How to Obtain v₁ and v₂ for the Circuit Shown in Fig 3.51
When studying basic circuit analysis, one of the most common exercises is to determine the unknown node voltages v₁ and v₂ in a given network. Fig 3.51 (as found in many introductory electronics textbooks) typically presents a two‑node circuit that contains a mixture of independent voltage sources, current sources, and resistors. Although the exact component values may vary between editions, the underlying method—nodal analysis—remains the same. This article walks you through a complete, step‑by‑step procedure to obtain v₁ and v₂, explains the theory behind each step, and offers practical tips to avoid common pitfalls. By the end, you will be able to tackle similar problems with confidence Easy to understand, harder to ignore. No workaround needed..
1. Introduction
The goal of nodal analysis is to express Kirchhoff’s Current Law (KCL) at each essential node in terms of the node voltages. Once the equations are set up, solving the resulting linear system yields the desired voltages. In Fig 3.
- An independent voltage source Vₛ connected between the reference node (ground) and node 1.
- A resistor R₁ tying node 1 to node 2.
- A resistor R₂ connecting node 2 to ground.
- An independent current source Iₛ injecting current into node 2 (direction indicated by the arrow).
Although the actual schematic may include additional elements, the core topology is a simple ladder that makes the mathematics transparent while still illustrating the power of nodal analysis.
Key phrase: obtain v₁ and v₂ – this is the quantity we will calculate throughout the discussion Simple, but easy to overlook..
2. Circuit Description (Assumed Values for Illustration)
To make the explanation concrete, we will assign typical numeric values that appear in many textbook examples. Feel free to replace them with the exact numbers from your Fig 3.51; the algebraic steps remain identical.
| Symbol | Description | Assumed Value |
|---|---|---|
| Vₛ | Voltage source from ground to node 1 | 12 V |
| R₁ | Resistor between node 1 and node 2 | 4 kΩ |
| R₂ | Resistor from node 2 to ground | 6 kΩ |
| Iₛ | Current source into node 2 (arrow pointing into the node) | 2 mA |
It sounds simple, but the gap is usually here Simple, but easy to overlook..
The reference node (ground) is at 0 V. With these values, we will obtain v₁ and v₂ using nodal analysis.
3. Nodal Analysis Method – Theory Overview
Nodal analysis relies on two fundamental principles:
- Kirchhoff’s Current Law (KCL): The algebraic sum of currents leaving a node equals zero.
- Ohm’s Law: The current through a resistor is proportional to the voltage difference across it: ( I = \frac{V_{node;a} - V_{node;b}}{R} ).
By writing KCL at each non‑reference node and substituting Ohm’s Law for each branch, we obtain a set of linear equations whose unknowns are the node voltages That's the part that actually makes a difference. Which is the point..
3.1 Choosing the Reference Node
The reference node (ground) is arbitrarily selected; all other node voltages are measured relative to it. In Fig 3.51, the bottom node is already grounded, simplifying the equations Which is the point..
3.2 Writing KCL Equations
For each essential node (nodes not connected directly to the voltage source that fixes its voltage), we sum the currents leaving the node and set the sum to zero Worth keeping that in mind..
4. Setting Up the Equations for Fig 3.51
4.1 Node 1
Node 1 is directly connected to the voltage source Vₛ, which fixes its voltage relative to ground:
[ v_1 = V_s = 12\text{ V} ]
Because Vₛ is an independent voltage source, we do not write a KCL equation for node 1; instead, we treat v₁ as known It's one of those things that adds up..
4.2 Node 2
At node 2, three branches meet:
- The resistor R₁ connecting to node 1.
- The resistor R₂ connecting to ground.
- The current source Iₛ injecting current into the node.
Assuming currents leaving the node are positive, we write:
[ \frac{v_2 - v_1}{R_1} ;+; \frac{v_2 - 0}{R_2} ;-; I_s = 0 ]
The minus sign before Iₛ appears because the current source is directed into node 2 (i.Here's the thing — e. , it is a negative contribution to the sum of currents leaving the node).
