# Nodal & Mesh Analysis: the same but different

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Nodal & Mesh Analysis: the same but different
An explanation for determining the current or voltage of complicated circuits. by Cap’n Tim Johnson, PE

Common concepts Circuits are constructed of branches.
The component(s) within a branch are known values expressed in ohmsi. Branches have unique current flows and voltage readings. Branches intersect at nodes. Nodes allow current and voltages to be fractured or split into parts. Conversely, nodes allow current and voltages to be summed together. Different methods of determining currents and voltages have been developed: Nodal Analysis allows you to directly determine voltages. Mesh Analysis allows you to directly determine currents. Knowledge of either the voltage or current across a branch allows you to determine the other using Ohm’s Law. Modern analysis is done using a computer model and no computation is required. i in more advanced circuits the ohms can represent reactance.

Nodal Analysis—determining the voltages of nodes
The nodes of a circuit are arranged so that they all connect to the same ground point (also a node) even if the node has to connect through another node. At the start the voltages are unknown so the naming convention for the voltages at the node are Vnode# . The ground point node is considered to be zero volts and could possible be named Vgnd or V0. Because of Ohm’s Law, a rule known as Kirchoff’s Current Law (KCL) is used in Nodal (aka voltage) analysis: KCL: The sum of the currents at any node is zero. Σi = Σiin - Σiout

Nodal Analysis—assigning current flow initially
Which current at a node is going in and which is going out? For branches with sources the following apply: Battery or voltage source: the current flows out of the positive terminal. Current source: the symbol has an indication of current flow direction in it. For addition branches the current direction can be assigned randomly provided that you respect previous designations at subsequent nodes. Signage: if current is into a node its (-), if current is out of a node its (+) Draw your current direction assignments on the schematic. Assign a number to each branch current. They will not necessarily match the node numbers. Each branch should have an assignment. If one node has all input currents, this is allowed at this point of the solution.

Nodal Analysis—converting your marked up schematic into formulas—Step 1
In Step 1 we create a list of the equivalent currents in each branch. You can write them right on the schematic. The branch currents will be in form of the Ohm’ Law format I=V/R The i’s are not known nor are the V’s but the R’s are known. The Voltages are either from the node to ground (ergo, V is Vx). Or the Voltage for the current formula is between nodes. In this case the Voltage is a difference between the voltages at either node (Vx – Vy). Use the current arrow you drew as your guide: The voltage at the source of the arrow is listed first The voltage where the arrow points is listed second after the negative sign. Remember to bring along the signage from the drawing into the formula (2 - = +)

Vx/Rx(1/Rx)Vx (Vx – Vy)/Rx  (1/Rx)Vx - (1/Ry)Vy
Nodal Analysis—converting your marked up schematic into formulas—Step 2 Using KCL at each node, list the currents coming into the node PLUS the currents going out of the node and set them equal to zero. This creates another list of formulas (using the step 1 current equivalents) all equal to zero. At this point we simply the formulas: Vx/Rx(1/Rx)Vx (Vx – Vy)/Rx  (1/Rx)Vx - (1/Ry)Vy and put them into columns and rows (like a matrix). Use one row for each node KCL equation The columns are sorted by the voltages keeping the signage with the associated voltages.

Nodal Analysis—convert your KCL equations into a matrix equation for solution
With the voltages aligned in columns write the coeffiences of the voltage in its respective column and row with zero for voltages not listed in that equation. Repeat writing the coefficients of the voltages for each row. Solve for the Vx’s values using any convenient simultaneous solution mathematical tool. If current branch values are desired, substitute the now known Vx’s values into your step 1 branch equations.

Mesh Analysis—making a mesh out of the circuit
By using graph paper, the schematic can be reproduced so that the circuit looks like an accumulation of rectangular or square blocks with an occasional triangle thrown in. It could even have octagon features which is where the mesh name comes from. A current loop path is drawn within a block or pane where the current could start at a source, flow through a connected series of branch circuits and come back to the original point. Additional current loop paths are drawn in multiple panes until all the component in the branches have a least one current loop passing through. A source does not need to be present in every path. Each of the current loop paths are numbered i1, i2, i3, … and are distinct and closed onto themselves. The current loop paths are imaginary and use solely for the purpose of developing equations to solve for branch currents. Current loop paths can share a component passing in the same or opposite directions. The outside perimeter can be one such current loop path.

Mesh Analysis—current loop equations
Equations are developed using Kirchoff’s Voltage Law (KVL) for each current loop path. KVL states that the sum of the voltages in a loop is zero: Σ Vi = 0. This allows you to write the sum of the voltage drops around a current loop path that has no source is equal to zero. A restatement of KVL is : Σ Vdrops = Vsource This allows you to write the sum of the voltage drops is equal to the source voltage for those current loop paths that include a source voltage. Components that share current loop paths are included in the equations twice (one for each current loop passing thru). Once with the current path as positive: (ix-iy) for one loop equation The second time with the 2nd current path positive (iy-ix) for the other loop equation. The complete voltage drop for these types of components is R*itotal = Vbranch

Mesh Analysis—solving equations
The current loop equations have components times multiple current loops. Rearrange the equations by multiplying out the voltage drops then organizing the equations so that the current loops have single coefficients. Sort the currents loops left to right and where a current loop doesn’t appear place zero for that current’s coefficient times the missing current loop. Once all the current loop equation have been sorted, this is the format that allows you to simply lift the coefficients of the current loops into a matrix equation for solution of the imaginary current loop values. Branch currents are computed by going back to the current loop equations and substituting in the current loop values: Components with only one loop current have the value of the loop current. The branch current for components sharing loop currents are the difference between the loop currents as written.

Sources with examples