Nodal Analysis

Question

Using nodal analysis, determine the voltage at node 𝑉𝑥V x in the circuit shown below. Assume 𝑉1=10𝑉V 1 =10V, 𝑉2=5𝑉V 2 =5V, and all resistors have known values.

SalarySmartSA · Answer

to determine the voltage at node

v_x

using nodal analysis, we need to follow these steps:

v_x

v_x

v_x

let's outline the steps based on the given circuit: circuit description: assume the circuit includes resistors and voltage sources connected to node

v_x

. steps to solve:

v_x

v_x

v_x

v_x

v_x

i1,i2,…i_1, i_2, \ldots

v_x

v_x

mohamedkhedrmohamed123 · Answer

nodal analysis is a systematic method to determine the voltages at various nodes in an electrical circuit. to solve for the voltage at node $ v_x $ using nodal analysis, we need to apply kirchhoff’s current law (kcl) at each node. kcl states that the sum of currents entering a node is equal to the sum of currents leaving the node.
let's assume a simple circuit with a node $ v_x $ connected to two voltage sources $ v_1 $ and $ v_2 $ through resistors $ r_1 $ and $ r_2 $ respectively, and grounded through another resistor $ r_3 $.
### circuit description
- $ v_1 $ is connected to $ v_x $ through resistor $ r_1 $.- $ v_2 $ is connected to $ v_x $ through resistor $ r_2 $.- $ v_x $ is connected to the ground through resistor $ r_3 $.- $ v_1 = 10v $- $ v_2 = 5v $- the resistances $ r_1, r_2, r_3 $ have known values.
we need to find the voltage at node $ v_x $.
### step-by-step solution using nodal analysis
1. **identify nodes and assign voltages:**   - let the voltage at node $ v_x $ be $ v_x $.   - voltage at the ground node is 0v.
2. **apply kirchhoff’s current law (kcl) at node $ v_x $:**   according to kcl, the sum of currents leaving the node $ v_x $ should be equal to zero.      let's denote the currents through resistors $ r_1, r_2, $ and $ r_3 $ as $ i_1, i_2, $ and $ i_3 $ respectively.      $$   i_1 + i_2 + i_3 = 0   $$      the current through a resistor can be determined using ohm's law, $ i = \frac{v}{r} $.
3. **express currents through resistors in terms of node voltages:**   - current $ i_1 $ through $ r_1 $:     $$     i_1 = \frac{v_x - v_1}{r_1}     $$   - current $ i_2 $ through $ r_2 $:     $$     i_2 = \frac{v_x - v_2}{r_2}     $$   - current $ i_3 $ through $ r_3 $:     $$     i_3 = \frac{v_x - 0}{r_3} = \frac{v_x}{r_3}     $$
4. **set up the kcl equation:**   $$   \frac{v_x - v_1}{r_1} + \frac{v_x - v_2}{r_2} + \frac{v_x}{r_3} = 0   $$
5. **substitute the known values:**   $$   \frac{v_x - 10}{r_1} + \frac{v_x - 5}{r_2} + \frac{v_x}{r_3} = 0   $$
6. **combine terms to solve for $ v_x $:**   multiply through by a common denominator (let's use $ r_1 r_2 r_3 $) to clear the fractions:   $$   r_2 r_3 (v_x - 10) + r_1 r_3 (v_x - 5) + r_1 r_2 v_x = 0   $$      expand and collect terms involving $ v_x $:   $$   v_x (r_2 r_3 + r_1 r_3 + r_1 r_2) = 10 r_2 r_3 + 5 r_1 r_3   $$      factor out $ v_x $:   $$   v_x (r_2 r_3 + r_1 r_3 + r_1 r_2) = 10 r_2 r_3 + 5 r_1 r_3   $$      solve for $ v_x $:   $$   v_x = \frac{10 r_2 r_3 + 5 r_1 r_3}{r_2 r_3 + r_1 r_3 + r_1 r_2}   $$
### interpretation of results
the expression for $ v_x $ shows that the voltage at node $ v_x $ is determined by the resistances $ r_1, r_2, r_3 $ and the voltage sources $ v_1 $ and $ v_2 $. the weights of the resistances in the denominator dictate the relative influence of each voltage source on $ v_x $.
### example calculation
let's consider an example with specific resistor values:- $ r_1 = 2 \omega $- $ r_2 = 3 \omega $- $ r_3 = 1 \omega $
substitute these values into the expression:
$$v_x = \frac{10 \cdot 3 \cdot 1 + 5 \cdot 2 \cdot 1}{3 \cdot 1 + 2 \cdot 1 + 2 \cdot 3}$$
calculate the numerator and denominator separately:
numerator:$$10 \cdot 3 \cdot 1 + 5 \cdot 2 \cdot 1 = 30 + 10 = 40$$
denominator:$$3 \cdot 1 + 2 \cdot 1 + 2 \cdot 3 = 3 + 2 + 6 = 11$$
therefore,$$v_x = \frac{40}{11} \approx 3.64 v$$
### conclusion
using nodal analysis, we determined the voltage at node $ v_x $ in the circuit by setting up and solving the kcl equation at node $ v_x $. the general solution for $ v_x $ is given by:
$$v_x = \frac{10 r_2 r_3 + 5 r_1 r_3}{r_2 r_3 + r_1 r_3 + r_1 r_2}$$
for specific resistor values, $ r_1 = 2 \omega $, $ r_2 = 3 \omega $, and $ r_3 = 1 \omega $, the voltage at node $ v_x $ was found to be approximately $ 3.64v $. this approach can be applied to any circuit with known resistor values and voltage sources to find the voltage at a particular node using nodal analysis.

