Op-amp Feedback

The questions below are due on Friday March 06, 2020; 10:00:00 AM.
 
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Single Pole Op-Amp

Consider a nearly ideal op-amp whose transfer function from input difference (V_+ - V_-) to output voltage V_o is given by

V_o(s) = H_{OPAMP}(s) (V_+(s) - V_-(s)) = \frac{10^6}{s+1} (V_+(s) - V_-(s)).

If we use what is usually referred to as non-inverting configuration, using resistors R_1 and R_2 to create the feedback network. This op-amp may be modeled using a feedback block diagram, with a 'forward-path' given by H_{OPAMP}(s) and a 'feedback path' scaled by a factor f.

This is not our usual tracking configuration, because the output is scaled by f before comparing to the input, but this form is more natural when describing operational amplifier circuits. To analyze this case, we know that based on how the circuit elements are connected,

V_{+} = V_{in}, \; \; \; V_{-} = f V_{out}
where f is LESS than one, and is determined by the resistor divider network.

The operational amplifier's output voltage, V_{out} is determined by the product of its transfer function and the differential input voltage,

V_{out} = H_{OPAMP} (V_{+} - V_{-}),
as is also shown in the block diagram above.

Combining,

V_{out} = H_{OPAMP} (V_{in} - f V_{out}),
resulting in a variant of Black's formula,
V_{out}(s) = \frac{H_{OPAMP}(s)}{1+f H_{OPAMP}(s))} V_{in}(s).
where we have put the s back in the the equation to remind ourselves that we have a tranfer function with a frequency response. Note that
V_{out}(s) \approx \frac{1}{f} V_{in}(s).
when |H_{OPAMP}(s)| >> 1.

Determine an expression for f in terms of the resistor values R_1 and R_2. Be sure to use 'R_1' and 'R_2' as your variables in the python expresssion.

What values of the feedback factor f are realizable using the resistor network? Open circuit (R=\infty) and short-circuit (R=0) resistor values are allowed.

Maximum f

Minimum f

Can this op-amp possibly be unstable for any value of f in the range you determined in the previous part of the problem?

Recall that the step response for a first-order differential equation system

\frac{d}{dt} y(t) = \lambda y(t) + b \; x(t)\;\;,
is given by (if \lambda < 0 )
y(t) = \frac{b}{-\lambda}(1 - e^{\lambda t}).
A plot of the step response for b = 10 and \lambda = -100 is given below. Note that the step response settles to \frac{b}{-\lambda} as t \rightarrow \infty, and has risen to about two-thirds of the steady-state at t \approx \frac{1}{-\lambda}.

For steady-state gains from one to one thousand (corresponding to \frac{1}{1000} \le f \le 1) give an approximate expression for the relationship between gain and the time for the step response to achieve two-thirds of steady-state.

Enter a python expression for 2/3's rise as a function of gain (use g for gain):

Suppose R_1 = 10^6 ohms (which we often refer to as a 1M\Omega or one megohm) and R_2 = 100 ohms, to within 0.1 percent, what is the steady-state gain?

Gain:

In lab, you will be building an op-amp PD controller, with the following topology.

Please analyze the op-amp making the "design" assumptions. That is, v_{out}(t) changes so as to make v_+(t) \approx v_-(t), i_+(t) =0, and i_-(t) = 0.

If nothing is connected the v_{in} terminal, what are the steady-state values of v_+ and v_{out}?

v_+ =

v_out =

If v_{in} = 2.4, as a function of R, what are the steady-state values of v_+ and v_{out}. Be sure to use 10000 to denote the 10k \Omega resistor value.

Enter a python expression for v_+:

Enter a python expression for v_out:

Assuming R = \infty and op-amp is ideal, 2.5 - v_{out} is related to v_{in} - 2.5 by a transfer function \mathcal{G}(s) = K_d s . In terms of C, what is K_d?

Enter a python expression for Kd:

Assuming R \ne 0 and op-amp is ideal, 2.5 - v_{out} is related to v_{in} - 2.5 by a transfer function \mathcal{G}(s) = K_p + K_d s . In terms of R, C, and s, what is \mathcal{G}(s)?

Enter a python expression for G(s):