# Op-amp dynamics

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The symbol for an op-amp with its terminals enumerated is shown below.

The model for a non-ideal version of this op-amp is now shown below.

In addition to the traditional dependent source, this model includes an output resistor R_{O}, and internal dynamics modeled by a second dependent source in a series loop with the resistor R_{D} and capacitor C. Note that, while not explicitly shown, v_{+} and v_{−} are both node voltages defined with respect to ground.

The op-amp is used to build the amplifier shown below.

Let C = 0 for this part only. Using the op-amp model described above, determine the Thevenin equivalent of the op-amp-based amplifier when it is viewed from its output port at which v_{OUT} is defined.

For problems below use `RA`

for R_A, `RB`

for R_B, `RO`

(with
letter O) for R_O, etc. as well as `vIN`

for v_{IN}, `VIN`

for
V_{IN}, `A`

for A, etc... and either `w`

or `omega`

for \omega
as needed. `sqrt`

, `atan`

, and `exp`

for natural base e are also
available as needed.

V_{Th} =

R_{Th} =

Let R_{O} = 0 for this part. Assume that the op-amp based amplifier is at rest for t \le 0 with v_{IN} = 0 and v_{C} = 0. Then, at t = 0, v_{IN} takes a step from 0 V to V_{IN} such that v_{IN}(t) = V_{IN}u(t). Using the op-amp model described above, determine v_{OUT}(t) for t \geq 0. For the following question use VIN for V_{IN}.

v_{OUT}(t) =

Again let R_{O} = 0. Assume that the op-amp based amplifier operates in the sinusoidal steady state with v_{IN} = V_{in} \cos(\omega t) and v_{OUT} = V_{out} \cos(\omega t+\phi). Determine the amplifier gain \dfrac{V_{out}}{V_{in}} and phase shift \phi, both as functions of frequency.

\dfrac{V_{out}}{V_{in}} =

\phi =