Question 89 There is no energy stored in the circuit in Fig. P13.89 at the time the impulse voltage is applied. a) Find \( i_{1} \) for \( t \geq 0^{+} \). b) Find \( i_{2} \) for \( t \geq 0^{+} \). c) Find \( v_{o} \) for \( t \geq 0^{+} \). d) Do your solutions for \( i_{1}, i_{2} \), and \( v_{o} \) make sense ir terms of known circuit behavior? Explain. 89 There is no energy stored in the circuit in Fig. P13.89 at the time the impulse voltage is applied. a) Find i1 for t≥0+. b) Find i2 for t≥0+. c) Find vo for t≥0+. d) Do your solutions for i1,i2, and vo make sense ir terms of known circuit behavior? Explain.
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An:- Given data?\rightarrow initially there is no enogry stored in the ciscuit.\rightarrow For\delta(t) \longrightarrow \begin{aligned}t<0 & \Rightarrow 0 \\t>0 & =0 \\t & =0 \Rightarrow 1\end{aligned}\therefore a) Find i_{1} for t \geq 0^{+}:-At t \geq 0^{t} \Rightarrow \delta(t)=1\rightarrow The above diagram can be redrawn asfor Transforma connections:-Notewfor \omega=1 \mathrm{rad} (sec'-\rightarrow Apply KVL in Loop -( :-20+\frac{1}{2} i_{1}+\frac{1}{2}\left(i_{1}-i_{2}\right)=0 \text { (1) }\rightarrow Apply KVL in loop -(2) :-\frac{1}{2} i_{2}+3 i_{2}-\frac{1}{2}\left(i_{1}-i_{2}\right)=0 \text { (2) }\rightarrow By solving equation (1) & (2) we get 'i, e iz :-\begin{array}{l}\Rightarrow \quad i_{1}-\frac{1}{2} i_{2}=20 \\\Rightarrow \quad-\frac{1}{2} i+4 i_{2}=0\end{array}multiply equation (1) by \frac{1}{2} and add resultant to equation (2) we gat?\begin{array}{c}\frac{1}{2} i_{1}-\frac{1}{4} i_{2}=10-\text { (1) } \\\frac{1}{2} i_{1}+4+i_{2}=0 \\4 \frac{1}{2}-\frac{1}{4} i_{2}=10 \\i_{2}\left(\frac{\frac{15}{4}}{4}\right)=10^{2} \\i_{2}=8 / 3 \text { Amps }\end{array}\rightarrow substitute (3) in (1) we get:-\begin{aligned}i_{1} & =20+\frac{1}{2} i 2 \\& =20+\frac{1}{2} \times(8 / 3) \\& =20+8 / 6 \\R_{1} & =\frac{128}{6} \text { Arrps }\end{aligned}\therefore a) R_{1} for t \geq 0^{+} \Rightarrow R_{1}=\frac{128}{6} Anys{ }_{q_{1}}=21.33 \text { Aorys }()\therefore b) i_{2} for t z 0^{+} \Rightarrow i_{2}=8 /_{3}=2.66 Aroys\therefore C) Vo for t \geq 0^{+} \RightarrowV_{0}=3 \times 12=3 \times 8 / 3=8 \text { volts } 11 ...