PHYS 3322 Modern Laboratory Methods I
Transfer Functions of Op Amp Active Filters
Purpose
This portion of this lab course develops your understanding of basic op amp behavior and
utilizes simple circuits to reinforce your understanding.
Background
+
V
_
vo
+
-
V
The operational amplifier (op-amp) is an active device in that it requires DC power to operate. It
has seven terminals, five of which are of primary interest. They are:
1) The inverting input; labeled “-“ in the diagram of Figure 1.
2) The non-inverting input; labeled “+” in the diagram of Figure 1.
3) The output terminal
4) Positive power supply V+, sometimes denoted +Vcc
5) Negative power supply V-, sometimes denoted -Vcc
6) The other two terminals are of little or no concern as they are used to compensate for
degradation in performance because of aging and imperfections and require a separate
auxiliary circuit to do this. Both are labeled Offset null in op amp diagrams.
Note: When drawing the circuit for the op amp the terminals labeled V+ and V- are usually
omitted.
The output of the amplifier vo is given by the formula:
vo = A(v + - v -)
(1)
+
Where A is the open-loop voltage gain of the amplifier, v is the non-inverting input voltage and
v- is the inverting input voltage. Both v+ and v- are node voltages with respect to ground.
Typically, the open-loop voltage gain A is on the order of 105 - 106. A resistor is placed between
the output node and the inverting input to provide feedback and adjust amplification. When an
op-amp circuit behaves linearly, the op-amp adjusts its output current such that the voltage
difference between the two inputs is nearly zero.
v -=v+
(2)
Another important feature of the op-amp is that its input resistance is very large and may be
taken as infinite in many applications. The most common type of op-amp is the 741, which has
Revised: 13 November 2002
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Transfer Functions of Op Amp Active Filters
an input resistance of 2 MΩ. This is large enough to be considered infinite in most applications.
Because of the high input resistance, only a very small current flows into either input of an opamp. In practical op-amp circuits, the current flowing into either of the inputs is usually on the
order of µA. In the case of an ideal op-amp, where the single assumption is made that the openloop voltage gain A goes to infinity,
ii = 0
(3)
where ii is defined to be the current entering the non-inverting input and exiting the inverting
input. Equations 2 and 3 can be used to analyze most of the properties of op-amp circuits.
The above can be summarized in the following rules for an ideal op amp
I.
The output attempts to do whatever is necessary to make the voltage difference
between the inputs zero.
II. The inputs draw no current.
R2
R1
VIN
_
A
VOUT
B
+
The circuit above is an inverting amplifier. To analyze its properties we must refer to the rules
stated above and assume this is an ideal op amp. So:
a) Since the non-inverting terminal is grounded, then point B is at zero potential or ground
potential. This means point A, the inverting input, is also at ground potential by Rule I.
b) Given (a), the voltage across R1 is VIN and the voltage across R2 is VOUT.
c) Since the current into the inverting input is zero (Rule II) then the net current through
junction A is also zero. A current junction rule at point A yields
VOUT
V
= − IN
R2
R1
The ratio of VOUT to VIN is the gain of the circuit. Explicitly,
VOUT
R
=− 2
VIN
R1
This relationship, which is the transfer function for the above circuit, says that the output voltage
is inverted with respect to the input voltage (the negative sign) and its amplitude (or gain) with
respect to the input voltage is governed by the ratio of the two resistors.
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Transfer Functions of Op Amp Active Filters
Procedure
The circuits below are filtering circuits with properties dependent on the values of the various
resistors and capacitors.
Circuit 1. For the circuit on the right determine:
a) The transfer function.
C
b) The maximum amplitude for the
transfer function.
c) The cutoff frequency.
R1
VIN
d) Without performing any formal
graphing analysis, do a
qualitative (rough) sketch of the
transfer function amplitude as a
function of frequency.
_
R2
VOUT
+
Circuit 2. For the circuit on the right determine:
R2
a) The transfer function.
b) The maximum amplitude for the
transfer function.
VIN
c) The cutoff frequency.
C
R1
d) Without performing any formal
graphing analysis, do a
qualitative (rough) sketch of the
transfer function amplitude as a
function of frequency.
_
VOUT
+
Circuit 3. For the circuit on the right determine:
C2
a) The transfer function.
b) The maximum amplitude for the
transfer function.
c) The cutoff frequencies.
d) Without performing any formal VIN
graphing analysis, do a
qualitative (rough) sketch of the
transfer function amplitude as a
function of frequency.
Revised: 13 November 2002
R2
C1
R1
_
VOUT
+
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