Operational Amplifiers I

We are presented with a sensor that outputs a signal up to +1 V. Because the signal is small, we want to insert a signal conditioning circuit in between to amplify the signal so the microcontroller can be powered.

In this model, the voltage divider on the left will represent the sensor, the 741 op-amp as the signal conditioning circuit, and Vout as the microcontroller's output. We wanted a gain of -10 through the op-amp and +12 V and -12 V on the top and bottom rails, respectively. We determined the following values for the resistors and voltage sources:

Component	Nominal Value
Ri	1 kΩ
Rf	10 kΩ
RX	1125 Ω
RY	Potentiometer (max 104.17 Ω)
V1	12 V
V2	-12 V

The Thevenin resistance, looking back form the inverting amplifier, was 96 Ω, which was at least 20 times smaller than the resistance for R_i. Therefore, there was no loading effect on our circuit.

We assembled the circuit and measured Vout, VRi, and VRf.

Our results were:

Vin	Vout	GAIN	VRi	IRi	VRf
0.0 V	0.0 V	N/A	0 V	0 mA	0 V
0.25 V	-2.762 V	-11.05	0.2785 V	0.279 mA	2.906 V
0.50 V	-5.06 V	-10.12	0.513 V	0.513 mA	5.17 V
0.75 V	-7.59 V	-10.12	0.753 V	0.753 mA	7.55 V
1.00 V	-10.22 V	-10.22	1.060 V	1.06 mA	10.20 V

Sure enough, we saw a gain of at least -10 in our output and the current drew no more than 1 mA as the sensor approached 1 V.

PSpice

The following is a demonstration on using PSpice.

The first exercise was to simulate this circuit:

Once the circuit was built, PSpice immediately gave us values for the current through each element and voltages through each node. 

The next part was to determine the Thevenin equivalent for the circuit:

Instead of just simply analyzing the circuit, we had to perform a DC sweep on the bottom current source (I3) . This puts I3 onto the x-axis of the graph. We wanted to observe the voltage as a function of current and obtained the following graph:

The graph tells us that when the current is zero (open circuit), the Thevenin voltage is 10 V. The slope of the graph gives us the Thevenin resistance, which is 3.33 Ω. The graph also tells us that when the voltage is zero (short circuit), the Norton current is 3 A.

The last part of the activity was to replace I3 with a variable resistor to show that maximum power transfer occurs when the variable resistor equals to the Thevenin resistance. This was our result:

As expected, our peak occurs when the variable resistance is 3.3 Ω.

As opposed to performing tedious calculations by hand, PSpice is a convenient tool in analyzing circuits and other properties of a circuit.

Maximum Power Transfer

The experiment was to verify the maximum power transfer theory of the following circuit:

We expected to see that when the potentiometer (load) is of the same value as the 5.6 kΩ, the maximum power of the load would be achieved. The first experimental setup was simple:

Our experimental data came out as follows:

We came across some bad data and simply omitted those points, giving us the linear function seen on the graph. Despite the encountered errors, we were able to determine the maximum power when the load resistance is 5120 Ω, thus supporting our hypothesis.

The next part of the experiment had us build the following circuit:

We ran the circuit through a software called Logger Pro in order to determine maximum power across the potentiometer, similar to the first part of the experiment. We were, however, unsuccessful because of the noise in the current graph. This prevented us from giving us an accurate reading of the power graph.

Perhaps the biggest cause of our problem were the adapters used to power the Logger Pro instruments. The AC source could attribute to the noises we see in our graphs because of the changes in current.

Thevenin Equivalents

Thevenin's theorem allows us to take a complicated system and simplify it down to a voltage source and an equivalent resistance.

We first found the Thevenin resistance and Thevenin voltage of a seemingly complex circuit. Our analysis yielded us values of

V_Th = 8.64 V
R_Th = 65.95 Ω

We then calculated the smallest value of R_L2 (the load resistance) when V_Th = 8 V.

R_L2 = 824.3 Ω

The next step was to take measurements of V_{Load 2}, the load voltage outside of the Thevenin circuit, for the original circuit and its Thevenin equivalent. We took load voltages for the smallest permissible resistance and for an open circuit. The Thevenin equivalent data are as follows:

Config	Theoretical Value	Measured Value	Percent Error
RL2 = RL2,min	VLoad 2 = 8.00 V	8.37 V	4.63 %
RL2 = ∞	VLoad 2 = 8.64 V	9.06 V	4.86 %

For the original circuit:

Config	Theoretical Value	Measured Value	Percent Error
RL2 = RL2,min	VLoad 2 = 8.00 V	8.55 V	6.88 %
RL2 = ∞	VLoad 2 = 8.64 V	9.06 V	4.86 %

At the end, we calculated some values of power supplied to the load resistor:

Config.	VLoad 2	PLoad 2
RL2 = 0.5RTh	2.88 V	0.252 W
RL2 = RTh	4.32 V	0.283 W
RL2 = 2RTh	5.76 V	0.252 W

The table shows that when R_L2 = R_Th, the maximum power is supplied.