Voltage Dividers
MEM 639 Real-Time Microcomputer Control 1
© Copyright Paul Oh, 2007
Digital-to-Analog Converters (DAC) – generate voltage using n-bit digital data
A. Voltage Dividers
V
R
R
A
B
in
Ohm’s Law, can prove that:
V
V
A
B
VA =
RA
Vin
RA + RB
and
VB =
RB
Vin
RA + RB
NB: VA + VB = Vin
B. An N-bit DAC has N voltage dividers
Example: Class circuit:
Vout =
WORD − 128
Vref
128
WORD = 255
4.961 V
WORD = 128
0.00 V
WORD = 129
0.039 V
WORD = 0
-5.00 V
Where
Vref = 5.0 V
Resolution = 40 mV
© Copyright Paul Oh, 2007
Op-Amps
MEM 639 Real-Time Microcomputer Control 1
© Copyright Paul Oh, 2005
Ideal Op-Amps: 2 rules and 5 properties
v
Rule 1: Voltage at “+” and “-” terminals are equal
Rule 2: No current at “+” and “-” terminals
1
R
R
d
a
v
v
2
a
+
+
Controlled Voltage Source
• Input impedance Rd ≈ ∞
• Output impedance Ra ≈ 0
• Open-loop gain is ∞
• Bandwidth is ∞
• va = 0 when v1 = v2
Example: Derive the op-amp’s input-output relationship
Solution:
Step 1: Label currents and voltage points
vα
if
ii
© Copyright Paul Oh, 2007
Step 2: Apply Rules 1 and 2
With Rule 2, observe that
vα
if
Or,
ii
vi − vα vo − vα
+
=0
Ri
Rf
With Rule 1, observe that
Hence
The gain is −
Rf
ii + i f = 0
vi
v
=− o
Ri
Rf
vα = 0
or
vo = −
Rf
Ri
vi
and hence is called an inverting op-amp. In practice, one cannot have infinite gain.
Ri
In fact, even if the input and feedback resistors are small and large respectively, the gain will be clamped.
Op-amps have rail voltages, often denoted as +V and –V. Instead of swinging to infinity, we have
v
-
If v1 > v2 then va = +V
1
R
R
d
a
va
+
-
v
2
+
If v1 < v2 then va = −V
Controlled Voltage Source
© Copyright Paul Oh, 2007
Impedance – what’s the big deal?
Illustrative Problem: Design an inverting op-amp that has a gain of -10
Solution (non-unique):
vo = −
Recall, we solved that:
Rf
Ri
Choose R f = 1000Ω and Ri = 100Ω
vi
Naïve solution! Gain is -10 if no impedance!
Impedance and Resistance obstruct alternating (AC) and direct current (DC) respectively
Impedance is zero only at absolute zero temperature (molecule don’t move). Otherwise,
everything in nature has impedance, including power supplies.
Re-examining the problem realistically:
R
R
This
v
i
R
f
R
i
is really
v
+
o
v
s
v
R
'
i
f
i
-
i
v
+
o
power supply
Thus have by voltage divider equation vi' =
Hence using a naive selection of resistors vi' =
Rule of thumb:
Ri vi
where Rs ≈ 25Ω typically
Ri + Rs
100
vi = 0.8vi
100 + 25
You’re only getting 80% of the input
voltage you think you’re putting in
Ri > 10 ⋅ Rs is a sloppy circuit
Ri > 100 ⋅ Rs is reasonable
© Copyright Paul Oh, 2007
Voltage Follower – what’s the big deal?
Illustrative Problem: Derive the input-output relationship for the following
v
i
+
vo
Solution: Rule 1 says that the “+” and “-” voltages are equal.
Thus have vo = vi
Question: Why would anyone want a unity gain amplifier?
Answer: Because op-amps have near zero output impedance.
Thus even though the power supply generating vi has impedance Rs
vo will give vi' with little output impedance
As such, voltage followers are great and often used in instrumentation amplifiers
© Copyright Paul Oh, 2007