGS from 0 to 1.5 V in steps of
0.25 V and VBS = 0 V. Overlay the plots for W/L =
between the exact expression and its linear approximation) in the resistor should be at most 5%. Cal-
3.6 µm/0.36 µm and W/L = 36 µm/3.6 µm. Comment on the results.
culate the gate bias Vbias and the dimensions of the
transistor. If a linear resistive material with a sheet
resistance of 10 Ω/sq. is available, what would be
4. Plot ID vs. VDS (0 to 1.8 V) for VBS from -1 V to
its dimensions? What is the motivation for using a
transistor instead of a resistive material?
2. Design a 2 pF capacitor using A square nMOS de-
0 V in steps of 0.25 V and VGS = 1.5 V. Overlay
the plots for W/L = 3.6 µm/0.36 µm and W/L =
36 µm/3.6 µm. Comment on the results.
5. Plot (log-log) ID vs.
VGS (18 mV to 1.8 V) for
vice (drain/source shorted). Plot its capacitance as
a function of voltage (0 to 1.8 V). What is the usable
VDS = 1 V and VBS = 0 V. Overlay the
plots for W/L = 3.6 µm/0.36 µm and W/L =
voltage range of this capacitor? (For this problem use
the process information given in the cadinfo page).
36 µm/3.6 µm and temperatures of {0, 27, 100}◦ C.
Comment on the results. Calculate the subthreshold
Repeat the above for a square pMOS device.
slope η. The current in a MOS transistor in the subthreshold region is proportional to exp(VGS /ηVt )
A square Metal1-Metal2 structure.
where Vt is the thermal voltage.
A square sandwiched structure with poly, M2, M4
tied together and M1, M3, M5 tied together.
For the last two structures, determine the bottom
6. Plot (log-log) ID vs. VBS (-1.5 V to -15 mV) for
VDS = 1 V and VGS = 1 V. Overlay the
plots for W/L = 3.6 µm/0.36 µm and W/L =
10
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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36 µm/3.6 µm and temperatures of {0, 27, 100}◦ C.
Comment on the results.
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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Topic 5
Noise and mismatch
Rf
Iin
referred and output noise spectral densities. How
−
does the simulated noise compare to analytical calculations? What fraction of noise is contributed by
Rf ? (The relative contribution of different compo-
Vo
+
nents can be printed out in the simulator)
Sv,opa
R
Rf/k
+
R
+
Vpcos(ωt)
-
Iin
R/k
−
−
+
Vo
+
Sv,opa
Figure 5.2: Problem 3
Sv,opa
3. The filter in Fig. 5.2 is driven by a sinusoid at ω =
Rf
Iin
Rx
C vo
-
1/RC. Calculate the output noise voltage, output
signal to noise ratio (ratio of mean squared signal to
−
Vo
mean squared noise voltages), and the power dissipated in the circuit. If the impedances of all com-
+
ponents are scaled up by a factor α, what happens
to the transfer function of the circuit, output noise
voltage, output signal to noise ratio, and the power
Sv,opa
dissipation?
Figure 5.1: Problem 1
Derive a relationship between the signal to noise ra1. Determine the output noise spectral density and input
referred (current) noise spectral density of the transimpedance amplifiers in Fig. 5.1. The opamp has
an input referred voltage noise spectral density of
tio, power dissipation, and the bandwidth of the circuit (in Hz). What tradeoffs does this relationship
represent?
4. Determine the rms signal, rms noise, signal to noise
Sv,opa V2 /Hz and is otherwise ideal.
2. Design a transimpedance amplifier with a gain of
10 kΩ and the highest possible bandwidth without
peaking using an OPA656 opamp. The photodiode
has a 5 pF capacitance. Simulate the frequency response, step response (100 µA step input), and input
ratio (as a ratio of mean squared quantities) at the
output of Fig. 5.3. Assume an low frequency input.
