Circuit with two resistances and inductance

In summary, the problem is trying to find two currents, i1 and i2, by solving a differential equation. The equation can be simplified by starting with a nodal equation and solving for the potential at the one independent node. Once the potential is known, the two currents can be found by solving the differential equation for the current through the inductor.
  • #1
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Homework Statement


http://img534.imageshack.us/img534/5788/gimpjl.png

I have this circuit and I want to know [itex]i_1[/itex] and [itex]i_2[/itex] (currents in [itex]R_1[/itex] and [itex]R_2[/itex]).
[itex]\varepsilon[/itex] is electromotive force and L is inductance.

I'd like to know if the following system of equations is correct and if I can get [itex]i_1[/itex] and [itex]i_2[/itex].

2. The attempt at a solution

\begin{cases} \varepsilon=R_1(i_1+i_2)+R_2i_2 \\ \varepsilon-L\frac{d}{dt}i_1=R_1(i_1+i_2) \end{cases}

I think I'm wrong, because i1 is the current in L, not in R1, but I must get current in R1!
How can I correct the system of equations?
 
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  • #2
Do you have to solve the differential equation for the circuit, or can you take advantage of the known properties of first order circuits (initial conditions at t = 0+; steady state conditions; the forms of their transient solutions) to write the solution?

As a general hint/suggestion, note that if you happened to have an expression for the potential where R1 and R2 (and L) meet, then it would be a simple matter to write expressions for the two currents that you're looking for.
 
  • #3
gneill said:
Do you have to solve the differential equation for the circuit, or can you take advantage of the known properties of first order circuits (initial conditions at t = 0+; steady state conditions; the forms of their transient solutions) to write the solution?

As a general hint/suggestion, note that if you happened to have an expression for the potential where R1 and R2 (and L) meet, then it would be a simple matter to write expressions for the two currents that you're looking for.


Thank you for your answer! My main problem is writing the equations of the system. Can you help me?
 
  • #4
The set of equations that you've already written will allow you to find a differential equation for the current through the inductor (i1 in your equations). If you solve for this current you can then go back and find i2, and then find the current through R1 since its the sum of the other two. The algebra may get tedious.

I think that if I were to do this problem I'd consider starting with a nodal equation and find the potential at the one independent node (so only one equation to deal with). This would involve an integral equation rather than a differential equation, but a quick differentiation would reduce it to familiar form. Solving it, having the expression for the node voltage would let me easily find the two currents. Also, I think I'd simplify the circuit first: a Norton equivalent would get rid of one of the resistances.
 
  • #5
thank you so much! :)
 

Related to Circuit with two resistances and inductance

1. What is the purpose of using two resistances and inductance in a circuit?

The use of two resistances and inductance in a circuit allows for the control and regulation of the flow of electricity. The resistances limit the current and the inductance stores energy, resulting in a more stable and efficient circuit.

2. How do the values of the resistances and inductance affect the circuit's behavior?

The resistance values determine the amount of current that can flow through the circuit, while the inductance value affects how quickly the current changes. Together, these values impact the circuit's frequency response and overall performance.

3. How can the total impedance of the circuit be calculated?

The total impedance of the circuit can be calculated using the formula Z = √(R^2 + (ωL)^2), where R is the total resistance and ωL is the total inductance. This calculation takes into account both the resistive and reactive components of the circuit.

4. What are some real-world applications of circuits with two resistances and inductance?

Circuits with two resistances and inductance are commonly used in electronic devices such as radios, amplifiers, and power supplies. They are also utilized in electric motors, generators, and other industrial equipment to control and manage the flow of electricity.

5. How can the values of the resistances and inductance be adjusted to optimize circuit performance?

The values of the resistances and inductance can be adjusted by changing the physical components in the circuit or by using variable resistors and inductors. This allows for fine-tuning of the circuit's behavior and can improve its efficiency and stability.

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