# Op-Amp

1. Show that gain rolls-off at -6 dB/octave for a passive LP filter.

2. (a) Write down the expression for the frequency-dependent open-loop gain of an operational amplifier. Sketch the magnitude of the open-loop gain of an operational amplifier as a function of frequency. Explain what is meant by a `single-lag' response of an op-amp.

(b) Using the example of a simple RC high-pass filter circuit, show how the inclusion of a coupling capacitor will limit the low frequency of any amplifier circuit.

(c) Derive the expression for the high-frequency closed-loop gain of a basic non-inverting amplifier (the feedback equation). Sketch the circuit of the amplifier to illustrate your derivation. What is the feedback fraction β for this circuit?

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#### Solution Preview

** Please see the attached document for the solution to the given problems **

1. A passive low-pass filter has the gain formula:

If X = R at the corner frequency, then the gain is -3dB. We have:

(please see the attached file)

From there, if we double the frequency from the corner frequency, which will half the value of X, we will see a trend that there is a -6dB/octave attenuation:

(please see the attached file)

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