Mastering Kirchhoff's Rules for Multi-Loop RC Circuits

1. Physics 121 - Electricity and MagnetismLecture 08 - Multi-Loop and RC Circuits Y&F Chapter 26 Sect. 2 - 5 • Definitions of Circuit Terms • Kirchhoff Rules • Problem solving using Kirchhoff’s Rules • Multi-Loop Circuit Examples • RC Circuits • Charging a Capacitor • Discharging a Capacitor • Discharging Solution of the RC Circuit Differential Equation • The Time Constant • Examples • Charging Solution of the RC Circuit Differential Equation • Features of the Solution • Examples

2. i1 i i i2 • Junction Rule aka Current rule aka Node Law • (charge conservation): At any junction (essential node) • the algebraic sum of the currents equals zero. • Loop Rule (energy conservation): The algebraic sum of • all potential changes is zero for every closed path • around a circuit. Corollary: voltage difference between • two nodes is the same for all paths connecting them. Solving Circuit Problems using Kirchhoff’s Rules CIRCUITS CONSIST OF: ESSENTIAL NODES (JUNCTIONS) ... and … ESSENTIAL BRANCHES (elements connected in series, one current/branch) The current through all series elements in an essential branch is the same; The number of currents in a circuit = the number of essential branches Currents are often the unknowns with all other elements specified. Analysis strategy: • 1) Use Kirchhoff Rules to generate N independent equations in N unknowns 2) Solve the resulting set of simultaneous equations (you need a strategy). • Linear, algebraic for resistances and EMFs only (now)

3. Procedure for Generating Circuit Equations: • Find and enumerate essential nodes and branches. • Name the currents (1 per branch) or other unknowns. Arbitrarily assign a direction to current in each branch. • - A negative current result  opposite flow. • Apply Junction Rule, create equations • Apply Loop Rule, create equations: • - Choose direction for traversing each closed loop. Individual branches • (several) may be traversed with or against assumed current directions. • When crossing resistances: • - Voltage drop (DV = - iR) is negative when following assumed current. • - Positive voltage change DV = +iR for crossing opposite to assumed current. • When crossing EMFs from – to +, DV = +E. Otherwise DV= -E. • Keep generating equations until you have N independent ones. Applying Kirchhoff’s Rules: using “active” sign convention: voltage drops considered negative with “passive” sign convention: voltage drops considered positive “branch” means “essential branch” “junction” means “essential node” • After solving equation set, calculate power or other quantities as needed. • Dot product i.E determines whether EMFs supply or dissipate power • For later: When following current across C write –VC= -Q/C. • When crossing inductance write VL= - Ldi/dt

4. i i R1= 10 W E2 = 3 V E1 = 8 V R2= 15 W + + - - A battery (EMF) absorbs power (charges up) when I is opposite to E E2 is opposite to Vdrop -3.0x0.2 EXAMPLE: MULTIPLE EMFs SINGLE LOOP +

5. E1 E2 + - R1 R3 R2 + - Apply Procedure: A E F C D B • Identify essential branches (3) & junctions (2). • Same current flows through all elements in any series branch. • Name all currents (3) and other variables. Assume arbitrary current directions. i1 i3 • At junctions, write current rule (junction rule, node rule) equations. i2 • Same equation at junctions A and B (not independent). • Junction Rule yields only 1 of 3 equations needed • Are points C, D, E, F junctions? (not essential nodes) Example: Multi-loop circuit with 2 EMFs • Given all resistances and EMFs in circuit: • Find currents (i1, i2, i3), then potential • drops and power dissipated by resistors • 3 unknowns (currents) • imply 3 independent equations needed

6. + - E1 E2 R1 R3 R2 + - A C B E F D Loop equations for the example circuit: • Only 2 of these three are independent • Now have 3 equations in 3 unknowns ADCBA - CCW ADCBFEA - CCW i3 i1 ABFEA - CCW i2 Procedure, continued: • Apply Loop Rule as often as needed to find • equations that include all the unknowns (3). • Clockwise or counter-clockwise traversal is OK. • When following the assumed current direction • voltage change = - iR. When going against assumed • current direction voltage steps up by +IR • EMF’s count positive when traversed from – to + side • EMF’s count negative when traversed from + to - sides Solution: (after a lot of algebra) Define:

7. i2 + i3 E = 12 Volts R1 = 3 W R2 = 8 W R3 = 6 W E R1 R2 R3 i3 + - F C G D B A H E i2 + i3 i i LOOP RULE: ABCDA - CW i2 i1 CEFDC - CW EGHFE - CW FINISH: POWER: Example: find currents, voltages, power 6 BRANCHES  6 CURRENTS. • JUNCTION RULE: Junctions C & E are the same point, as are D & F -> 4 currents left. • Remaining 2 junction equations are dependent • -> 1 junction equation

