Finding the steady state expression of a circuit

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Discussion Overview

The discussion revolves around finding the steady-state expression for the current in a circuit with two sinusoidal voltage sources. Participants explore the implications of the circuit's configuration, the use of superposition, and the application of phasor analysis to derive the required current expression.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about the definition of Io and its role in the problem, suggesting that it is not clearly defined in the problem statement.
  • There is a discussion about the steady-state solution of circuits, with some participants indicating that capacitors act as open circuits and inductors as short circuits under DC conditions, while others emphasize the need for sinusoidal analysis.
  • Some participants propose using superposition to analyze the circuit with two voltage sources, suggesting that each source should be considered separately to find the current through the inductor.
  • There is mention of using phasor analysis and the need to derive voltages and currents in the frequency domain, with some participants questioning how to approach the calculations.
  • Participants discuss the notation of I' and I'', suggesting they represent different currents in the circuit rather than derivatives.
  • Some participants share their calculations and seek validation, expressing uncertainty about whether their approach is correct.
  • There are indications of frustration from participants who feel lost and are seeking clearer guidance or examples to follow.

Areas of Agreement / Disagreement

Participants generally express confusion and uncertainty regarding the problem, with no consensus on the interpretation of Io or the best approach to find the steady-state expression. Multiple competing views on analysis techniques and circuit behavior remain unresolved.

Contextual Notes

Limitations include unclear definitions of variables, assumptions about circuit behavior under sinusoidal inputs, and unresolved mathematical steps in deriving the current expression.

Bluestribute
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Homework Statement


For the circuit in (Figure 1) , suppose
va=10cos16,000tV
vb=20cos4000tV.
Suppose that R = 700Ω .

Write the steady-state expression for io(t) as io=I′ocos(ω′t+ϕ′)+I′′ocos(ω′′t+ϕ′′), where −180∘<ϕ′≤180∘, −180∘<ϕ′′≤180∘, and ω′>ω′′. Find the numerical value of I′o.

Find the numerical value of ϕ′.

Find the numerical value of ω′.

Find the numerical value of I′′o.

Find the numerical value of ϕ′′.

Find the numerical value of ω′′.

Homework Equations


Um . . .

The Attempt at a Solution


You lost me at Io . . . Seriously, I don't know what it's even asking . . .
 

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For starters: what do you know about the steady state solution of a circuit?
 
Bluestribute said:
You lost me at Io . . . Seriously, I don't know what it's even asking . . .
Since Io is not defined in the problem statement or the circuit diagram there is no solution to the problem as given.
 
axmls said:
For starters: what do you know about the steady state solution of a circuit?
When you put it like that . . . sinusoidal? Capacitors are open and inductors shorts?

This is all that they give me. The first question asks what analysis technique should be used (aka superposition). Then it asks about the I equation . . .
 
gneill said:
Since Io is not defined in the problem statement or the circuit diagram there is no solution to the problem as given.

I'm assuming Io is the current caused by the voltage vo.

Bluestribute - capacitors acting as open circuits and inductors acting as short circuits is something that happens in DC circuits, but you have a sinusoidal input. Do you know anything about phasor analysis (i.e. Frequency domain analysis)?
 
axmls said:
I'm assuming Io is the current caused by the voltage vo.

Bluestribute - capacitors acting as open circuits and inductors acting as short circuits is something that happens in DC circuits, but you have a sinusoidal input. Do you know anything about phasor analysis (i.e. Frequency domain analysis)?
Yeah . . . I guess it just seems like a really broad question and I'm not understanding what you're looking for. Or what I'm supposed to do to find I'o
 
It gives you a circuit with two sources and a few elements. It gives you equations to describe the sources, and it wants you to use all of that information to find an equation that describes the current in (I assume) the middle branch of the circuit (after the transient solution has died out).
 
I think it's a case of using superposition to complete the expression for the current through the inductor.
 
So what is I'o? What are these prime terms? I know the Cdv/dt formulas. But . . . do I use that to solve for i(t) and then . . . do . . . something?
 
  • #10
I' and I'' are an alternative notation to I1 and I2.

At least, that's how I think they are using it here. Nothing to do with derivatives.

And, despite my new April 1 avatar, the aforewritten lines are meant to be taken seriously. :smile:
 
  • #11
So is I1 and I'o the . . . left mesh?
 
  • #12
Two meshes. One carries I'o and the other I''o.
 
  • #13
So derive Va and multiply by 625E-9 and derive Vb, then add those together?
 
  • #14
I also tried doing V/Z (Z, for Va, being 700-100j). No luck.
 
  • #15
Why don't you show details of your work for one of the cases?

You've recognized that you will need to use superposition here because the two voltage sources have different frequencies. So call "case a" the situation where you've suppressed vb and only source va is active. Show how you solve for the resulting current through the inductor.
 
  • #16
Inductor is shorted.

1/(625nf*16000) = 100
Z = 700 - 100j (resistor + conductor)
Va = 10cos16000t

Divide Va by Z with a calculator
 
  • #17
It's the inductor voltage which you are needing to find, so I can't see why you would want to short it.
 
  • #18
I want to short it because I have no idea what I'm doing and after 17 posts, I still have no idea what I'm doing. It'd be great if someone could just show me some work instead of having me run circles around my own head, because I CLEARLY am not getting ANYWHERE with anything that's been said. I CLEARLY don't know what's going on, what's being asked, or how to solve it.
 
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  • #19
The circuit has two unrelated voltage sources, and you are required ultimately to find the current through the inductor. If it had only one voltage source then I think you would be able to solve that, correct?

So, take the sources one at a time, and "pretend" that the other source is zero volts. (That doesn't mean it's not present, it means it is present and is supplying a voltage of 0.00V.)

Determine the inductor current under each of those two conditions.
 
  • #20
So:

(Va-Vo)/-j100 + -Vo/700 = Vo/j400
Vo = 12.8665+2.4508j

like that for Va? Is that even remotely correct? Then what do I do with it?
 
  • #21
Bluestribute said:
So:

(Va-Vo)/-j100 + -Vo/700 = Vo/j400
Vo = 12.8665+2.4508j

like that for Va? Is that even remotely correct? Then what do I do with it?
That's good! Might as well determine the current it causes through the inductor at this point. Convert the result back to the time domain and set it aside for now while you do the same procedure for vb active.
 
  • #22
The approach looks right. I can't do any math checks at the moment.

Once you know the inductor voltage, divide by the reactance to determine current. Then change to polar so you can calculate the phase angle as required in the general expression you are trying to find values for.
 
  • #23
Z = 700 + 300j (because -100j + 400j)
V/Z = 0.0168-0.00370j

That seems incredibly small for the current . . .
 
  • #24
Nevermind. Thanks. I was just playing stupid. No, reactance is that inductor value. Now I'm getting the right numbers . . .
 

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