What is the amplifier’s transfer function? The opamp
can be either (i) class A (Fig. 5.3(b)): In this case
a constant current Ibias , equal to the highest possible output current) is drawn from the amplifier; or
12
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
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Analog Integrated Circuit Design; Assignment problem set
13
C
Vdd
Ibias=max(iout)
R
R/k
−
+
Vi=(Vpp/2k)cos(ωt)
+Vs/2
−
Vo
+
-
+
-Vs/2
+Vs/2
Ibias
I0/n
I0/n
Iout
iout
-Vs/2
W/4n
M
L 2
W
M4
W/n
(a)
(b)
+Vs/2
W
M1
M3
L
gm4, gds4
gm3, gds3
+Vs/2
−
+
L
L
−
Figure 5.5: Problem 6
+
-Vs/2
-Vs/2
(c)
and ∆β24 , ∆VT 24 between M2 and M4 ). Which of
the mismatches is more critical?
Figure 5.3: Problem 4
Vdd=1.8V
(ii) class B (Fig. 5.3(c)): In this case, currents out of
M1
M2 100µA
the opamp are drawn from the positive supply and
currents into the opamp are pushed into the negative
max. voltage=1.1V
supply. In each case, calculate the power dissipation.
Relate the power dissipation to amplifier specifications: gain, bandwidth, and signal to noise ratio.
I0/n
1µA
(a)
Vdd=1.8V
M1
M2 100µA
Iout
W/n
W
L
L
+
−
Vout
M3
M4
Vbiasp2
max. voltage=1.1V
1µA
Figure 5.4: Problem 5
5. You are required to design a current mirror that can
operate with an output voltage Vout (Fig. 5.4). The
total current drawn from the supply must be Itot . Determine W/L and n which will maximize the ratio of
signal (load current) to noise (rms current in a bandwidth fB )? Consider only the thermal noise spectral
density. Think about why this value of n is optimal
for signal to noise ratio.
6. Determine the output current in Fig. 5.5. Determine
the output noise current in terms of small signal parameters of M3 and M4 . Which of the devices primarily contribute to the noise? Determine the output current error due to current factor and threshold mismatches (∆β13 , ∆VT 13 between M1 and M3 ,
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
(b)
Figure 5.6: Problem 7
7. Fig. 5.6 shows a simple current mirror and a cascode
current mirror delivering 100 µA from a 10 µA reference. The maximum voltage at the output can be
1.1 V.
(a) Design the simple mirror with L = 2 µm.
(b) Design the cascode current mirror for the same
output voltage constraint with L = 2 µm for M1,2 .
Choose M3,4 as you wish subject to the constraints
that the output impedance should be as high as possible at all frequencies and that the output thermal
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14
noise spectral density should not increase by more
than 3 dB when compared to the simple current mirror. Provide an arrangement to generate Vbiasp2 .
Plot the output impedance and output current noise
spectral density in for the two mirrors (Terminate the
output with a 1.1 V dc source). What is the relative
noise contribution from different devices? Plot the
dc output current as the output voltage is varied from
0 to 1.8 V.
ID-∆ID/2
ID+∆ID/2
+ Vout
−
+ Vout
−
+
VGS
-
Figure 5.7: Problem 8
8. Two transistors carrying a current ID are required to
have a current mismatch ≤ σID and operate in saturation with an output voltage Vout (Fig. 5.7). Compute the transistor dimensions and its fT in terms of
the mismatch constants AV T and Aβ , ID , σID and
Vout . Comment on the tradeoffs implied by this relationship.
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 6
Miscellaneous
1. The circuit in Fig. 6.1(b) is the miller equivalent
of Fig. 6.1(a). Determine the transfer functions of
Fig. 6.1(a) and Fig. 6.1(b)? Are they the same?
Determine the transfer function of Fig. 6.1(c). Replace Fig. 6.1(c) by its miller equivalent Fig. 6.1(d)
and determine its transfer function. Are the results
the same? If not, what are the differences and why?
