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Solution: R-2R Ladder
Vout  
D D D D 
Vref  0  1  2  3 
R
4
2 
 16 8
Rf
Vout
• Using just R and 2R removes the
problems of the weighted adder
• R values are reasonable
• Precision requirements can be met
• RC time constants are small
These are digital switches
R-2R Ladder: Thevenin analysis
Vout
 D0 D1 D2 D3 
  Vref 


 
R
4
2 
 16 8
Rf
How do we end up with these weights? Apply Thevenin's...
Announcements
• Assignment 8 posted
– Due Tomorrow. A bit longer than others.
• Project progress?
• Dates
– Tuesday 12/6 project demonstrations in the lab
(no presentations)
– Sunday 12/11 project reports due to me by
email
– Tuesday 12/13 final exam, 1pm-3pm here.
Lecture 25: Review
Exam preparation – use assignments, go over the mid-term (solutions
posted on Assignments page). Use this review. Practice with EWB
Can’t review everything – focus on some older stuff today
• Simple circuits: equivalent circuits (work for both AC & DC)
• KCL KVL; mesh or nodal analysis. Works for AC & DC
• Thevenin & Norton circuits
• Transient circuits; capacitors & inductors
• These notes contain (much) more, and will be posted on the web
Simple circuits
• Voltage/Current sources: provide prescribed voltage/current regardless of
load
• Kirchoff's current law: The sum of currents into a node=0
• Kirchoff's Voltage Law: The sum of voltages round a closed loop=0
•Voltage divider:
R
v1  1 vtotal
REQ
•Current divider
i1 
REQ
R1
iS
Circuit analysis method 1:
Apply element combination rules
Series resistors
Parallel resistors
Series voltage sources
Parallel current sources
Mesh Analysis
Example: 2 meshes
(Mesh is a loop that does not contain other loops)
Step 1: Assign mesh currents clockwise
Step 2: Apply KVL to each mesh
•
The self-resistance is the effective resistance of the resistors in series within
a mesh. The mutual resistance is the resistance that the mesh has in common
with the neighbouring mesh
• To write the mesh equation, evaluate the self-resistance, then multiply by the
mesh current
• Next, subtract the mutual resistance multiplied by the current in the
neighbouring mesh for each neighbour.
• Equate the above result to the driving voltage: taken to be positive if it tends
to push current in the same direction as the assigned mesh current
Mesh1: (R1+R2)I1
Mesh2:
- R2I2
=ε1-ε2
-R2I1+ (R2+R3)I2
=ε2-ε3
Step 3: solve currents: use substitution or Cramer's rule
Cramer's Rule
Step 3: solve currents: use substitution or Cramer's rule
PRACTICE!!! http://sole.ooz.ie/en
I - I2 - I3 = 0
4I + 5I2 + 0I3= 3
0I - 5I2 + 10I3= 0
Mesh Analysis with 3 loops
Mesh analysis with a current source
Magnitude of current in mesh containing current source is IS ,
(although if the current flow is opposite to the assigned current
direction the value will be negative).
This works only if the current source is not shared by any other mesh
For a shared current source, label it with an unknown voltage.
Example
• In this circuit, find the value of Is that will reduce the voltage across the 4Ω
resistor to zero.
Example
• In this circuit, find the value of Is that will reduce the voltage across the 4Ω
resistor to zero.
Mesh equation:
when 4Ω voltage=0:
• What if the 2 Ω and the 6 Ω resistors are swapped?
Example II
• Which of the two circuits has the larger terminal voltage, A or B?
• Which has the larger current through the 9V battery?
• Practical batteries are modelled as voltage sources in series with a resistor.
Example II
• Which of the two circuits has the larger terminal voltage, A or B?
• Which has the larger current through the 9V battery?
• Practical batteries are modelled as voltage sources in series with a resistor.
Mesh equations:
i1= current through 9V battery
solve to give i1=0.41A
Mesh equations:
i1 = current through 9V battery
solve to give i1= -0.56A
Example II
• Which of the two circuits has the larger terminal voltage, A or B?
• Which has the larger current through the 9V battery?
• Practical batteries are modelled as voltage sources in series with a resistor.
x
x
+
+
VR
-
VR
- i1
+
i1
+
y
i1= current through battery
solve to give i1=0.41A
y
i1 = current through battery
solve to give i1= -0.56A
Thevenin and Norton Equivalent Circuits
load
Any network of sources
and resistors will appear
to the circuit connected
to it as a single voltage
source and a series
resistance
vTH= open circuit voltage at
terminal (a.k.a. port)
load
RTH= Resistance of the
network as seen from port
(Vm’s, In’s set to zero)
Thevenin and Norton Equivalent Circuits
Any network of sources
and resistors will appear
to the circuit connected
to it as a single current
source and a parallel
resistance
How do we calculate RT, VT, iN, RN ?
Calculation of RT and RN
• RT=RN ; same calculation
• Set all sources to zero (‘kill’ the sources)
– Short voltage sources
– Open Current sources
• Calculate equivalent resistance seen by the load
Calculation of VT
• Remove the load and calculate the open circuit voltage
• The Thevenin equivalent is then VT in series with RT
Example
• Find the Thevenin equivalent
Example
• Find the Thevenin equivalent
Example
• Find the Thevenin equivalent
Y
X
X
Y
AC circuit elements
• Capacitors in series:
1 V V1  V2  V3 1
1
1
 