8. Multiple EMF Example: find currents, voltages, power JUNCTION RULE at A & B: LOOP ACDBA: LOOP BFEAB: USE JUNCTION EQUATION: For power use: EVALUATE NUMERICALLY: R2 = 4 W R1 = 2 W E2 = 6 V E1 = 3 V MULTIPLE EMF CIRCUIT USE THE SAME RULES

9. RC Circuits: Time dependance i a R + + • Capacitance, Resistance, EMF given • Use Loop Rule + Junction Rule • New term in Loop Rule: Vc= Q/C Vc b - - E C • Find Q, i, Vc, U for capacitor • as functions of time i Charging: “Step Response” Switch to “a” then watch. Loop equation: • Assume clockwise current i through R. • At t = 0: expect largest current. • As t  infinity: expect zero current • Vcap E, Q  Qinf = CE • Energy is stored in C, some is dissipated in R Discharging: Switch to “b”. no EMF, Loop equation: • Energy stored in C is now dissipated in R • Arbitrarily choose current still CW • Assume Vcap= Eat t =0 • Q0= full charge = CE • It must die away to zero as t  infinity • Result: i through R is actually CCW 27.8 Can current flow through the capacitor indefinitely? First charge up C (switch to “a”) then discharge (switch to “b”)

10. Q0 Q t 2t 3t t Voltage across C also decays exponentially: Current also decays exponentially: RC Circuit: solution for discharging Circuit Equation: Loop Equation is : Substitute : First order differential equation, form is Q’ = -kQ  Exponential solution Charge decays exponentially: • t/RC is dimensionless RC = t = the TIME CONSTANT Q falls to 1/e of original value

11. RC is constant RC = time constant = time for Q to fall to 1/e of its initial value Time t 2t 3t 4t 5t Value e-1 e-2 e-3 e-4 e-5 % left 36.8 13.5 5.0 1.8 0.67 After 3-5 time constants the action is over Solving for discharging phase by direct integration Initial conditions (“boundary conditions”) exponentiate both sides of above right exponential decay

12. Units for RC 8-1: We defined  = RC, which of the choices best captures the physical units for the time constant  ? [] = [RC] =[(V/i)(Q/V)]=[Q/Q/t]=[t] • F (ohmfarad) • C/A (coulomb per ampere) • C/V (ohmcoulomb per volt) • VF/A (voltfarad per ampere) • s (second)

13. a) When has the charge fallen to half of it’s initial value Q0? set: take log: b) When has the stored energy fallen to half of its original value? recall: and where at any time t: evaluate for: take log: c) How does the power delivered by C vary with time? power: C supplies rather than absorbs power Drop minus sign recall: power supplied by C: Examples: discharging capacitor C through resistor R

14. Solution: Charge starts from zero, grows as a saturating exponential. Qinf Q • RC = t = TIME CONSTANT • describes time dependance again • Q(t)  0 as t  0 • Q(t)  Qinf as t  infinity t 2t 3t t RC Circuit: solution for charging Circuit Equation: Loop Equation is : Substitute : • First order differential equation again: form is Q’ = - kQ + constant • Same equation as for discharge, but with i0 = E/ R added on right side • At t = 0: Q = 0 when i = i0. Large current flows (C acts like a plain wire) • As t  infinity: Current  0 (C acts like a broken wire) • Q  Qinf = CE = limiting charge

15. Current in the charging circuit: Current decays exponentially just as in discharging case Growing potential Vc on C blocks current completely at t = infinity At t=0 C acts like a wire. At t=infinity C acts like a broken wire Voltage drop VR across the resistor: Voltage across R decays exponentially, reaches 0 as t infinity Form factor: 1 – exp( - t / t ) After 3-5 time constants the action is over Factor .63 .865 .95 .982 .993 .998 Time t 2t 3t 4t 5t 6t RC Circuit: solution for charging, continued Voltage across C while charging: Voltage across C also starts from zero and saturates exponentially

16. RC circuit – multiple resistors at t = 0 8-2: Consider the circuit shown, The battery has no internal resistance. The capacitor has zero charge at t = 0. Just after the switch is closed, what is the current through the battery? • 0. • /2R. • 2/R. • /R. • impossible to determine C R R

17. RC circuit – multiple resistors at t = infinity 8-3: Consider the circuit shown. The battery has no internal resistance. After the switch has been closed for a very long time, what is the current through the battery? • 0. • /2R. • 2/R. • /R. • impossible to determine C R R

18. Use: Set: Take natural log of both sides, evaluate at t = 2 s: Define: 1 MW = 106W Discharging Example: A 2 mF capacitor is charged and then connected in series with a resistance R. The original potential across it drops to ¼ of it’s starting value in 2 seconds. What is the value of the resistance?

