2. (15%) For the \( 2^{\text{nd}} \) order circuit shown in Fig. 2, find the complete response \( v(t) \), including both the components of natural response and forced response, of the circuit for \( t > 0 \) when \( v(0) = 9\mathrm{V} \) and \( i_{\mathrm{L}}(0) = 0\mathrm{A} \).

3. (10%) Wheatstone bridge based circuits are often applied in sensor devices and offer possible ways to transfer resistive sensor measurements into electrical signals. A Wheatstone bridge based Op-amp circuit, shown as Fig. 3, was devised to transfer the resistive sensor \( R{1} \) to the output voltage \( v{0} \). Determine the value of \( v{0} \) when \( R{1} = 10\mathrm{k}\Omega \).

4. (15%) While \( V{C} \) denotes the DC component of the output voltage, \( v{0} \) and \( v{1} \) denote the AC components of the output and input voltages. Determine the small-signal voltage gain \( A{\nu} = v{0} / v{1} \) of the BJT amplifier shown in Fig. 4. Assume the thermal voltage \( V{\mathrm{T}} = 25\mathrm{mV} \), \( V{\mathrm{BE}} = 0.7\mathrm{V} \), \( \beta = 100 \).

5. (10%) Determine the output voltage \( v{0} \) of the circuit shown in Fig. 5. Assume the threshold voltage of the NMOS transistor \( V{\mathrm{TH}} = 1\mathrm{V} \), its transconductance parameter \( k{n} = \mu{n}C_{\mathrm{ox}}(W / L) = 1\mathrm{mA} / \mathrm{V}^{2} \), and neglect the channel-length modulation effect \( \lambda = 0 \).

6. (20%) Determine the small-signal voltage gain \( A{s} \) of the MOS amplifier shown in Fig. 6. The transistor has \( V{\mathrm{TH}} = 3\mathrm{V} \), \( k{n} = \mu{n}C{\mathrm{ox}}(W / L) = 0.25\mathrm{mA} / \mathrm{V}^{2} \) and \( \lambda = 0.02\mathrm{V}^{-1} \), and assume that the coupling capacitors \( C1 \) and \( C2 \) are sufficiently large to shorten the AC signals at their frequencies of interest, and \( A{s} \equiv -g{m}R{L} \), where \( R_{L} \) is AC equivalent resistance.

7. (10%) For the Op-Amp differentiator shown in Fig. 7, show the relation between the input and output as:$$V{\mathrm{out}} = -RC \frac{dV{\mathrm{in}}}{dt}$$

8. (10%) Suppose the sinusoidal voltage and current across an circuit component are $v(t) = V{\mathrm{in}} \cos(\omega t + \theta{\mathrm{v}})$ and $i(t) = I{\mathrm{in}} \cos(\omega t + \theta{\mathrm{i}})$, respectively. Show the average power (over a complete period) delivered to the component is$$P = \frac{V{\mathrm{in}} I{\mathrm{in}}}{2} \cos(\theta{\mathrm{v}} - \theta{\mathrm{i}})$$

1. For a pn-junction in Fig. 1, explain (a) why are there free electrons (holes) on the n-side (p-side), (b) why are there positive (negative) ions on the n-side( p-side) of the depletion region. (10%)

2. Draw the small-signal model and prove the dc gain and -3dB frequency of the circuit in Fig. 2. (10%)

3. Q₁, QREF and QF in Fig. 3 have the same current gain β. Q₁ and QREF have emitter area n•AE and AE, respectively. Assume that the collector current of QF is equal to its emitter current. IREF is the input current, compute the output current Icopy? (10%)

4. Determine the transfer function \( V_{out}(s)/V_{in}(s) \) of the circuit shown in Fig. 4, find the zero and pole frequency? (10%)

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