A simple active filter is shown in figure. Assume ideal op-amp. Derive the transfer function $${v_0}/{v_i}$$ of the circuit, and state the type of the filter (i.e., high-pass, low-pass, band-pass, or band-reject). Determine the required values of $${R_1},{R_2}$$ and $$C$$ in order for the filter to have a $$3$$-$$dB$$ frequency of $$1$$ $$kHz,$$ a high frequency input resistance of $$100\,\,k\Omega $$ and a high frequency gain magnitude of $$10.$$

The input voltage $${V_i}$$ in the circuit is a $$1$$ $$kHz$$ sine wave of $$1$$ $$V$$ amplitude. Assume ideal operational amplifiers with $$ \pm 15V\,\,DC$$ supply. Sketch on a single diagram the wave-forms of $${V_i},$$ $${V_1}$$ and $${V_0}$$ show, indicating the peak value of$${V_1}$$ and the average value $${V_0}$$.

In the circuit of figure $${R_s} = 2\,k\Omega ,$$ $${R_L} = 5\,k\Omega $$ For the op-amp $$A = {10^5},$$
$${R_i} = 100\,k\Omega ,\,\,{R_0} = 50\,k\Omega .$$ For $${V_0} = 10V.$$
Calculate $${V_S}$$ and $${{{V_0}} \over {{V_S}}}$$ and estimate the input resistance of the circuit,

Figure below shows an op-amp amplifier. Find the output voltage in steady state condition where $$(i)$$ switch is open $$(ii)$$ $$S$$ is closed