Solved Problems

Tag: KCL

Solving a Circuit Using KVL and KCL
Use Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) to find $V_x$ and $I_y$ .

Applying Kirchhoff’s Voltage Law (KVL)

KVL states: The sum of all voltages in a closed loop is zero.

Loop 1: 20V Source, 2Ω Resistor, and 5A Current Source

We start with Loop 1 which is shown below.

The KVL equation is:

$-20V - (2\Omega \cdot 2A) + V_{5A} = 0$

Simplify:

$-20V - 4V + V_{5A} = 0$

$V_{5A} = 16V$

Loop 2: Voltage Across $V_{5A}$ , 3V Source, and 7Ω Resistor

Next, we analyze Loop 2 shown below:

The equation is:

$-16V + 4V - (6\Omega \cdot I_y) = 0$

Simplify:

$-12V -6\Omega \cdot I_y = 0$

$I_y = -2A$

KCL: Applying Kirchhoff’s Current Law (KCL)

KCL states: The total current entering a node equals the total current leaving the node.

At the middle node, the currents are:

$2A + 5A +(-2A) - I_{3\Omega} = 0$

Simplify:

$I_{3\Omega} = 5A$

Finding $V_x$

Finally, we calculate $V_x$ using KVL in Loop 3:

The equation is:

$-16V + (3\Omega \cdot 5A) - V_x = 0$

Simplify:

$-16V + 15V - V_x = 0$

$V_x = -1V$

Using Python to Solve the Circuit

One powerful way to solve electrical circuits is by using the Lcapy library in Python. Lcapy is an open-source Python package designed for symbolic linear circuit analysis and signal processing.

Below is an example of how to use Lcapy to solve the given circuit and find $I_y$ (the current through $V_2$ ) and $I_{R_1}$ (the current through the 2Ω resistor).

—

Python Code to Solve the Circuit
```
```python
from lcapy import Circuit
cct = Circuit("""
V1 1 0 20; down,
R1 1 N_A 2; right=1.5, i=I_{R_1},
I1 N_A 0_2 5; down, 
W N_A 5; up=0.7, 
R2 N_A 3 3; right=1.5, 
V_x 0_3 3 -1; up, 
V2 5 4 4; right=2, i=I_y,
R4 0_4 4 6; up,
W 0 0_2; right, 
W 0_2 0_3; right, 
W 0_3 0_4; right, 
""")
print(f"Current through R1 (I_R1): {cct.R1.i}")
print(f"Current through V2 (I_y): {cct.V2.i}")
```
which prints
```
Current through R1 (I_R1): 2
Current through V2 (I_y): -2
```
January 24, 2025
Nodal Analysis – Five-Node Circuit

Use nodal analysis to solve the circuit shown below and determine the power of node $V_{s_2}$ .

I. Identify all nodes in the circuit.
The circuit has five nodes as shown below.
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December 25, 2024
Solving a Simple Circuit of Three Elements

Solved the circuit to determine $I_x$ and power absorbed or supplied by each element.

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October 22, 2019
Circuit Containing Only Sources

We go through solving a circuit which only containes independent sources: two voltage sources and two current sources. KVL and KCL are used to determine voltages and currents.

Determine the amount of power absorbed or supplied by each source.

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August 28, 2019
Superposition method – Circuit with two sources

Find $I_x$ using superposition rule:

Solution

Superposition

The superposition theorem states that the response (voltage or current) in any branch of a linear circuit which has more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are turned off (made zero).
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October 15, 2013
Solve By Source Definitions, KCL and KVL

Find the voltage across the current source and the current passing through the voltage source.

Assume that $I_1=3A$ , $R_1=2 \Omega$ , $R_2=3 \Omega$ , $R_3=2 \Omega$ , $I_1=3A$ , $V_1=15 V$ ,

Solution
$R_1$ is in series with the current source; therefore, the same current passing through it as the current source:
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September 27, 2013
Problem 1-12: Using Voltage Sources to Determine Node Voltages

Determine the power of $R_1, R_2$ and $Vs_1$ . (Hint: there is no need to use nodal analysis; voltages between nodes can be easily found by the voltage sources.)

Solution

$V_{R_1}= Vs_1 = 10v \rightarrow P_{R_1}=\frac{V_{R_1}^2}{R_1}=50 W$
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April 23, 2010
Problem 1-11: Solving a Circuit with Three Nodes by the Nodal Analysis

Solve the circuit using nodal analysis and determine the power of $I_{s_1}$ .

Solution
I. Identify all nodes in the circuit. The circuit contains 3 nodes, as illustrated below.
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April 23, 2010
Problem 1-10: Solving by Nodal Analysis – Circuit with Four Nodes

Let’s use nodal analysis to solve this circuit and determine $V_a$ .

Solution
I. Identify all nodes in the circuit. There are four nodes in the circuit, as indicated below
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April 23, 2010