Explain the concept of nodal analysis in ac circuit ---**Figure 2.11 Description:**
* **Type:** Electrical circuit diagram.
* **Components:**
* Left: Current source I₁ = 1 A ∠0°, oriented downwards.
* Middle Left: Resistor R = 2 Ω.
* Middle: Inductor X_L = 2 Ω.
* Middle Right: Capacitor X_C = 4 Ω.
* Right: Current source I₂ = 2 A ∠0°, oriented upwards.
* **Connections:** All components are connected in parallel between the top and bottom horizontal wires. The bottom wire is connected to ground.
* **Labels:** I₁, R, X_L, X_C, I₂.
**Text:**
Figure 2.11 Circuit for Example 2.5
Solution The circuit is redrawn in Figure 2.12, showing the nodes and a simplified representation of the admittances.
Y₁ = 0.5 S Y₂ = −j0.5 S and Y₂ = j0.25 S
**Figure 2.12 Description:**
* **Type:** Electrical circuit diagram, likely representing nodal analysis setup.
* **Nodes:** Two labeled nodes V₁ (top left) and V₂ (top right), and a ground reference node at the bottom.
* **Components:**
* Left Branch: Current source I₁ = 1 A ∠0°, connected between V₁ and ground, oriented downwards.
* Middle Left Branch: Admittance Y₁, connected between V₁ and ground. An arrow labeled I_A points downwards through Y₁.
* Middle Branch: Admittance Y₂, connected between V₁ and V₂. An arrow labeled I_B points downwards through Y₂.
* Middle Right Branch: Admittance Y₃, connected between V₂ and ground. An arrow labeled I_C points downwards through Y₃.
* Right Branch: Current source I₂ = 2 A ∠0°, connected between V₂ and ground, oriented upwards.
* **Labels:** I₁, V₁, Y₁, I_A, Y₂, I_B, V₂, Y₃, I_C, I₂. Highlighted areas around V₁ and V₂ connections.
**Text:**
Figure 2.12 Circuit for Step 1 of Example 1.8
The nodal equations are written as
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Nodal analysis is a fundamental technique in electrical circuit analysis that systematically applies Kirchhoff's Current Law at each node. This method is particularly powerful for AC circuits with multiple sources and complex impedances, as it reduces the circuit to a manageable set of equations. The key concept involves selecting a reference node, typically ground, and expressing all other node voltages relative to this reference.
In AC circuit analysis, we use phasor representation with complex numbers to handle sinusoidal voltages and currents. The key concept is converting impedances to admittances, where admittance Y equals one over impedance Z. For a resistor, admittance is simply one over resistance. For an inductor, it becomes negative j over omega L, and for a capacitor, it's j omega C. Using our circuit values: the 2-ohm resistor gives Y1 equals 0.5 siemens, the 2-ohm inductor gives Y2 equals negative j 0.5 siemens, and the 4-ohm capacitor gives Y3 equals j 0.25 siemens.
Now we transform the original circuit into nodal analysis format. We identify two essential nodes: V1 on the top left and V2 on the top right, with ground as our reference. The circuit contains two current sources - I1 equals 1 ampere at 0 degrees flowing downward, and I2 equals 2 amperes at 0 degrees flowing upward. We replace all impedances with their corresponding admittances: Y1 for the resistor, Y2 for the inductor, and Y3 for the capacitor. This systematic redrawing prepares us for applying Kirchhoff's current law at each node.
Now we apply Kirchhoff's Current Law at each node. At node V1, current I1 enters the node, while currents IA and IB leave the node. Therefore, I1 equals IA plus IB. Expressing these currents in terms of admittances and node voltages: I1 equals Y1 times V1 plus Y2 times the voltage difference V1 minus V2. At node V2, current IB enters from V1 and current I2 enters from below, while current IC leaves through Y3. This gives us: Y2 times V1 minus V2 plus I2 equals Y3 times V2. These equations form the foundation for our nodal analysis solution.