Solved Problems

Tag: KVL

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
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
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
Find currents using KVL

Find resistor currents using KVL.

Solution:

$R_1$ and $V_1$ are parallel. So the voltage across $R_1$ is equal to $V_1$ . This can be also calculated using KVL in the left hand side loop:

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September 26, 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 1-10: 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