Ok! I got this equations and answers:

For V0:
8i0 = 6; therefore i0 = 6/8 = 0.75A in the resistance of 8 Ohm's

For i0:
6 = i0 + i0/4 = 6/1.25 = 4.8A

I am assuming that the book is rounding the current on the 8Ohm's resistor to 1. And not rounding the current of i0 to 5.

I'll look over your equations once you use units properly (or at least make an honest attempt to).

 

Ok! I got this equations and answers:

So is i0=0.75A or is i0=4.8A? ....or does 6=4.8?

 

is = i0 + i1 + i2

So you wrote the KCL node equation where the sum of currents at a node equal zero; however, in nodal analysis the branch currents must be expressed in terms of the node voltage (and current sources are whatever they are).
This problem has a single node, Vo, plus the ground node. Therefore in the KCL node equation express all currents which are not a given constant (e.g., 6A) in terms of Vo. Note that current leaving the node is positive. Solve for Vo then find the currents as required.

 

After using units so nicely and properly, why is your last equation the sum of a pure number and three voltages?

 

One of the benefits of using nodal analysis exclusively and always following the same simple rules for writing the node equations is that units become superfluous. Write the node equations correctly and you can't go wrong; but get creative and try an unfamiliar approach - that is when unit tracking can save you from silly mistakes.

"...why is your last equation the sum of a pure number and three voltages?" When you solve the equation, Vo is just a number.

 

One of the benefits of using nodal analysis exclusively and always following the same simple rules for writing the node equations is that units become superfluous. Write the node equations correctly and you can't go wrong; but get creative and try an unfamiliar approach - that is when unit tracking can save you from silly mistakes.

"...why is your last equation the sum of a pure number and three voltages?" When you solve the equation, Vo is just a number.

Vo is NEVER just a number. It is a voltage and a voltage is the combination of a numerical value and a unit.

"Write the node equations correctly and you can't go wrong"?

Really?

That's like saying, "Don't make any mistakes and you can't go wrong". It's an obvious and completely meaningless claim. What if you DON'T write the node equations correctly? Just look at the TS's first attempt (in Post #12) and you will see that he wrote the following node equation:

\(
i \; - \; i_o \; - \; \frac{i_o}{4} \; - \; 8i_o \; = \; 0
\)

As written, that looks like a perfectly fine node equation. But that '8' is not a pure number (unlike the 4), it is '8Ω'. So (completely separate from the other error in the final term), tracking units would have immediately shown that this equation simply cannot be correct. But because he couldn't be bothered to track units, he just blindly chugged ahead.

I can't even begin to count the number of times that people have spent hours and pages working a problem (both in academic and in industrial settings) that, had they put the units in the very first line where they did the initial set up, they would have seen that any time they spent from that moment on was going to be wasted effort.

Also, are you really claiming that people only make "silly mistakes" when trying an unfamiliar approach? We ALL make silly mistakes on a regular basis, no matter how familiar we are with the approach. If you are working through the algebra to solve a problem, you ARE going to make occasional mistakes. The vast majority of those mistakes will mess up the units and allow you to catch them immediately or, if you don't, to quickly track them down once you do realize that an error has been made.

 

Vo is NEVER just a number. It is a voltage and a voltage is the combination of a numerical value and a unit.

My symbolic algebra package has the capability to automatically track units. I turn that feature off because it is too distracting. If you think that Vo is the combination of a numerical value and a unit, then that can be corrected by turning the feature off.

"Write the node equations correctly and you can't go wrong"?
Really?

Sure. That's why its important to learn how to write node equations correctly. It's easy if you follow the simple rules, but if you have to think about what you're doing then you're breaking one of the rules.

That's like saying, "Don't make any mistakes and you can't go wrong". It's an obvious and completely meaningless claim.

It's more like saying, "Learn how to write node equations without having to think about it, because having a correct node equation means that tracking units is superfluous." It is an obvious claim because it is true, but certainly not meaningless although the meaning may be hidden deeply.

I can't even begin to count the number of times that people have spent hours and pages working a problem (both in academic and in industrial settings) that, had they put the units in the very first line where they did the initial set up, they would have seen that any time they spent from that moment on was going to be wasted effort.

So you understand there is a difference between not having to think about what you are doing, versus not thinking about what you are doing.

Also, are you really claiming that people only make "silly mistakes" when trying an unfamiliar approach?

Working in unfamiliar territory naturally leads us to making conceptual errors which would be deemed "silly mistakes" by those more familiar with the methodology (even by ourselves, eventually).

We ALL make silly mistakes on a regular basis, no matter how familiar we are with the approach. If you are working through the algebra to solve a problem, you ARE going to make occasional mistakes. The vast majority of those mistakes will mess up the units and allow you to catch them immediately or, if you don't, to quickly track them down once you do realize that an error has been made.