KholoudHassan · Answer

the first thing we should do is we need to set up and solve a system of linear equations based on the circuit diagram.
we have:
v1 = 10 vv2 = 5 vall resistor values are known
let's define the nodes in the circuit:
node 1: node connected to v1node 2: node connected to v2node x: the node where we want to find the voltage vxusing kirchhoff's current law (kcl):at node 1:(v1 - vx) / r1 + (vx - v2) / r2 = 0at node x:(vx - v1) / r1 + (vx - v2) / r2 = 0
so :vx = (r2 * v1 + r1 * v2) / (r1 + r2)vx = (r2 * 10 + r1 * 5) / (r1 + r2)
after this just use the resistors values to solve for vx.

bu.gnam · Answer

to determine the voltage at node

v_x

using nodal analysis, we need to follow these steps: circuit description assume the circuit consists of several components including resistors and voltage sources connected to node

v_x

. steps for nodal analysis:

v_x

v_x

v_x

∑iin=∑iout\sum i_{ ext{in}} = \sum i_{ ext{out}}

iini_{ ext{in}}

v_x

iouti_{ ext{out}}

v_x

\frac{v}{r}

v_1 = 10v

v_2 = 5v

v_x

example calculation outline: let's outline a hypothetical circuit: suppose

v_1

is connected across a resistor to node

v_x

.suppose

v_2

is connected across another resistor to node

v_x

. assume resistors

r_1

and

r_2

connect

v_1

and

v_2

to node

v_x

, respectively. example equations (hypothetical): from kcl at node

v_x

:

v1−vxr1+v2−vxr2=0\frac{v_1 - v_x}{r_1} + \frac{v_2 - v_x}{r_2} = 0

substitute

v_1 = 10v

,

v_2 = 5v

, and solve for

v_x

:

10v−vxr1+5v−vxr2=0\frac{10v - v_x}{r_1} + \frac{5v - v_x}{r_2} = 0

this equation will allow you to solve for

v_x

given the values of

r_1

and

r_2

. conclusion: to find the exact voltage at node

v_x

, you need to apply the above steps with the specific circuit details, including resistor values and how

v_1

and

v_2

are connected to node

v_x

. this methodical approach ensures that you correctly apply nodal analysis to determine

v_x

based on the given circuit configuration.

mouloud-abaragh · Answer

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