Using superposition, obtain an expression for the current [latex] i_x [/latex] in the circuit shown in Fig. 3.44(a).

We first find the current i due to the voltage source [latex] ν _0 [/latex] acting alone; that is, with the source current [latex] i_0[/latex] set to zero (replaced by an open circuit), as illustrated by Fig. 3.44(b). This gives [latex] \left.i\right|_{i_0=0}=\frac{ν_0}{R_1+R_2+R_3} [/latex] Next we find the current i due to the current source [latex] i_0 [/latex] acting alone; that is, with the source voltage [latex] ν_0 [/latex] set to zero (replaced by a short circuit), as illustrated by Fig. 3.44(c). We obtain [latex] \left.i\right|_{ν_0=0}=\frac{R_2 i_0}{R_1+R_2+R_3} . [/latex] There are no other sources, so the (total) current [latex] i_x [/latex] is given by [latex] i=\left.i\right|_{i_0=0}+\left.i\right|_{ν_0=0}=\frac{ν_0+R_2 i_0}{R_1+R_2+R_3} [/latex]

Question 3.16: Using superposition, obtain an expression for the current i...

Introduction to Circuit Analysis and Design

Using superposition, obtain an expression for the current $i_x$ in the circuit shown in Fig. 3.44(a).

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We first find the current i due to the voltage source $ν _0$ acting alone; that is, with the source current $i_0$ set to zero (replaced by an open circuit), as illustrated by Fig. 3.44(b). This gives

$\left.i\right|_{i_0=0}=\frac{ν_0}{R_1+R_2+R_3}$

Next we find the current i due to the current source $i_0$ acting alone; that is, with the source voltage $ν_0$ set to zero (replaced by a short circuit), as illustrated by Fig. 3.44(c). We obtain

$\left.i\right|_{ν_0=0}=\frac{R_2 i_0}{R_1+R_2+R_3} .$

There are no other sources, so the (total) current $i_x$ is given by

$i=\left.i\right|_{i_0=0}+\left.i\right|_{ν_0=0}=\frac{ν_0+R_2 i_0}{R_1+R_2+R_3}$