Carry out this exercise by first omitting Cgs and CL ,
and then including them in the analysis.
Vdd
RL
Vdd
RL
Vo
Vdd
Vi
I0
Vo
Vo
Vi
VG
Rs
Vi
I0
(a)
I0
(b)
I0
(c)
Figure 6.2: Problem 2
2. Determine the spectral density of output noise voltage and input referred noise voltage of the stages in
Fig. 6.2.
15
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Analog Integrated Circuit Design
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Nagendra Krishnapura (nagendra@iitm.ac.in)
16
C
Rs
Rs C(1+A)
-A
+
Vi
-
+
Vo
-
C(1+1/A)
-A
+
Vi
-
Co
C
Rs
+
Vi
-
+
Vo
-
CL
Cgs
+
Vo
-
Rs
+
Vo
-
Ci
+
Vi
-
gm, gds
gm, gds
Figure 6.1: Problem 1
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 7
Opamp design at the transistor level
Vdd
M3
Vdd
M4
M3
vo
Vcm+vcm
Vdd
M4
R1
vo
Vcm+vcm
M1
M2
M1
R2
Vcm+vi/2
M2
Vcm-vi/2
vx
I0
I0/2
I0
gds0
gds0
(a)
(b)
Figure 7.3: Problem 3
Figure 7.1: Problem 1
3. Calculate the small signal tail node voltage vx in
Fig. 7.3. vi is a small signal increment. The tran-
1. The common mode gain of a differential amplifier is
measured by applying a small signal common mode
input vcm as shown in Fig. 7.1. Fig. 7.1(a) has a cur-
sistors can be modeled using gm and gds .
rent mirror load and Fig. 7.1(b) has a current source
load which is independently biased. What is the
common mode gain of these two configurations? Ex-
4.
φ2
press the answer in terms of the small signal parameters of: M0 (gm0 , gds0 ), M1,2 (gm0 , gds1 = ∞),
M3,4 (gm3 , gds3 )
I0/2
Vcm+vi/2
φ1
Vcm ± vi /2
I0/2
- vo +
φ1
Vcm+vi/2
φ2
8 pF
φ2
+v
o
-
vi = Vip cos(2πfin t)
I0
φ1
φ2
(b)
(a)
Vcm
φ1
Vcm-vi/2
I0
8 pF
+
vout
−
8 pF
I0/2
Vcm-vi/2
φ2
8 pF
0
1/fs
t
Figure 7.2: Problem 2
Figure 7.4: Problem 5: Sample and hold circuit
2. Determine the small signal dc gains of the two amplifiers in Fig. 7.2. The transistors can be modeled
using gm and gds . Explain the results.
5. Sample and hold: Design the sample and hold circuit in Fig. 7.4 using the fully differential folded cas-
17
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Nagendra Krishnapura (nagendra@iitm.ac.in)
18
code opamp designed above. Use ideal switches with
1 kΩ on resistance. Use fs = 4 MHz and fin =
{1/4, 9/4} MHz (sinusoidal input with 1.6 Vppd1
amplitude) and plot the output waveforms. Provide a
plot that shows the settling behavior of the opamp.
10 kΩ
8 pF
10 kΩ
vip
10 kΩ
Vcm
vin
10 kΩ
10 kΩ
8 pF
Vcm = 0.9 V
10 kΩ
vip
Vcm
Vcm
vin
differential step
vip , vin
common mode step
Figure 7.5: Problem 6: Inverting amplifier
6. Inverting amplifier: Design the inverting amplifier
in Fig. 7.5 using the fully differential two stage amplifier designed above. Show the output waveforms
for a 1 V differential step and a 0.5 V common mode
step.
1 Vppd:
volts, peak-peak differential
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 8
Oscillators
Vdd
Rs
L
Vdd
Rs
Rdiff
Rs
L
vom
C
vop
there anything special about it? Model the transistor
using only its gm .
