[[The Bare Essentials of Electrical Engineering.pdf#page=93&offset=99,702|The Bare Essentials of Electrical Engineering, Analysis Techniques Using Kirchhoff's Laws]] # Nodal Analysis - Used to determine voltage at every node in a circuit. After picking GND node, systematically apply KCL at every essential node, except GND - By finding every nodal voltage, we can find every branch current, enabling us to determine power absorbed/delivered by every element - **Reminders for using KCL** (see analysis of [[The Bare Essentials of Electrical Engineering.pdf#page=94|Figure 3.1]]) - Every element directly touching that node contributes a current term - A node is *not defined by geometry*, it is defined by continuous conductor with no element in between - Key idea: don't focus on currents flowing through branches, but rather on *nodal voltages* - To find currents across branches w/ resistance, can use formula $I=\frac{V_{from}-V_{to}}{R}$ - Basic steps 1. Number all [essential] nodes of given circuit 2. Write KCL for every [essential] node, keeping in mind that only nodal voltages should be used (no currents). If other unknowns are involved (e.g dependent source equations), express them as a function of the unknown nodal voltages (e.g Ohm's Law) 3. Group resulting equations together in matrix form 4. Solve for unknown nodal voltages by inverting the resulting linear eq. 5. Calculate any qty. of interest from the known nodal voltages - Treat each node independently - e.g. if a current source is between 2 nodes, it will show up in both equations of these nodes but carry opposite directions - See [[The Bare Essentials of Electrical Engineering.pdf#page=95|Figure 3.3]] for example - See [[The Bare Essentials of Electrical Engineering.pdf#page=101|Example 3.2.2]] for **nodal analysis of a resistor & voltage source between a node & ground**; see [[The Bare Essentials of Electrical Engineering.pdf#page=103|Example 3.2.3]] for **analysis of a voltage source & resistor between two nodes** - Even if not defined in the problem, floating voltage sources must be surrounded by all unknown voltages, rather than being connected to any components -> "non-essential node" ### Nodal Analysis w/ Dependent Sources - See [[The Bare Essentials of Electrical Engineering.pdf#page=99|Example 3.2.1]] - If a voltage source connects a node to GND, the value of the node equals the source voltage. Negative if node is connected to negative terminal - Floating voltage source: power source whose terminals are connected between two non-reference nodes -> neither terminal is tied to GND - To solve this problem, add a dummy variable which becomes an additional unknown - Treat a dependent source the same way as an independent one, except you need an equation linking its controlling qty. to the nodal voltages # Mesh Analysis ![[Pasted image 20260208022930.png|center|450]] - Used to determine every loop current in a circuit. Use KVL around meshes, meaning loops, to find the mesh currents, which we can use to calculate any voltage or branch current. - "Basic" mesh analysis can only be applied to planar circuits -> can be drawn on 2D plane w/ no overlap between unconnected elements - Element currents flow through individual elements of a circuit - If we look at the junction of $R_{a},R_{b},R_{x}$ in Figure 3.12, we see that in (a) $I_{b}-I_{a}+I_{x}=0$, so $I_{x}=I_{a}-I_{b}$; in (b), $I_{y}=I_{b}-I_{a}$ - Instead of directly calculating element currents, mesh analysis first calculates other currents, called mesh or loop currents ($I_{1},I_{2}$ in Figure 3.12) - The loop current $I_{1}$ is the current that flows thru all elements in its closed path, same for $I_{2}$ - $I_{1}$ & $I_{2}$ flow thru $R_{x}$ in opposite directions - If we combine the equations above, we find that $I_{x}=I_{1}-I_{2}$ and $I_{y}=I_{2}-I_{1}$. We now see that elements not shared among meshes will carry the same current as the mesh current in their loop, i.e. $I_{a}=I_{1},I_{b}=I_{2}$ - If mesh current goes in same direction as current element, let current element be positive & vice versa - Elements included in many loops carry all the currents of the loops to which they belong. It's critical to explicitly define both element and mesh currents (directions) - KVL reminder: traverse with current -> voltage drop; traverse against current -> voltage rise - See step-by-step alongside [[The Bare Essentials of Electrical Engineering.pdf#page=108|Figure 3.13]] - Mesh analysis rules 1. If resistor only belongs to a specific mesh, it is positive in that KVL equation 2. Shared resistor between two meshes is positive in its respective KVL equation 3. For sources, use $-V$ for $-\to+$ and $+V$ for $+\to-$ - If element current is in same direction as mesh current, it is positive - In mesh analysis, all currents used in KVL eqs. should *only* be mesh currents - See analysis of [[The Bare Essentials of Electrical Engineering.pdf#page=112|Figure 3.15]] for analysis of more complex *basic* circuit - See analysis of [[The Bare Essentials of Electrical Engineering.pdf#page=113|Figure 3.16]] for analysis using a known mesh current. - Let the mesh current $I_{1}$ be equal in magnitude to the current source $I_{s1}$ but carrying a negative sign to indicate that it's going in the opposite direction of the current source - Can just replace the entire KVL equation for that mesh with $-I_{s1}$ - If there's a dependent source in a mesh, find KVL for all other meshes first ## Current Source Between Two Meshes - See [[The Bare Essentials of Electrical Engineering.pdf#page=114|Figure 3.17]] - We find $I_{x}=I_{2}-I_{4}$ because when $I_{2}$ is at $R_{3}$, it's pushing current L -> R, in the same direction as $I_{x}$; when $I_{4}$ is at $R_{3}$, it's pushing current R -> L, opposite direction as $I_x$ - When a current source exists between two meshes, assign a dummy voltage across that current source w/ an arbitrary polarity - After finding the KVL equations, we can eliminate the dummy voltage by combining equations - assuming that the mesh currents interacting w/ the current source are going in the same direction - Don't necessarily have to use a dummy voltage, can also just choose a different path or loop that doesn't include it in order to avoid summing its voltage, i.e. combining the two loops that share the current source *or* ignoring them completely. See [[The Bare Essentials of Electrical Engineering.pdf#page=117|Figure 3.19]] for better visualization - Often useful to find equation relating the current source to the meshes it interacts with. E.g. in [[The Bare Essentials of Electrical Engineering.pdf#page=118|Example 3.4.1]], we can use equation $I_{2}-I_{1}=7$ in solving