Detailed Analysis of Node Voltage and Test Results in Electrical Engineering Lecture 12

EE 261, Spring 2025, Lecture 12 Goal/Topic: Test results More on node-voltage analysis Dependent sources Supernodes Announcement: Homework 4 posted (due Sunday by midnight).

Test 1 Results: Fall 2024 # of Students 130 B (although not definitive on its own) B (although not definitive on its own) Average 79.3 Std. Dev. 15.2 High by historical standards. Median 80 High Score (18 x) 100 Solution available at Canvas and course website. If you missed anything, please check out the solution. Understand your mistakes and learn from them. The remainder of the course builds on the foundation laid going into the first exam.

All yield same expression for ?. My vote for best approach. + + Easier to think in terms of voltage drops? Easier to think in terms of KVL? + + ?? ?? ?? ??= ??1+ ?1+ ??2 ?? ??= ??1+ ?1+ ??2 ?? ??= ??1+ ?1+ ??2 ?? ??= ??1+ ?1+ ??2 ??+ ??1+ ?1+ ??2+ ??= 0 ??+ ??1+ ?1+ ??2+ ??= 0 ??+ ??1+ ?1+ ??2+ ??= 0 ??+ ??1+ ?1+ ??2+ ??= 0 ??+ ??1+ ?1+ ??2+ ??= 0 ??+ ??1+ ?1+ ??2+ ??= 0 What is the current ?? Consider voltage increase from ? to ?. ??= ??+ ??2+ ?1+ ??1 ??= ??+ ??2+ ?1+ ??1 ??= ??+ ??2+ ?1+ ??1 ??= ??+ ??2+ ?1+ ??1 ??= ??+ ??2+ ?1+ ??1 Group terms and rearrange. ?? ?1 ?? = ? ?1+ ?2 ? = ?? ?1 ?? Independent of order of elements. ?1+ ?2 This is expression used for ? in KCL eq. for node ?. (Change sign if Series resistors. considering current from node ?.) Voltage source doesn t break their series relationship.

Given resistances, what is ??? Consider: Express any control parameter in terms of node voltages. ??= ???2 Could leave ?? as an unknown in node voltage equations, but then need another equation in our system of equations. Instead, we will express control parameters in terms of node voltages from the start. Now write KCL at nodes a, b, and c. Always think in terms of outgoing currents (treat as positive). (In terms of node voltages.)

KCL at node a. 4?? One equation; three unknowns. At a: 8 +?? 8 +?? 8 +?? 8 +?? 8 +?? 8 +?? +?? ?? ?6 ?6 ?6 ?6 ?6 ?6 +?? ?? +?? ?? +?? ?? +?? ?? +?? ?? ?? ?? ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? + 4?? + 4?? + 4?? + 4?? + 4?? + 4?? + + + + + + = 0 = 0 = 0 = 0 = 0 = 0 ?1 ?1 ?1 ?1 ?1 ?1 ?2 ?2 ?2 ?2 ?2 ?2

KCL at node b. ?? Two equations; three unknowns. ?? ?? ?6 ?6 ?6 ?6 ?? ?? ?? ?? ?? ?? +?? ?2 ?2 ?2 ?2 +?? +?? +?? +?? ?? ?7 ?7 ?7 ?7 +?? ?? +?? ?? +?? ?? At b: = 0 = 0 = 0 = 0

KCL at node c. If location of ?8 and voltage source swapped, obtain same equation. 10 V Three equations; three unknowns. 4?? 4?? 4?? 4?? 4?? 4?? ?? ?? ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?4+ ?5 ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? +?? ?? +?? ?? ?7 ?7 ?7 ?7 ?7 ?7 +?? ?? +?? ?? +?? ?? +?? ?? +?? +?? +?? +?? +?? +?? +?? 10 +?? 10 +?? 10 ?8 ?8 ?8 ?8 ?8 ?8 +?? 10 +?? 10 +?? 10 At c: + + + + + + = 0 = 0 = 0 = 0 = 0 = 0 ?2 ?2 ?2 ?2 ?2 ?2 ?3 ?3 ?3 ?3 ?3 ?3

Using these three equations, group coefficients of ??, ??, and ??. Move remaining terms (constants) to right-hand side. Three equations; three unknowns. Assuming value of resistors known, solve using MATLAB or calculator or (whatever tool works for you). See HW 4 (we ve essentially solved one of the problems).