Rs
ix
L
L
vom
Vdd
C
vx
vop
L
ix
I0
Rs
I0
(a)
C1
C2
I0
(b)
Figure 8.3: Problem 3
Figure 8.1: Problem 1
3. Calculate the small signal impedance vx /ix . What is
1. Calculate the current flowing in each transistor in
Fig. 8.1(a) in the quiescent condition. Calculate the
small signal differential resistance Rout looking into
the condition for this to be infinity? What is the frequency at which this happens? Model the transistor
using only its gm .
the drains of the two transistors.
N inverters(N: odd)
In Fig. 8.1(b), calculate (vop − vom )/ix . What is
the condition for this to be infinity? What is the frequency at which this happens?
Vdd
Figure 8.4: Problem 4
Zin
C1
C2
4. In Fig. 8.4, assume that all nodes are at the self bias
voltage of the inverter. Model the small signal gain
of each inverter as A0 /(1 + s/p1 ) and calculate the
I0
condition for instability (i.e. when the loop gain becomes −1). Hint: Among the roots of −1, pick the
one which satisfies the above for the lowest value of
Figure 8.2: Problem 2
2. Calculate the input impedance Zin in Fig. 8.2. Is
A0 .
19
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 9
Bandgap reference
Bias a 1x sized diode connected PNP1 at 5 µA as shown
in Fig. 9.1(a) and sweep the temperature from 0 to 100◦ C.
Vdd
Determine dVBE /dT at 27 C.
Design the bandgap shown in Fig. 9.1(c). Choose R1 for
a quiescent current of 5 µA and R2 to get zero tempera-
C1
◦
5µA
+
Vbe
ture coefficient at Vbg . Choose R3 = R2 . What is the role
of R3 ? Simulate the bandgap reference with the model
-
adjust R1,C1,2 and gm1 to model the pole-zero doublet
(a)
and the transconductance of the single stage opamp
(b)
Vdd
W/L
W/L
Vbg
R2
−
Vx
Vy
+
Cc
R1
and Vbg change? What is the purpose of this modification? Resimulate with the opamp model as before and
output.
model of the
single stage opamp
R3
of R1 , R2 , R3 (= R2 ) if necessary to get zero TC at 27◦ C.
Modify the circuit as in Fig. 9.1(d). How should Vx , Vy ,
vious assignment and simulate the temperature sensitivity
of Vbg and the transient response to a current step at the
C2
1.0
model of the single stage opamp
and plot Vbg . Test the transient response by applying a
1 uA pulse to the output of the opamp. Adjust the values
test the temperature sensitivity, transient response and the
loop gain.
Substitute the differential pair opamp designed in the pre-
+
−
R1
-
of a single stage opamp assuming that the single stage
opamp is made like the first stage of the previous problem. (Fig. 9.1(b)-model the gm, and the pole zero doublet). Choose Cc for ringing ≤ 10%. Test the bandgap
reference by sweeping the temperature from 0 to 100◦ C
gm1
1GΩ
+
1x
1µA pulse for
transient test
8x
(c)
Vdd
W/L
W/L
single stage opamp
Vz or Vx
Vbg
Vz
−
Vw or Vy
Vw
+
R2
R3
Vx
Cc
1µA pulse for
transient test
Vy
R1
connect the opamp inputs to
1x
8x
Vx and Vy OR Vz and Vw depending on the
input common mode range
(d)
Figure 9.1: Bandgap reference
1 Use
the model ideal pnp in ideal diode.lib
20
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 10
Low dropout regulator
Vdd
Bandgap
reference
1.2V
+
output voltage is constant over time. These are departures from conventional amplifiers.
Isup
IL
Vout
−
Fig. 10.1(b) shows a “pass transistor” M1 enclosed in a
Load
feedback loop. For simplicity, a unity gain case is shown.
