MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.301 Solid State Circuits
Final Exam
December 15, 2008
180 minutes
1. This examination consists of four problems. Work all problems.
2. This examination is closed book.
3. Please summarize your solutions in the spaces provide in this examination packet. Draw
all sketches neatly and clearly where requested. Remember to label ALL important
features of any sketches.
4. All problems have equal weight.
5. Make sure that your name is on this packet and on each examination booklet.
Good luck.
General equations, worst case OCT:
R2Ο
R3
R1
rπ
va
gmva
R1Ο
R2
R10
�
� (R1 + R2 )
= rπ �
�
1 + gm R2
R20 = R1 �[rπ + (β + 1)R2 ] +R3 +
�
��
�
R�
2
gm rπ R� R3
rπ +(β+1)R2
Problem 1 OCT’s and transfer functions
Consider the following amplifier:
Q4
VB
ro/2
Vout
Q3
ro/2
Co
Vcasc
Q2
Q1
Cin
iin
• β→∞
• Cπ = 0
• Cµ = 100fF
• Co = Cin = 1pF
• Assume Ic1 = Ic2 = Ic3 = Ic4
• Do not ignore the Early effect.
(a) Solve for the following transfer functions
•
vout
iin
•
vf
vout
• vin (vf , iin )
(b) Approximate the 3dB frequency (f3dB ) using the method of open circuit time constants. Make reason­
able approximations.
3
Problem 2 Op amps
A three-stage BJT op-amp is shown below. Assume VBE,on = 0.7V, β = 100, VA = 2.5V,
Q9
100�A
Vi-
Q2
Q1
= 25mV.
Q11
Q 10
Vi+
kT
q
Q3
Q4
Q5
Q6
st
1 stage
Vo
C1
Q7
Q8
7k�
2
nd
stage
rd
3 stage
(a) Calculate all collector currents
(b) Calculate the low-frequency gain of each stage. Include rπ ’s and ro ’s in your calculation.
(c) Explain the function of the 2nd stage.
(d) The dynamics of the amplifier are dominated by C1 . Estimate the location of the lowest frequency pole
in terms of C1 and quantities involved in part (b).
4
Problem 3 Op Amp applications
Note: For parts (a) and (b) you may assume the op amps and diodes are ideal.
(a) Derive the voltage vo as a function of vI and vA in the following circuit
R/2
R
vA
vI
+
R
vO
(b) Derive vA as a function of vI in the following circuit when:
1. vI > 0
2. vI < 0
R
R
+
vI
vA
(c) What function is implemented by the following circuit? Give both a qualitative and quantitative ( vvoI )
answer.
R
R
R
+
vI
R/2
+
vO
R
(d) Now consider the diodes to be nonideal (i.e., they have a forward drop of 0.6V when conducting). What
is the output voltage vA of the circuit in part (b) under this condition?
R
R
vI
+
5
vA
Problem 4 Charge Control
An inverter circuit is shown below
+10V
SW2
SW1 R
100�
vO
ic
vI
20pF
The transistor has negligible space-charge layer capacitances. In this case, the charge control equations
are
iC
=
iB
=
iE
=
� 1
qF
1 � dqR
− qR
+
−
τF
τR
τBR
dt
qF
dqF
qR
dqR
+
+
+
τBF
dt
τBR
dt
� 1
qF
1 � dqF
− qF
+
−
τBF
τF
τBF
dt
When the transistor is in saturation
iB − iBo =
qS
dqS
+
τS
dt
The transistor parameters are
τF = 1ns
βF = 100
τBF = 100ns
τR = 2ns
βR = 5
τBR = 10ns
τS = 15.1ns
(a) The circuit is initially configured with switch 1 open and switch 2 closed. Input voltage vI (t) is a 10.7
volt step and the initial value of the capacitor voltage is zero. Find iC (t) valid for all time slightly
greater than zero.
(b) Switch 1 is now closed. Other conditions remain the same as for part (a). Find the value of R that
makes iC a constant for all time slightly greater than zero.
(c) Switches 1 and 2 are now both opened. With vI (t) a 10.7V step and zero initial capacitor voltage, the
voltage vo (t) is given by the following graph
6
fa
st
10V
vO(t)
10V
τ1
t
TD
What are the values of delay time TD and τ1 ? You may express TD in terms of logarithms if you wish.
(d) The circuit is now operated with switch 1 closed and switch 2 open, R = 2.5kΩ, and vI a constant 10.7
volts. After equilibrium is reached switch 2 is closed. What is the value of iC slightly after the switch
is closed? What is the final value of iC ?
7
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6.301 Solid-State Circuits
Fall 2010
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