Add resistor ?9. What happens to node eq.? A) Add a new term B) Modify an existing term C) Modify multiple terms D) No change KCL at node a. Current into node ? still 8 A. No change in eqs. for ?? and ?? as well. ?9 No change to any node voltage. (Voltage across current source does change. Now ??+ 8?9.) No change. 8 +?? +?? ?? ?6 ?? ?? ?4+ ?5 + 4?? At a: + = 0 ?1 ?2

(wire) KCL at node c. Replace ?8 with a short. What happens to KCL eq.? ?? now known! ??= 10 V. Not an unknown in nodal analysis! ?? But, does KCL for node ? even make sense? Of course!But Need new variable. Don t know current through voltage source. But this equation would provide it after finding 10 V 10 V 10 V 4?? ?? ?? ?4+ ?5 +?? ?? ?7 +?? +?? 10 ?8 ?? At c: + = 0 ?? and ??! ?2 ?3

Node-voltage analysis: Uses KCL to obtain equations for voltages. Reminders: A current source tells us current between two nodes. We can express the current through a resistor in terms of the voltage across the resistor and its resistance. A voltage source between ground and a node tells us the voltage at that node. (Eliminates an unknown.) the voltage at that node. (Eliminates an unknown.) A voltage source between ground and a node tells us Cannot easily express current through a voltage source directly between two non-reference nodes. Supernode. Combine nodes that are not independent (i.e., tied by voltage source) and consider currents out of combination.

Special case: Voltage source directly between two non-reference nodes. Supernode. Combine nodes that are not independent (voltage source fixes relationship between two nodes). Supernodes are nice! They remove one unknown for free.

Recall: We want equations for currents flowing between nodes in terms of node voltages. Consider: 4 How many essential nodes? (numeric poll) Upper voltage source directly connects two of them. Select reference and assign labels. (For purpose of this lecture, will select ground as bottom node. This is not the only choice nor necessarily the best choice.)

Consider: ? ? ? ? ? ? If ?? is given and the sources are as shown, but resistances are unknown, can ?? be determined? Because known voltage source 10 V connects ??and ??, if you know one voltage you can determine the other. A) Yes B) No ??= ??+ 10 V

??? Consider: ? ? ? ? ? If ?? is given and the sources are as shown, but resistances are unknown, can current between ??and ?? be determined? Know 8 A flowing into ? node, but without resistances don t know anything else about current flow. At this point, somewhat unclear what equation for ?? would be. If ?? is given and the sources are as shown, but resistances are unknown, can current between ??and ?? be determined? A) Yes B) No

Consider: Voltages ?? and ??not independent. If you know one voltage, you know the other. But, don t know current through voltage source, so can t write KCL at nodes a and b without introducing a new unknown (current). Create supernode that combines these nodes.

Consider: KCL holds for a super node (KCL holds for any portion of circuit). Sum of currents out of supernode must equal zero. (Charge isn t accumulating anywhere in circuit.)

i = ? Consider: Obtain one KCL equation from this supernode. ?? and ??not independent. (N&R approach.) Can replace ?? with ?? 10. Or, can replace ?? with ??+ 10. Only have two unknown voltages. (??and??) or (??and ??) KCL at node c provides additional equation using either approach. Or, leave ?? and ??as unknowns. KCL at super node yields one eq. (With ??,??, and ??.) Voltage source provides another eq. ?? ??= 10 V Constraint equation.

Aside: Could write KCL for ground, but that doesn t yield independent equation. First approach: Write KCL for super node and node c. Replace ?? with ??+ 10. +?? ?? ?4 ?4 ?4 ?4 ?4 ?? 1 ?1 8 +??+ 10 8 +??+ 10 8 +??+ 10 8 +??+ 10 8 +??+ 10 +?? ?2 ?2 ?2 ?2 ?2 +?? +?? +?? +?? +?? ?? +?? ?? +?? ?? +?? ?? = 0 = 0 = 0 = 0 = 0 At SN: Group and arrange terms. ?1 ?1 ?1 ?1 ?1 One equation; two unknowns. 1 ?1 +1 +1 ?4??= 8 10 ?2 ?4