M1 should have a high enough W/L to remain in saturation with the desired dropout and the highest output
R2
R1
(a)
Bandgap
reference
1.2V
current. Miller compensation around M1 is usually not
used because it severely compromises power supply re-
Vdd
jection (Incremental voltage gain from Vdd to the output
voltage).
−
M1
+
Vout
20µA
IL
Load
CL
(b)
Use the model in Fig. 9.1(b) for the single stage opamp.
Use a 50 µA quiescent current in M1 . Adjust the
width (with minimum length) of M1 for a dropout of
300 mV with a 50 mA current. You can use a 1.2V voltage
source in place of the bandgap reference. Compensate the
Zout
loop using a load capacitor CL for a phase margin of 45◦
at IL = 0 and IL = 50 mA and choose the higher one. Do
Figure 10.1: Low dropout regulator
A voltage regulator is nothing but a noninverting amplifier the following (except the last one) for two cases (IL = 0
whose input is the bandgap voltage from a reference. In and IL = 50 mA—you can use a current source for the
Fig. 10.1(a), the output voltage is (R2 /R1 )Vbg . By mak- load):
ing R2 variable, one can get a variable voltage output.
1. Vary Vdd from 1.4 V to 1.8 V and plot Vout
• The output impedance should be very low: This is
accomplished by realizing a very high loop gain over
as wide a bandwidth as possible.
• The efficiency ((Vout IL )/(Vdd Isup )) should be very
2. Plot Zout from 1 kHz to 10 MHz
3. Plot the transfer function from Vdd to Vout from
1 kHz to 10 MHz
high: For this, the current Isup −IL consumed by the
circuit should be minimized (This makes it hard to
4. Plot the small signal step response for a 10 µA step
in the output current
satisfy the previous condition). The “dropout” Vdd −
Vout should be minimized.
5. Plot the large signal step response (IL switching
from zero to 50 mA and 50 mA to zero)
• Usually only a positive IL needs to be driven. The
21
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 11
Continuous-time filters
(a) Compute the transfer functions Vo /Vi in terms
(b) Using the parameterized subcircuits for the bi-
of the parameters (Q, ωp , b0 , b1 , b2 ) for the circuits in Fig. 11.1(a, b).
(b) Turn these circuits into parameterized subcir-
linear and the biquadratic filters, simulate the
four filters (using the cascade structure) in a circuit simulator. Use the rules of cascading dis-
cuits “bilinear” and “biquad” in a circuit simulator1 with the required parameters. You can
cussed in the lectures. Clearly state the order of
cascade and the pole zero pairing.
then use these subcircuits to realize ideal cascade realizations of any transfer function.
For the Bessel filter, simulate the frequency
1.
0dB
0dB
-1dB
-1dB
-40dB
-40dB
response of the prototype (last column of Table 11.1). If this filter were scaled such that it
had an attenuation As = 40 dB at 2 MHz (the
stopband edge), what would be its attenuation
at the passband edge (1 MHz)?2 Does it meet
the specs in Fig. 11.2(a)?
(a)
2 rad/s
1 rad/s
2MHz
1MHz
Now simulate the scaled Bessel filter.
Plot their magnitude and phase responses3 , and
the group delay.
(b)
(c) For each filter, determine the maximum transfer function magnitude from the input to each
of the stage (first or second order) outputs. If
each output were limited to 1 V, what is the
Figure 11.2: Problem 2
2. You are required to realize a filter that meets
the specifications shown in Fig. 11.2(a). You
maximum input voltage that could be applied
to each without having distortion?
are given (Table 11.1) the poles and zeros of 4
types (Excluding Bessel) of filters which satisfy the
(d) For each of the 5 filters list the maximum
quality factor of the biquad stages used, the
maximum resonant frequency, and the maxi-
prototype specifications in Fig. 11.2(b).
(a) Tabulate the order, the resonance frequencies,
the quality factors of the poles, and the location of transmission zeros (if present) of the different types of filters that satisfy the specs. in
Fig. 11.2(a).
