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$$
\begin{array}{l}{\text { The op amp in the circuit shown is ideal. }} \\ {\text { a) Calculate } v_{o} \text { if } v_{a}=1 \mathrm{V} \text { and } v_{b}=0 \mathrm{V}} \\ {\text { b) Repeat (a) for } v_{a}=1 \mathrm{V} \text { and } v_{b}=2 \mathrm{V}} \\ {\text { c) If } v_{a}=1.5 \mathrm{V}, \text { specify the range of } v_{b}} \\ {\text { that avoids amplifier saturation. }}\end{array}
$$

$$
V^{+}=V^{-}
$$

$$
V_{b}=V_{1} \longrightarrow (1)
$$

$$
i_{1}=0, \quad i_{2}=0
$$

KCL

$$
\sum I=0 \Rightarrow i_{1}-i_{2}=0
$$

$$
i_{1}=i_{2}
$$

$$
i_{1}=\frac{فرق الجهد}{المقاومة}=\frac{V_a-V_1}{25}=\frac{V_a-V_b}{25} \longrightarrow (2)
$$

$$
i_{2}=\frac{فرق الجهد}{المقاومة}=\frac{V_{1}-V_{0}}{100} \longrightarrow (3)
$$

$$
i_{1}=i_{2} \Rightarrow \frac{V_{b}-V_{0}}{100}=\frac{V_a-V_b}{25}
$$

(a) $$
V_{0}=? ?, V_a=1, V_b=0
$$

$$
\frac{0-V_{b}}{100}=\frac{1-0}{25}
$$

$$
V_{0}=-4 volt
$$

$$
-10<V_{0}<10
$$

liner mode

(b) $$
V_{a}=1, V_{b}=2, V_{0}=? ?
$$

$$
V_0=6 V
$$
     liner mode

(c) $$
V_{a}=1.5 V,
$$
range of $$V_b=??$$

Liner mode $$
-V_{c c} \leq V_{0} \leq V_{c}
$$

$$
\frac{V_{b}-V_{0}}{100}=\frac{V_a-V_b}{25}\Rightarrow 100*\frac{V_b-V_0}{100}=\frac{1.5-V_b}{25}*100
$$

$$
V_{b}-V_{0}=4\left[1.5-V_{b}\right] \Rightarrow V_{0}=V_{b}-6+4 V_{b}
$$

$$
V_{0}=5 V_{b}-6 \rightarrow -V_{cc} \leq 5V_b-6 \leq V_{cc}
$$

$$
6-10 \leq 5V_ b-6 \leq 10+6
$$

$$
\frac{-4}{5} \leq \frac{5 V_{b}}{5} \leq \frac{16}{5}
$$

$$
-0.8 \leq V_ b \leq 3.2 V
$$

to avoid  saturation

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