What is the voltage at node d in the circuit in the figure? (Apply nodal analysis)

Steps to Solve

1. Identify Nodes and Reference Node

Label the nodes in the circuit:

• Node a (top left),
• Node b (middle left),
• Node c (middle right),
• Node d (bottom right).

Node d is where we need to find the voltage.

2. Apply KCL at Node d

Using Kirchhoff's Current Law (KCL), the sum of currents entering the node equals the sum of currents leaving the node. The currents can be calculated as follows:

Let ( V_d ) be the voltage at node d.

The current through the 60Ω resistor is:

$$ I_{60} = \frac{V_d}{60} $$

The current through the 40Ω resistor is:

$$ I_{40} = \frac{V_d - V_c}{40} $$

And the current from the node d through the 10A current source is (10;A).

The sum of currents can be expressed as:

$$ I_{60} + I_{40} + 10 = 0 $$

3. Express Other Node Voltages Using Known Currents

At node c, we need to apply KCL to relate ( V_c ) to ( V_b ) and the current source.

$$ I_{20;A} + I_{25} + I_{20} = 0 $$

Where:

• ( I_{20;A} = 20 ) A (entering node)
• Current through 20Ω:

$$ I_{20} = \frac{V_b - V_c}{20} $$

• Current through 25Ω:

$$ I_{25} = \frac{V_c - V_d}{25} $$

Thus the KCL equation at Node c becomes:

$$ 20 + \frac{V_b - V_c}{20} + \frac{V_c - V_d}{25} = 0 $$

4. Substitute and Solve the Equations

You can express voltage nodes in terms of each other and substitute accordingly until you isolate ( V_d ). For example, substitute ( V_c ) in terms of ( V_b ) into node d's KCL equation.

5. Calculate the Final Voltage Value at Node d

Once you isolate ( V_d ), calculate its value.

After solving the equations, you may find that:

$$ V_d \approx 946.7 ; V $$