19. E b) When does VCap (voltage on C) reach 1 Volt? c) Find the current in the resistor at that time Example: Discharging C = 500 mF R = 10 KW E = 12 V Capacitor C is charged for a long time to E, then discharged from t=0 onward. a) Find current at t = 0

20. Charging Example: How many time constants does it take for an initially uncharged capacitor in an RC circuit to become 99% charged? Use: Require: Take natural log of both sides: Did not need specific values of RC

21. R E C b) What’s the current through R at t = 2 sec? Example: Consider charging a 100 mF capacitor in series with a 10,000 W resistor, using EMF E= 5 V. • How long after voltage is applied does Vcap(t) reach 4 volts? Take natural log of both sides:

22. Example: Multiple loops and EMFs What is the CHANGE in charge on C? First: E2 charges C to have: Second: Close switch for a long time At t = infinity current i3 though capacitor  zero Find outer loop current i = i1 = i2 using loop rule, CW path Now find Voltage across C, same as voltage across right hand (or left hand) branch Final charge on C: • Open switch S for a long time. • Capacitor C charges to potential of battery 2 • Then close S for a long time

23. SUPPLEMENTARY MATERIAL

24. i1 i 2 1 i2 Generic symbol: Basic circuit elementshave two terminals. i Branch: A path that connects at least two nodes (essential or not). Includes 1 or more elements. Path: A route (or trace) through adjacent basic circuit elements with no element included more than once. A path may pass through essential and/or non-essential nodes and continue from branch to branch. It may or may not be closed. Essential Branch: • A path that connects two essential nodes without passing through another essential node. • The circuit elements of an essential branch are all in series. • There is exactly one current per essential branch. Loop: A closed path whose last node is the same as the starting node. Mesh: A loop that does not enclose any other loops. Planar Circuit: A circuit whose diagram can be drawn on a plane with no crossing branches. ECE Definitions of Circuit Terms(those in blue are central for Physics 121) Node: A point where 2 or more circuit elements are joined. Essential Node (Junction): A node where at least 3 circuit elements are joined.

25. b a R1 i1 i2 R5 i7 v1 d e c R3 R2 i3 i4 R7 v2 R6 f g R4 i5 Ideal, Independent Voltage Sources (ideal EMFs) : v1, v2 - - Ideal, Independent Current Source: i6 + + i6 Nodes: a, b, c, d, e, f, g (nodes b & g are drawn extended) Essential Nodes (Junctions): b, c, e, g Branches: v1, v2, R1, R2, R3, R4, R5, R6, R7, i6 Paths: Many, traversing 1, 2, …..n branches Essential Branches (7): v1-R1, R2-R3, v2-R4, R5, R6, R7, i6 Currents: 1 per essential branch, total of 7: i1 . . . i7 Meshes (subset of loops): v1-R1-R5-R3-R2, v2-R2-R3-R6-R4, R5-R7-R6, R7-i6 Loops: Meshes + V1-R1-R5-R6-R4-V2, V1-R1-R7-R4-V2, v1-R1-i6-R4-V2 , R6-R5-i6 Source: Nilsson & Reidel Electric Circuits Examples of Circuit Terms Optional for Physics 121

26. CONCLUSION: The equivalent resistance for a series combination is the sum of the individual resistances and is always greater than any one of them. Example: Equivalent resistance for series resistorsusing Kirchhoff Rules Junction Rule: The current through all of the resistances in series (a single essential branch) is identical. No information from Junction/Current/Node Rule. Loop Rule: The sum of the potential differences around a closed loop path equals zero. Only one loop path exists: The equivalent circuit replaces the series resistors with a single equivalent resistance Req:same E, same i as above. inverse of series capacitance rule

27. Example: Equivalent resistance for resistors in parallel Loop Rule: The potential differences across each of the 4 parallel branches are the same. There are four unknown currents. Apply loop rule to 3 paths. i not in these equations Junction Rule: The sum of the currents flowing in equals the sum of the currents flowing out. Combine equations for all the upper junctions at “a” (same at “b”). The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above. CONCLUSION: The reciprocal of the equivalent resistance for a parallel combination is the sum of the individual reciprocal resistances and is always smaller than any one of them. inverse of parallel capacitance rule