Solving a Nodal Analysis: Find Va(t)

[eehelp150]

Homework Statement

Find Va(t) using Nodal Analysis:

Homework Equations

Zc = 1/(jwC)
Zl = jw*L

The Attempt at a Solution

W = 250
Z36mH = j9 ohm
Z80mH = j20 ohm
Z0.25mF = -j16 ohm (is capacitor impedance always negative? It says so in my instructor's notes but I forgot why)

20cos(250t) -> 20 <0° V
1.2cos(250t+45°) -> 1.2<45° A

Are the following Nodal equations correct? Is Node1 simply Va?
Node1:
$$\frac{Node1-20<0}{8+9j} + \frac{Node1}{-j16}=0$$

Node2:
$$-1.2<45° + \frac{Node2-4Va}{j20}=0$$

 

[The Electrician]

Node1 and Node2 are just one node. Nodes are not defined by black dots. Those two dots are connected by a wire of perfect conductivity and the voltages at the two dots are the same, so they are really only one node.

 

[eehelp150]

Node1 and Node2 are just one node. Nodes are not defined by black dots. Those two dots are connected by a wire of perfect conductivity and the voltages at the two dots are the same, so they are really only one node.

I assume Va is the voltage across the Capacitor?

 

[The Electrician]

That looks OK; now all you have to do is solve for Va.

By the way, the reason that capacitor impedance is negative is because 1/j = -j

 

[eehelp150]

That looks OK; now all you have to do is solve for Va.

By the way, the reason that capacitor impedance is negative is because 1/j = -j

Of course. I feel like an idiot now.

 

[eehelp150]

That looks OK; now all you have to do is solve for Va.

By the way, the reason that capacitor impedance is negative is because 1/j = -j

I ended up getting
Va = 4528.6 - 911.4i
Is that correct?

 

[The Electrician]

No, I don't get that. As a sanity check, for Va to be 4000 volts plus seems a little high to me.

What happened to your equation in post #3?

If you'll show your work, we can probably find your error.

 

[eehelp150]

No, I don't get that. As a sanity check, for Va to be 4000 volts plus seems a little high to me.

What happened to your equation in post #3?

If you'll show your work, we can probably find your error.

Don't know what happened to it

I plugged the equation into MyAlgebra.com and it spit out the answer. Here is the equation again:
$$\frac{Va-20}{8+j9}-\frac{Va}{j16}-\frac{3Va}{j20} = \frac{1.2\sqrt{2}}{2}+ \frac{1.2\sqrt{2}}{2}i$$

 

[gneill]

I ended up getting
Va = 4528.6 - 911.4i
Is that correct?

It's not what I get, if that helps

 

[eehelp150]

It's not what I get, if that helps

I tried another calculator and got:
1.89293-12.2816 i
is that correct?

 

[gneill]

I tried another calculator and got:
0.848528+0.848528 i
is that correct?

Nope.

Sometimes plugging in all the complex values into a node equation and lugging them around in the reduction is more tedious and error prone than using symbols. Why not assign variables to the branch impedances and the current and solve? That way we can follow your steps.

 

[eehelp150]

Nope.

Sometimes plugging in all the complex values into a node equation and lugging them around in the reduction is more tedious and error prone than using symbols. Why not assign variables to the branch impedances and the current and solve? That way we can follow your steps.

I read the output wrong, it was 1.89293-12.2816 i.

 

[The Electrician]

Don't know what happened to it

I plugged the equation into MyAlgebra.com and it spit out the answer. Here is the equation again:
$$\frac{Va-20}{8+j9}-\frac{Va}{j16}-\frac{3Va}{j20} = \frac{1.2\sqrt{2}}{2}+ \frac{1.2\sqrt{2}}{2}i$$

I think you may be having a problem with your complex constant j. In Mathematica there is a difference between j9 and j 9.

Without a space between the j and the 9 (in other words, like this j9), Mathematica treats that as a variable name. With a space, like j 9, that is treated like j*9, the product of j and 9. Also at the very end of your equation you have i instead of j.

What does MyAlgebra use for the complex constant, j or i? Whichever it is, put the right one in your equation and instead of j9 and j20, etc., put j*9 and j*20 (or possibly i*9 and i*20 if i is the symbol for the complex constant) and run the equation again.

 

[eehelp150]

I think you may be having a problem with your complex constant j. In Mathematica there is a difference between j9 and j 9.

Without a space between the j and the 9 (in other words, like this j9), Mathematica treats that as a variable name. With a space, like j 9, that is treated like j*9, the product of j and 9. Also at the very end of your equation you have i instead of j.

What does MyAlgebra use for the complex constant, j or i? Whichever it is, put the right one in your equation and instead of j9 and j20, etc., put j*9 and j*20 (or possibly i*9 and i*20 if i is the symbol for the complex constant) and run the equation again.

 

[The Electrician]

I read the output wrong, it was 1.89293-12.2816 i.

This is correct.