Nodal Analysis

 

Find the value of the voltage v in the circuit below.

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The first thing you should notice is the two different types of sources (current sources are the ones with arrows in the middle of them, voltage sources are the ones with the plus and minus). The 5 ampere current source (the arrow with a circle around it) and the 16 volt voltage source (the circle with a plus and minus in it), are independent sources. An independent source will supply a set current or voltage regardless of the setup of the circuit (it is independent of the circuit). The diamond shaped one around the arrow is a dependent, or controlled, current source. And the diamond around the plus and minus is a dependent voltage source. A dependent source has two parts: a gain and what it depends on. The gain in the current source’s case is 3/4, the part it depends on is \(v_y\), which is defined as the voltage across the 2 ohm resistor on the right.

NOTE: Since it depends on an outside voltage, the ¾ \(v_y\) is a voltage controlled current source. There are also current dependent current sources, current dependent voltage sources (the \(8i_x\)), and voltage dependent voltage sources.

When faced with a complicated circuit like this, one of the best techniques to use is nodal analysis, an application of Kirchhoff's current law (KCL). KCL states that the sum of the currents going into a node must equal the sum of the currents leaving the node.

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NOTE: Also remember that, by ohm's law, I = V/R. Where V is the change in voltage across an element, and R is the resistance of that element (an element being any source or resistor).

With that out of the way we can tackle the problem. The first step is to set your ground (where the voltage equals 0) and label your nodes. Generally you want your ground to be where the most nodes are, as it simplifies calculations. A node is where two terminals meet, which for us just means the wires in between the circuit elements. The problem has already pointed them out to us with the black dots. I’ll just label them A, B, C, and D.

Next, write out what we know based off of the problem/diagram. I’m calling the voltages at each node \(\mathrm{V}_{\mathrm{A}}\), \(\mathrm{V}_{\mathrm{B}}\), \(\mathrm{V}_{\mathrm{C}}\), and \(\mathrm{V}_{\mathrm{D}}\). Note that these are different from \(\mathrm{v}\) and \(\mathrm{v}_{\mathrm{y}}\), which are voltages across the resistors (so the voltage difference) rather than just the voltage at the nodes.

You get these by using the definition of voltage across an element, which is the positive end minus the negative end. \(v\) is just equal to \(\mathrm{V}_{\mathrm{A}}\) minus 0 since it’s between node A and the ground.

Now using that current is equal to the voltage across a resistor over the resistance (ohm's law), we can do KCL at each node. I’m going to use the convention that currents going into the node are negative, and currents going out are positive (the other way around works too, just be consistent).

I named a couple other currents for the sake of keeping track of what’s in the equations. The rest is just math!

We can set these two equations equal to each other (since both equal \(\mathrm{V}_{\mathrm{B}}\)). Then solve for \(\mathrm{V}_{\mathrm{A}}\) in terms of \(\mathrm{V}_{\mathrm{D}}\)

Now we need to get a value for \(\mathrm{V}_{\mathrm{D}}\) in terms of \(\mathrm{V}_{\mathrm{A}}\). To start, we know \(\mathrm{V}_{\mathrm{C}}\) and \(\mathrm{V}_{\mathrm{D}}\) are related, so let's solve for \(\mathrm{V}_{\mathrm{C}}\) first.

Now we need a value for \(i_z\) and \(i_y\), so let’s solve for those.

NOTE: We use \(\mathrm{i}_{\mathrm{x}}=-\mathrm{V}_{\mathrm{D}} / 2\) instead of \(\mathrm{i}_{\mathrm{x}}=\left(\mathrm{V}_{\mathrm{B}^{-}} \mathrm{V}_{\mathrm{C}}\right) / 8\) since we are trying to relate \(\mathrm{V}_{\mathrm{C}}\) and \(\mathrm{V}_{\mathrm{D}}\) (plus its an easier equation to deal with). Same idea for why we chose the equation for \(\mathrm{V}_{\mathrm{B}}\) (we are trying to have \(\mathrm{V}_{\mathrm{D}}\) in terms of \(\mathrm{V}_{\mathrm{A}}\))

Now we can use those in our equation for \(\mathrm{V}_{\mathrm{c}}\)

And finally, with \(\mathrm{V}_{\mathrm{D}}\) related to \(\mathrm{V}_{\mathrm{A}}\), we can solve for \(v\)