Substituting the known values:
[ \frac{v_2 - 12}{4,\text{k}\Omega} ;+; \frac{v_2}{6,\text{k}\Omega} ;-; 2\text{ mA} = 0 ]
5. Solving the Equations
5.1 Convert Units for Consistency
It is convenient to express resistances in kilo‑ohms and currents in milliamps, which yields voltages in volts directly:
[ \frac{v_2 - 12}{4} ;+; \frac{v_2}{6} ;-; 2 = 0 ]
5.2 Clear the Denominators
Multiply every term by the least common multiple of 4 and 6, which is 12:
[ 12\left(\frac{v_2 - 12}{4}\right) + 12\left(\frac{v_2}{6}\right) - 12\cdot 2 = 0 ]
[ 3(v_2 - 12) + 2v_2 - 24 = 0 ]
5.3 Expand and Collect Like Terms
[ 3v_2 - 36 + 2v_2 - 24 = 0 ]
[ (3v_2 + 2v_2) - (36 + 24) = 0 ]
[ 5v_2 -
5.3 Expand and Collect Like Terms
[ 3v_2 - 36 + 2v_2 - 24 = 0 ]
Combine the constant terms:
[ 3v_2 + 2v_2 - (36 + 24) = 0 ;\Longrightarrow; 5v_2 - 60 = 0 ]
5.4 Solve for (v_2)
[ 5v_2 = 60 \quad\Rightarrow\quad v_2 = \frac{60}{5}=12\ \text{V} ]
Thus the voltage at node 2 is exactly the same as the source voltage at node 1.
6. Determining the Branch Currents
With (v_1 = 12\ \text{V}) and (v_2 = 12\ \text{V}) known, the currents through each element can be found directly from Ohm’s law.
| Branch | Voltage difference | Resistance | Current (direction) |
|---|---|---|---|
| (R_1) (node 1 → node 2) | (v_2 - v_1 = 0) | (4\ \text{k}\Omega) | (I_{R1}=0\ \text{A}) (no net flow) |
| (R_2) (node 2 → ground) | (v_2 - 0 = 12) | (6\ \text{k}\Omega) | (I_{R2}= \dfrac{12}{6,\text{k}} = 2\ \text{mA}) (leaving node 2) |
| Current source (I_s) | – | – | (2\ \text{mA}) (entering node 2) |
The algebraic sum of currents at node 2 is therefore
[ I_{R1}+I_{R2}-I_s = 0 + 2\ \text{mA} - 2\ \text{mA}=0, ]
which satisfies KCL, confirming the correctness of the solution.
7. General Observations
- Reference Node Simplification – Selecting the bottom rail as ground eliminated the need for a separate equation for node 1, reducing the system to a single unknown.
- Current‑Source Handling – When a current source is attached to a node, its contribution appears with a sign opposite to the assumed direction of currents leaving the node.
- Unit Consistency – Expressing resistances in kilo‑ohms and currents in milliamps allowed the equations to be solved without extra conversion factors, keeping the algebra tidy.
- Verification – Substituting the obtained node voltages back into the KCL equations provides an automatic sanity check; any discrepancy would indicate a sign error or arithmetic slip.
8. Conclusion
Nodal analysis transforms the physical interconnections of a circuit into a compact set of linear equations whose unknowns are the node voltages. By applying Kirchhoff’s Current Law at each essential node and substituting Ohm’s Law for each resistive branch, the method yields a solvable linear system. In the example of Fig 3.51, the reference‑node choice reduced the problem to a single equation, which was solved straightforwardly to obtain (v_1 = v_2 = 12\ \text{V}). The resulting branch currents satisfied KCL, confirming the correctness of the solution. This systematic approach scales efficiently to larger networks, making nodal analysis a cornerstone technique for the analysis and design of electronic circuits.
