Finding the steady state expression of a circuit

[Bluestribute]

Homework Statement

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

Write the steady-state expression for i_o(t) as i_o=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 . . .

 

[axmls]

For starters: what do you know about the steady state solution of a circuit?

 

[gneill]

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.

 

[Bluestribute]

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 . . .

 

[axmls]

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)?

 

[Bluestribute]

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

 

[axmls]

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).

 

[NascentOxygen]

I think it's a case of using superposition to complete the expression for the current through the inductor.

 

[Bluestribute]

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?

 

[NascentOxygen]

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.

 

[Bluestribute]

So is I₁ and I'_o the . . . left mesh?

 

[NascentOxygen]

Two meshes. One carries I'_o and the other I''_o.

 

[Bluestribute]

So derive Va and multiply by 625E-9 and derive Vb, then add those together?

 

[Bluestribute]

I also tried doing V/Z (Z, for Va, being 700-100j). No luck.

 

[gneill]

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 v_b and only source v_a is active. Show how you solve for the resulting current through the inductor.

 

[Bluestribute]

Inductor is shorted.

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

Divide Va by Z with a calculator

 

[NascentOxygen]

It's the inductor voltage which you are needing to find, so I can't see why you would want to short it.

 

[Bluestribute]

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.

 

[NascentOxygen]

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.

 

[Bluestribute]

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?

 

[gneill]

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.

 

[NascentOxygen]

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.

 

[Bluestribute]

Z = 700 + 300j (because -100j + 400j)
V/Z = 0.0168-0.00370j

That seems incredibly small for the current . . .

 

[Bluestribute]

Nevermind. Thanks. I was just playing stupid. No, reactance is that inductor value. Now I'm getting the right numbers . . .