We all make transcription errors that are not silly mistakes, but can lead to silly results. Tracking units may have some value during the initial setup of the node equation; a better approach is result validation - substitute the result back into the problem to determine whether KVL or KCL have been violated.

 

As the person judiciously applies due diligence, the transcription errors and other "silly errors" disappears. Due diligence in design work is the application of the old "measure twice, cut once" saying in carpentry.

 

My symbolic algebra package has the capability to automatically track units. I turn that feature off because it is too distracting. If you think that Vo is the combination of a numerical value and a unit, then that can be corrected by turning the feature off.

So your notion of what a physical quantity is is determined by some software package?

Since you claim that units are not a fundamental part of a physical quantity, let's say that Tom's height is 32 while Bob's height is 75. Who's taller?

Sure. That's why its important to learn how to write node equations correctly. It's easy if you follow the simple rules, but if you have to think about what you're doing then you're breaking one of the rules.

Ah. It's becoming a lot clearer now. You don't like to be bothered to think. You just want to be a monkey letting some tool do your thinking for you.

It's interesting that you completely ignored the clear example from this very thread about how tracking units would have caught a simple mistake right at the very beginning.

 

As the person judiciously applies due diligence, the transcription errors and other "silly errors" disappears. Due diligence in design work is the application of the old "measure twice, cut once" saying in carpentry.

Yes. And properly tracking units plays a key role in applying due diligence.

 

It's interesting that you completely ignored the clear example from this very thread about how tracking units would have caught a simple mistake right at the very beginning.

Your 'clear' example was completely ignored because it was totally uninteresting. Even a monkey, if it followed the simple method for writing a node equation, would not have come up with such an answer. Nevertheless, the error would have been exposed by result validation (without the need for unit tracking.) However, if someone should fail to use the simple method which will guarantee having a correctly formed set of node equations, then perhaps they should be required to perform unit tracking as penance. One sin begets another!

 

I solved the problem by converting the three parallel branches to their conductive equivalent then divided the current accordingly. To check it, I converted the CCCS load to it's equivalent resistance and computed the currents.

I did NOT calculate Vo as one could figure that out strictly by inspection once they had all the currents.

I did write the formulae down, well, typed them into Word in case I had to produce them. Once I wrote the formulae, I used Excel to manipulate the math.

 

I solved the problem...

Whatever works for you. But it seems much more complicated than writing the node equation. I'd probably make a mistake(s) doing it your way.

 

I solved the problem by converting the three parallel branches to their conductive equivalent then divided the current accordingly. To check it, I converted the CCCS load to it's equivalent resistance and computed the currents.

I did NOT calculate Vo as one could figure that out strictly by inspection once they had all the currents.

I did write the formulae down, well, typed them into Word in case I had to produce them. Once I wrote the formulae, I used Excel to manipulate the math.

Just do the part in red to solve the network. Notice that the controlled current source is carrying a downward directed current just like the resistors do. If the ccs were replaced by a resistor that carried the same current, what would the value of that resistor be? Well, the ccs is carrying 1/4 as much current as the 2 ohm resistor, so replacing the ccs with a 4 * 2Ω = 8Ω resistor would leave all the currents and voltages unchanged.

So, to solve the circuit, replace the controlled current source with an 8Ω resistor and then do the rest in your head.

 

That's exactly how I did it ... I know that to draw one-fourth the current, you need four times the resistance.

 

Whatever works for you. But it seems much more complicated than writing the node equation. I'd probably make a mistake(s) doing it your way.

There's always more than one way to skin a cat. Whatever makes it click in the TS's head. If the TS was constrained on what he had to use to solve it, that is a different story, but the alternate methods could "check" the answer, or as WBahn asks "Does it make sense?"

 

Your 'clear' example was completely ignored because it was totally uninteresting. Even a monkey, if it followed the simple method for writing a node equation, would not have come up with such an answer. Nevertheless, the error would have been exposed by result validation (without the need for unit tracking.) However, if someone should fail to use the simple method which will guarantee having a correctly formed set of node equations, then perhaps they should be required to perform unit tracking as penance. One sin begets another!

This "guarantee" of yours is nothing of the sort -- it's another fallacy based on the notion that IF the person doesn't make any mistakes THEN they won't get a wrong answer. But given an example that proves that people DO make mistakes, you call it uninteresting. Since you are so willing to assume that people using a simple method won't make mistakes, why do you recommend doing result validation. After all, the simple method guarantees that there aren't any errors in your world.