1 In
circuit simulators, to realize a current controlled voltage source,
you also usually need to have a 0 V voltage source through which the
desired current is flowing.
2 You don’t need to rescale the filter and simulate. You should be able
to answer this by looking at the prototype response.
3 Plot the magnitude responses of the 5 filters in the same plot; same
for the phase response and the group delay. Plot the magnitude response (in dB) twice—once showing the whole picture and once zoomed
in on the passband. Use sensible scales so that the details of the response
can be seen. e.g. with notches, the response goes down to −∞ dB and
the default scale may be totally unsuitable.
22
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
+
Vi
-
+
C=1/ωp R=1
+
-
b1IC
+
-
b0IR
Vo
Vi
IC
23
transconductance=1S
transconductance=1S
Analog Integrated Circuit Design; Assignment problem set
+
Vi
-
-
+
C=1/ωp R=Q
L=1/ωp
Vi
IR
IC
(a)
IR
+
-
b2IC
+
-
b1IR Vo
+
-
b0IL
-
IL
(b)
"bilinear"
"biquad"
Figure 11.1: Problem 1
mum group delay variation in the passband (<
1 MHz). This gives you a comparison of different types of filters that are designed to meet a
given specification (Fig. 11.2).
Table 11.1: Prototype zeros and poles
Butterworth
Chebyshev
Inverse Chebyshev
Elliptic
Bessel
poles
poles
zeros
poles
zeros
poles
poles
−1.1031 ± j0.2194
−0.0895 ± j0.9901
±j3.0671
−0.2811 ± j1.1013
±j3.5251
−0.3643 ± j0.4786
−0.3868 ± j1.0991
−0.9351 ± j0.6248
−0.2342 ± j0.6119
±j1.8956
−0.9461 ± j0.8751
±j1.6095
−0.1053 ± j0.9937
−0.6127 ± j0.8548
−0.6248 ± j0.9351
−0.2895
−1.4202
−0.2194 ± j1.1031
−0.7547 ± j0.6319
−0.8453 ± j0.4179
−0.8964 ± j0.2080
−0.9129
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 12
Switched-capacitor filters
1. A continous time first order filter has a transfer function
Hc (s) =
3. Simulate each of the filters designed in problem 3
in a circuit simulator. Plot the magnitude and phase
responses.
2
1 + s/ωp
4. A second order filter has a transfer function has the
form
ωp = 2π × 20 krad/s. Transform Hc (s) into discrete
time transfer functions Hd (z) using bilinear transformation. The sampling frequency fs = 1 MHz.
Plot the magnitude and phase responses of Hc and
H(s) =
N (s)
1 + (s/Qp ωp ) + (s/ωp )2
Hd with the real frequency (Hz) from 0 to 1 MHz
along the x axis. Are the magnitude and phase responses the same for all the cases? Comment on the
(a) What is N(s) for lowpass, bandpass, highpass,
results.
(b) Transform each of these into a discrete time fil-
and band stop filters? (In each case, assume that
the gain in the center of the passband is unity)
ter using bilinear transformation. Assume that
Qp = 4 and ωp = fs /10, where fs is the sam-
Repeat for ωp = 2π × 200 krad/s.
2. Design the above filter (Hc (s) or Hd (z)) as
pling frequency.
(a) a continuous time opamp-RC filter
(b) bilinear transformed switched capacitor filter (for this, assume that both the input Vi and
its inverted form −Vi are available)
(c) switched capacitor version of a) with the resistor replaced by a switched capacitor
(c) Sketch the pole zero plots of the continumous
time filters and their discrete time counterparts.
5. Compute the transfer function V1 /Vi in the Fleischer
Laker biquad. Fig. 12.1. The output is defined at the
end of φ1 and the input Vi changes on the rising edge
of φ2 .