 

Ceq q
q
C1 C2 C3
• Capacitors in parallel:
Ceq 
q q1  q2  q3

 C1  C2  C3
V
V
• Capacitive impedance: ZC=1/jωC
• Inductors in series:
Leq  L1  L2  L3
• Inductors in parallel:
1
1 1 1
  
Leq L1 L2 L3
• Inductive impedance: ZL= jωL
• Circuit analysis tools for DC circuits work on AC circuits, but replace resistance
with complex impedance
AC circuit analysis example
If V1=10cos(1000t) (volts) and V2=5cos(1000t) (volts), what is the current through the capacitor?
AC circuit analysis example
If V1=10cos(1000t) (volts) and V2=5cos(1000t) (volts), what is the current through the capacitor?
Mesh 1: (100+ZC)I1-ZCI2=V1
Mesh 2:
-ZcI1+(Zc+ZL)I2=-V2
1
1

 100 j
jC j  1000  10 5
Z L  jL  1000  0.1  j  100 j
ZC 
I1 
10 100 j
5
0
100  100 j 100 j
100 j
0

500 j
 0.05 j
10000
IC(jω)=I1(jω)-I2(jω)=0.05j+0.05+0.05j=0.05+0.1j
φ=tan-1(0.1/0.05) = 63 degrees
A=√(0.052+0.12)=0.11
IC(jω)=0.1163
iC(t)=0.11cos(1000t+63)
I2 
100  100 j
10
100 j
5
100  100 j 100 j
100 j
0

 500  500 j  1000 j
 0.05  0.05 j
10000
Charging a capacitor
0.37
Time constant τ=RC. Time needed to charge capacitor to 63% of full charge
Larger RC means the capacitor takes longer to charge
Larger R implies smaller current flow
The larger C is, the more charge the capacitor can hold.
Solution is only true for simple circuit with resistor and capacitor in
series, but more complicated circuits can be reduced to this using
Thevenin's Theorem
Example
A battery with an emf of 1.5V and an internal resistance of 0.6Ω is used to
charge a 5F capacitor when a switch is closed. How long does it take to reach
a voltage across the capacitor of 1V?
5F
Example
A battery with an emf of 1.5V and an internal resistance of 0.6Ω is used to
charge a 5F capacitor when a switch is closed. How long does it take to reach
a voltage across the capacitor of 1V?
5F
Example
How long does it take if we attach an additional battery with an emf of 9V and
an internal resistance of 18Ω as shown?
5F
Example
How long does it take if we attach an additional battery with an emf of 9V and
an internal resistance of 18Ω as shown?
5F
RTH=0.58 Ohm
i= 0.40 A
VTH=1.74 V
Time to 1V= 2.5 seconds
5F
Op Amps
Remember the Golden Rules:
1) iin=0: no current flows into the opamp.
2) v+=vThese are only valid when there is negative feedback
In many circuits, one input to the opamp is connected to ground, so v+=v-=0
A simple example:
Op Amp circuits
Inverting
Amplifier
Summing
Amplifier
vout
R
 F
vS
RS
Non-Inverting
Amplifier
vout
R
 1 F
vS
RS
R

R
R
vout   F vS1  F vS 2  .....  F vSN 
RS 2
RSN 
 RS1
Differential
Amplifier
vout 
R2
(v2  v1 )
R1
Integrator
1
vout (t )  
RS CF
Differentiator
t
v
S
(t )dt

And two without negative feedback:
Comparator
dvS
vout (t )   RF CS
dt
Schmitt Trigger
Example
What does this circuit do? Derive an expression for the gain and give the
circuit a suitable name.
i1  i2  0
i2
vout  v 
v1  v 

R1
R2
R2
i1
R1
R1
R1=R2
R2
v1  v   (vout  v  )
vout  2v   v1
v  v
voltage
divider
R2
v 
v2  v 
R1  R2

1
v  v2
2
vout  v2  v1

Example
Design an opamp circuit to convert the triangular waveform v1 in the following
figure into the square wave v0 shown. Use a 0.1μF capacitor. (Hint: first
quantitatively determine the mathematical expression of v0 in terms of v1)
v0 is v1 differentiated
vo   K
dv1
dt
Simple Filter analysis: Which of the following is a
low-pass filter?
•What happens to the output voltage when ω→0 (DC condition)?
•In DC circuits, capacitors are open, inductors are shorts.
•or when ω→∞
•At very high frequencies, capacitors are shorts, inductors are open
Answer: (c)
For a more quantitative solution, find the
complex transfer function:
Vo ( j )
H V ( j ) 
Vi ( j )
Vo ( j ) 
ZC
Vi ( j )
ZC  Z R
1
H V ( j ) 
• RC low-pass filter: preserves lower
frequencies, attenuates frequencies
above the 3dB cutoff frequency
ω0=1/RC.
0 
1
RC
 X 