If I'm paying someone engineer's wages, I expect them to exercise due diligence and that means not only checking their result after spending a lot of time to get it, but also using simple and highly effective error detection techniques that will catch most of their mistakes BEFORE they spend a lot of time getting an answer that is guaranteed to be wrong right from the start. Also, by the time they get to result validation, about the only conclusion they can usually come to is that they messed something up. They typically have no idea what they messed up and no trail to trace back from the answer to where the mistake was made. If they track their units, not only do they catch their mistake much earlier (most of the time), but they have a shiny path leading them right back to where the mistake was made -- and if they are checking the units consistently, that mistake is almost always within a few lines of work.

 

This "guarantee" of yours is nothing of the sort -- it's another fallacy based on the notion that IF the person doesn't make any mistakes THEN they won't get a wrong answer. But given an example that proves that people DO make mistakes, you call it uninteresting. Since you are so willing to assume that people using a simple method won't make mistakes, why do you recommend doing result validation. After all, the simple method guarantees that there aren't any errors in your world.

It is apparent that what we have here is a lack of appreciation for the power of the simple method of writing node equations. Use of the simple method truly is a guarantee against making the conceptual mistakes that unit tracking might help uncover. But anyone can make simple errors with the procedure such as transcription errors (using the wrong subscript; transposing numbers; etc) or failing to complete the procedure (e.g. forgetting to include an adjacent node), yet unit tracking is useless for these simple errors. That is why unit tracking is so laughable; it can only help the incompetent who are prone to making serious conceptual mistakes, whereas anyone applying the simple method for writing node equations is insulated from making these conceptual mistakes.

Consider the following circuit as an example. Some would consider it a mistake waiting to happen. But give the circuit to a trained monkey able to follow the simple method, and the node equations will be derived with uncanny accuracy.


Before writing the node equations, Step #1 is to identify the nodes and super-nodes. Designate the ground node if that has not been given already. Color-code the diagram with highlighters.


Label the nodes with their voltage name as it will appear in the node equations. Label the diagram with any super-node offset voltages. If there are control currents present (i.e. currents which control something else in the circuit), realize that no current variables can appear in the node equations. But constant current sources or voltage-controlled current sources are OK, so express any control currents in terms of the node voltages and mark that on the diagram as shown.


Write a node equation for each identified node & super-node (but never the ground node). It may help to realize that a node equation is merely KCL (the sum of currents at a node equals zero) but there are no current variables shown in the node equation. For each node identify the adjacent nodes connected to it by a circuit branch. Each adjacent node will correspond to a term in the node equation. Each term in the node equation is formed by first writing that node's voltage, subtracting the adjacent node's voltage, and dividing the voltage difference by the resistance between the node and adjacent node. However, if the branch to the adjacent node contains a current source, then use the value of the sourced current instead (current leaving the node is considered positive). If the adjacent node is the ground node, I usually neglect subtracting zero from the node voltage. Add all the terms and set it equal to zero. Do the same procedure for the next node equation until the set of node equations is complete. All the rest is just an algebra problem. Note that impedance would be substituted for resistance when working in the complex frequency s-domain.

It is important to fully annotate and color-code the working diagram in order to minimize transcription errors. Be the Artist-Engineer. Note that the opportunity for making a conceptual mistake in writing the node equations is quite limited. If the trained monkey needs to make a conceptual decision when using the simple method, then it is likely they are not correctly following the procedure.

In order to determine that the above node equations are correct, I used the following circuit values to calculate the node voltages, then used result validation to ensure that KCL was not violated. No errors, and no unit tracking necessary.

R1=20, R2=33, R3=150, R4=27, R5=82, R6=100, R7=82, R8=120, RO=50, V1=4, V2=5, V3=8, a=0.17, b=0.39

V_A = 2.682402465
V_B = -0.4911339342
V_C = 1.043518510
V_O = 3.310480393

 

I have never said that using a systemic approach to develop a system of equations is not a powerful tool. In fact, I have stated repeatedly over the years that that is exactly what Node Voltage Analysis and Mesh Current Analysis are -- they are systematic ways of applying KCL and KCL, respectively, so that the equations can generally be written down by inspection with a much lower risk of making a mistake. But they provide NO guarantee that mistakes will not be made -- as the original post in this thread PROVES!

Most, not all, algebraic errors that are made along the way of working toward a solution will mess up the units. If units are tracked properly, these mistakes can be found and corrected immediately, instead of wasting time pursuing an answer that is guaranteed to be wrong requiring you to start from scratch. They also prevent adding 3 and 30 to get 33 when it was 3 A and 30 mA. I realize that you don't see any value in catching mistakes quickly, since apparently you don't make mistakes. But mere mortals DO.

 

View attachment 111580

What software did you use to produce the schematic?