Use the superposition theorem to find the current through R2 of Figure 8-17.

Question

Use the superposition theorem to find the current through [latex]R_{2}[/latex] of Figure 8-17.

Accepted Answer

Step 1: Replace [latex]V_{S2}[/latex] with a short and find the current through [latex] R_{2} [/latex] due to voltage source [latex]V_{S1} [/latex], as shown in Figure 8-18. To find [latex]I_{2}[/latex], use the current-divider formula (Equation 6-6 : [latex]I_{\chi }= \left(\frac{R_{T}}{R_{\chi}} \right) I_{T}[/latex]). Looking from [latex]V_{S1}[/latex],

[latex]R_{T(S1)}= R_{1}+ \frac{R_{3}}{2} = 100 \ \Omega + 50 \ \Omega = 150 \ \Omega [/latex]

[latex]I_{T(S1)}= \frac{V_{S1}}{R_{T(S1)}} = \frac{10 \ V}{150 \ \Omega } = 66.7 \ mA[/latex]

The current through [latex]R_{2}[/latex] due to [latex]V_{S1}[/latex] is

[latex]I_{2(S1)}= \left(\frac{R_{3}}{R_{2}+ R_{3}} \right) I_{T(S1)} = \left(\frac{100 \ \Omega }{200 \ \Omega } \right) 66.7 \ mA = 33.3 \ mA[/latex]

Note that this current is upward through [latex]R_{2}[/latex].

Step 2: Find the current through [latex]R_{2}[/latex] due to voltage source [latex] V_{S2}[/latex] by replacing [latex]V_{S1}[/latex] with a short, as shown in Figure 8-19. Looking from [latex]V_{S2}[/latex].

[latex]R_{T(S2)}= R_{3}+ \frac{R_{1}}{2} = 100 \ \Omega + 50 \ \Omega = 150 \ \Omega [/latex]

[latex]I_{T(S2)}= \frac{V_{S2}}{R_{T(S2)}} = \frac{5 \ V}{150 \ \Omega } = 33.3 \ mA[/latex]

The current through [latex]R_{2}[/latex] due to [latex]V_{S2}[/latex]is

[latex]I_{2(S2)}= \left(\frac{R_{1}}{R_{1}+ R_{1}} \right) I_{T(S2)} = \left(\frac{100 \ \Omega }{200 \ \Omega } \right) 33.3 \ mA = 16.7 \ mA[/latex]

Note that this current is upward through [latex]R_{2}[/latex].

Step 3: Both component currents are upward through [latex]R_{2}[/latex], so they have the same algebraic sign. Therefore, add the values to get the total current through [latex]R_{2}[/latex].

[latex]I_{2(tot)}[/latex]=[latex]I_{2(s1)}[/latex] +[latex]I_{2(s2)}[/latex] = 33.3 mA + 16.7 mA = 50 mA

Question 8.6: Use the superposition theorem to find the current through R2...

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