Group I gives two possible choices for the impedance Z in the diagram. The circuit elements in Z satisfy the condition $${R_2}{C_2} > {R_1}{C_{1.}}$$ The transfer $${\textstyle{{{V_0}} \over {{V_1}}}}$$ function represents a kind of controller. Match the impedances in Group I with the types of controllers in Group II. Group - I

Group - II
1. PID controller
2. Lead compensator
3. Lag compensator

The impulse response h(t) of a linear time invariant system is given by h(t) = $${e^{ - 2t}}u(t),$$ where u(t) denotes the unit step function.

The output of this system to the sinusoidal input x(t) = 2cos(t) for all time 't' is

The logic function implemented by the following circuit at the terminal OUT is

In the following circuit, the comparator output is logic “I” if V₁ > V₂ and is logic “0” otherwise. The D/A conversion is done as per the relation $$${V_{DAC}} = \sum\limits_{n = 0}^3 {{2^{n - 1}}{b_n}\,\,} Volts,$$$ where b₃(MSB), b₂, b₁ and b₀ (LSB) are the counter outputs.

The counter starts from the clear state.

The magnitude of the error between V_DAC and V_in at steady state in volts is