X dB  20 log 10 
(For voltage)
X
 0
1
X dB  20 log 10
 3dB
2
jC
1
R
jC
1
1
e j0


j tan 1 RC 
2
1
1  jRC
1  (RC ) e
1
 j tan 1 RC

e
2
1  (RC )

1
1  (RC )
2
e
 j tan 1  / 0
Example
Design a high-pass RC filter with a 3dB frequency cutoff of 80Hz using a
capacitor of 2μF
H V ( j ) 
H 
Vout
R

Vin R  1
1
1 1
(RC ) 2
0  1 / RC
R  1 /( 2 80  2 10 6 )
R  980

j C
1
1 j
RC
Active Filters
Vout
Z
 F
VS
ZS
Vout
ZF
 1
VS
ZS
Example
Given an input signal Vi=10mV(sin10t+sin10,000t), design a circuit such that the
output signal is VO= -100mV(sin10t). The high frequency component of the
output signal must be <1% of the low frequency part.
So, we want a circuit which amplifies the voltage, but only at low frequencies:
need an active filter.
A( j ) 
Vout
Z
 F
VS
ZS
1
1
1


1
Z F RF
j C F
ZF 
RF
1  jRF C F
A( j )  
RF / RS
1  jRF C F
Example
Given an input signal Vi=10mV(sin10t+sin10,000t), design a circuit such that the
output signal is VO= -100mV(sin10t). The high frequency component of the
output signal must be <1% of the low frequency part.
So, we want a circuit which amplifies the voltage, but only at low frequencies:
need an active filter.
Low frequency ω1:
Want a gain of 10
V
R
Vout
ZF
A( j )  out  F  10
ω1RFCF<<1 so
A( j ) 

VS
RS
VS
ZS
set RS=1kΩ
set RF=10kΩ
1
1
1
ZF

ZF 
RF

1
j C F
RF
1  jRF C F
A( j )  
RF / RS
1  jRF C F
High frequency ω2:
Low frequency gain=10,
so for <1%, need
high frequency gain <1/10
Check low frequency:
Combinational logic design steps:
1. Derive the Truth Table
2. Fill the Karnaugh map
3. Use the map to find the logic
4. Implement the logic in a circuit
Another Example: 7 segment displays
Karnaugh Map for "a"
Truth Table
"x" represents a "don't care"
condition - the value can be
either 0 or 1
• Box the ones for sum-of-products
This subcube: B
This subcube: A·C
This subcube: D
This subcube: A'·C'
• Realization (sum-of-products) is B+D+ A·C+ A'·C'
Sequential logic design steps:
1. List the states - assign each state a symbol
2. Draw the finite state diagram (Moore - states inside
nodes)
3. Derive the symbolic state transition table
4. Assign each state a binary code (also each output,
if more than one)
5. Derive the actual state transition table
6. Write out the Karnaugh map for each "next state"
and output(s)
7. Solve the maps to find the logic
8. Implement the logic in a circuit
3-bit binary up-counter:
List the states: 0 to 7
Derive the symbolic transition table
Draw the finite state diagram
001
1
000
010
2
011
3
100
4
3-bit up-counter
0
111
7
110
6
101
5
current state
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
next state
001
1
010
2
011
3
100
4
101
5
110
6
111
7
000
0
Derive the actual state transition table
Assign the states a binary code
001
000
010
011
100
3-bit up-counter
111
110
101
current state
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
next state
001
1
010
2
011
3
100
4
101
5
110
6
111
7
000
0
Derive the actual state transition table
current
C3
0
0
0
0
1
1
1
1
C2
0
0
1
1
0
0
1
1
next
C1
0
1
0
1
0
1
0
1
N3
0
0
0
1
1
1
1
0
N2
0
1
1
0
0
1
1
0
Solve the maps
N1
1
0
1
0
1
0
1
0
notation to show
function representing
input to D-FF
N1 := C1'
N2 := C1C2' + C1'C2
:= C1 xor C2
N3 := C1C2C3' + C1'C3 + C2'C3
:= C1C2C3' + (C1' + C2')C3
:= C1C2C3' + (C1C2)'C3
:= (C1C2) xor C3
Karnaugh maps for each "next state":
N3
C3
C3C2
C1
00 01
11 10
N2
C3
N1
C3
0
0
0
1
1
0
1
1
0
1
1
1
1
C1 1
0
1
0
1
C1 1
0
0
1
C1 0
0
0
0
C2
C2
C2
current
C3
0
0
0
0
1
1
1
1
C2
0
0
1
1
0
0
1
1
C1
0
1
0
1
0
1
0
1
next
N3
0
0
0
1
1
1
1
0
N2
0
1
1
0
0
1
1
0
N1 := C1'
N2 := C1C2' + C1'C2
:= C1 xor C2
N3 := C1C2C3' + C1'C3 + C2'C3
:= C1C2C3' + (C1' + C2')C3
:= (C1C2) xor C3
N1
1
0
1
0
1
0
1
0
Implement the logic: each state bit
requires one memory element
(flipflop)