(d) Noninverting delayed switched capacitor integrator whose magnitude response is equal to
6. Transform a second order CT (continuous time)
that of the bilinear transformed filter at dc and
the 3 dB frequency (i.e., the pole of the SC in-
bandpass filter into a DT(discrete time) bandpass filter using bilinear transformation. The gain at center
tegrator should be adjusted such that its -3dB
frequency is the same as that of the LDI trans-
frequency and the quality factor of the CT prototype
are both 10. The resonant frequency fp (in Hz) is
20% of the sampling frequency fs (in Hz).
formed filter).
Do it for both ωp = 2π × 20 krad/sand ωp = 2π ×
200 krad/s. In each case, give the schematic and the
component values.
7. Compute the values of the capacitors in the
Fleischer-Laker biquad to realize the above filter.
Assume B = D = 1 and A = C. Also, usually,
24
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Analog Integrated Circuit Design; Assignment problem set
25
E
φ2
φ2
C
φ1
φ1
φ2
D
Vi
φ1
φ2
G
φ2
φ1
φ1
φ2
φ1
φ2
φ1
φ2
φ2
φ1
φ1
B
n+2
φ2
n+1
φ1
A
+
φ1
H
φ1
n
φ2
V1
n-1
φ2
−
φ2
F
φ2
φ1
I
φ2
−
φ1
φ2
φ2
φ1
J
φ1
+
φ2
V2
Vi[n-1]
Vi[n]
Vi[n+1]
Vi[n+2]
V1[n-1]
V1[n]
V1[n+1]
V1[n+2]
V2[n-1]
V2[n]
V2[n+1]
V2[n+2]
(input)
φ1
Figure 12.1: Problem 5
you can set one of G, H, I, J to zero. Try each of the
following cases
(a) V1 as output; F circuit (E = 0)
(b) V1 as output; E circuit (F = 0)
(c) V2 as output; F circuit (E = 0)
(d) V2 as output; E circuit (F = 0)
What is the spread in capacitor values (The ratio of
the largest to the smallest capacitor) in each case?
8. Simulate the magnitude and phase responses of
the first case above in a circuit simulator. See
the handout below on guidelines to simulating switched capacitor filters in a circuit simulator:
http://www.ee.iitm.ac.in/∼nagendra/E4215/2004/handouts/scfsim.pdf
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL
Topic 13
Appreciating approximations
Approximations are key to understanding anything complicated. Exact expressions, even when possible, may be
too complicated to give any insight to the problem. Approximating is not the same as being sloppy. On the contrary, a greater understanding of the problem is required
to judiciously use approximations than plug in the whole
formula (e.g. see the quadratic eq. example below).
Evaluate the conditions for 1% and 10% accuracy for the
quantities mentioned using the approximations below.
√
1. You are required to calculate 1 + x and you approximate it by 1 + x/2.
2. You are required to solve the quadratic equation
ax2 +bx+c and you approximate the roots by −b/a,
−c/b. This works for widely separated real roots.
How widely do they have to be separated (ratio)?
3. You have a two stage amplifier in feedback loop with
loop gain L(s) = A0,loop /(1 + s/p1 )(1 + s/p2 ),
p2 > p1 , p1 = ωu,loop /A0,loop and you approximate
it by moving the lower frequency pole to the origin—
i.e. use the transfer function L(s) ≈ (ωu,loop /s)(1 +
s/p2 ) instead. You have to calculate (a) natural frequency ωn , (b) damping factor ζ. Compare the expressions for the two quantities. Calculate A0,loop to
get the above errors (Assume p2 = 2ωu,loop ).
The above are rather simple examples to show how much
you can get away with, if you use judicious approximations. See the book below for an extensive treatment of
approximation techniques.
Sanjoy Mahajan, Street-Fighting Mathematics: The Art of
Educated Guessing and Opportunistic Problem Solving,
The MIT Press, 2010.
26
Nagendra Krishnapura, Dept. of EE
Indian Institute of Technology, Madras
Analog Integrated Circuit Design
A video course under the NPTEL