Amplifiers with Offsets

The questions below are due on Monday October 28, 2024; 10:00:00 PM.
 
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Consider the following op-amp circuit, which can serve as an inverting or as a non-inverting amplifier depending on the choice of inputs.

The circuit has two input voltages (v_1 and v_2) and produces an output voltage (v_o). We want to derive an algebraic expression that relates the output voltage to the input voltages. We will be making the ideal op-amp assumption here. All voltages are expressed relative to ground.

You will be asked to enter algebraic expressions for voltages in the problems below. For each one, enter an expression the way you would type it into Python. Use the variable names exactly as specified in the question. Note that the name of the output voltage is v_o (not v_0), that is, Vee followed by Oh (lower-case letter o) - not by a zero. Note that, like all Python variables, these are case-sensitive.

1) Non-Inverting Amplifier

1.1) Forms

If v_2 = 0, it is possible to write an expression for the output voltage, v_o, that has the form:

v_o = k\cdot v_1
where k is in terms of R_1 and/or R_2. Enter a Python expression for k below, in terms of R_1 and/or R_2, assuming that this is an ideal op-amp:
k =~
When v_2 is not necessarily zero, we can also generate a more generic equation with v_o, in the form:
(v_o - v) = k \cdot (v_1 - v)

Assuming this is an ideal op-amp, enter expressions for v and k below, in terms of R_1, R_2, v_1, and/or v_2.

v =~
k =~

1.2) Values

So, assuming that v_2 is 5V and that v_1 is in the range 0V to 10V, how does v_o behave?

A real op-amp can never produce voltage values that are outside of the range of voltages from its power supplies. So, if we have +10V and 0V provided to the op-amp, the output voltages can never be less than 0 or greater than 10. If some combination of inputs would have produced values outside of this range, the actual output would only reach the closest limit voltage (0V, 10V).

Keeping this in mind, indicate the output voltage (rounded to an integer) for each of the combinations of inputs below (assume R_2 = 10k \Omega).

v_1 (Volts) v_2 (Volts) R_1 (Ohms) v_o (Volts)
10 5 100
7 5 100
5 5 100
3 5 100
0 5 100
10 5 10,000
7 5 10,000
5 5 10,000
3 5 10,000
0 5 10,000

 

2) Inverting Amplifier

Now consider the circuit again (the same circuit as above).

2.1) Forms

If v_1 = 0, we can write an expression for the output voltage, v_o, that has the form

v_o = k\cdot v_2

Enter an expression for k below, in terms of R_1 and/or R_2.

k =~
Now, letting v_1 be an arbitrary value, write an expression for the output voltage, v_o, in the form
(v_o - v) = k \cdot (v_2 - v)

Assume that this is an ideal op-amp. Enter expressions for V and k below, in terms of v_1, v_2, R_1 and/or R_2.

v =~
k =~

2.2) Values

So, assuming that v_1 is 5V and that v_2 is in the range 0V to 10V, how does v_o behave?

A real op-amp can never produce voltage values that are outside of the range of voltages from its power supplies. So, if we have +10V an 0V provided to the op-amp, the output voltages can never be less than 0 or greater than 10. If some combination of inputs would have produced values outside of this range, the actual output would only reach the closest limit voltage (0V, 10V).

Keeping this in mind, indicate the output voltage (rounded to one decimal place) for each of the combinations of inputs below (assume R_2 = 10 k\Omega).

v_1 (Volts) v_2 (Volts) R_1 (Ohms) v_o (Volts)
5 10 5,000
5 7 5,000
5 5 5,000
5 3 5,000
5 0 5,000
5 10 20,000
5 7 20,000
5 5 20,000
5 3 20,000
5 0 20,000