Understanding Linear Circuits & Superposition: Lecture 7

1. Lecture #7 Warning about breadboards in the lab: to stop blowing fuses … Slides from Lecture 6 with clearer markups New Topics: Potential plots for resistive circuits Water models for voltage source, resistors The capacitor The inductor Reading -- Chapter 3

4. Superposition A linear circuit is one constructed only of linear elements (linear resistors, and linear capacitors and inductors,linear dependent sources) and independent sources. Linear means I-V charcteristic of elements/sources are straight lines when plotted. Principle of Superposition: • In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone.

5. Superposition • Procedure: • Determine contribution due to one independent source • Set all other sources to 0: Replace independent voltage • source by short circuit, independent current source by open • circuit • Repeat for each independent source • Sum individual contributions to obtain desired voltage • or current

6. Superposition Example • Find Vo 4 V 2 W + – + Vo – – + 24 V 4 A 4 W -5.3 V

8. Calculating a Thévenin Equivalent • Calculate the open-circuit voltage, voc • Calculate the short-circuit current, isc • Note that isc is in the direction of the open-circuit voltage drop across the terminals a,b ! a network of sources and resistors + voc – b a network of sources and resistors isc b

9. Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

12. Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

14. Alternative Method of Calculating RTh For a network containing only independent sources and linear resistors: • Set all independent sources to zero voltage source  short circuit current source  open circuit • Find equivalent resistance Req between the terminals by inspection Or, set all independent sources to zero • Apply a test voltage source VTEST • Calculate ITEST network of independent sources and resistors, with each source set to zero Req ITEST network of independent sources and resistors, with each source set to zero – + VTEST

15. RTh Calculation Example #1 Set all independent sources to 0:

16. Potential Plots for a Single Resistor and Two Resistors in Series (Potential is Plotted Vertically) Arrows represent voltage drops

17. Potential Plot for Two Resistors in Parallel Arrows represent voltage drops

18. Schematic Symbol and Water Model of DC Voltage Source (assumes gravity acting downward)

19. Resistor (top left), its Schematic Symbol (top right), and Two Water Models of a Resistor

20. The Capacitor Two conductors (a,b) separated by an insulator: difference in potential = Vab => equal & opposite charge Q on conductors Q = CVab where C is the capacitance of the structure, • positive (+) charge is on the conductor at higher potential (stored charge in terms of voltage) • Parallel-plate capacitor: • area of the plates = A (m2) • separation between plates = d (m) • dielectric permittivity of insulator =  (F/m) • => capacitance F (F)

21. C Symbol: Units: Farads (Coulombs/Volt) Current-Voltage relationship: C or C Electrolytic (polarized) capacitor (typical range of values: 1 pF to 1 mF; for “supercapa- citors” up to a few F!) ic + vc – If C (geometry) is unchanging, iC = dvC/dt Note: Q (vc) must be a continuous function of time

22. Voltage in Terms of Current; Capacitor Uses Uses: Capacitors are used to store energy for camera flashbulbs, in filters that separate various frequency signals, and they appear as undesired “parasitic” elements in circuits where they usually degrade circuit performance

24. Schematic Symbol and Water Model for a Capacitor

25. Thus, energy is . Stored Energy You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor. CAPACITORS STORE ELECTRIC ENERGY Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2 (1pF) = 12.5 pJ (A 5F supercapacitor charged to 5 volts stores 63 J; if it discharged at a constant rate in 1 ms energy is discharged at a 63 kW rate!)

26. A more rigorous derivation ic + vc –

27. Example: Current, Power & Energy for a Capacitor i(t) v (V) v(t) – + 10 mF 1 t (ms) 0 1 2 3 4 5 vc and q must be continuous functions of time; however, ic can be discontinuous. i (mA) t (ms) 0 1 2 3 4 5 Note: In “steady state” (dc operation), time derivatives are zero  C is an open circuit

28. p (W) i(t) v(t) – + 10 mF t (ms) 0 1 2 3 4 5 w (J) t (ms) 0 1 2 3 4 5

29. Capacitors in Parallel i1(t) i2(t) + v(t) – i(t) C1 C2 + v(t) – i(t) Ceq Equivalent capacitance of capacitors in parallel is the sum

30. 1 1 1 = + C C C 1 2 eq Capacitors in Series + v1(t) – + v2(t) – + v(t)=v1(t)+v2(t) – C1 C2 i(t) i(t) Ceq

31. Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors is changed by Dv. How will this affect the voltage across each individual capacitor? DQ1=C1Dv1 Note that no net charge can can be introduced to this node. Therefore, -DQ1+DQ2=0 Q1+DQ1 + v1+Dv1 – C1 -Q1-DQ1 v+Dv + – Q2+DQ2 + v2(t)+Dv2 – C2 -Q2-DQ2 DQ2=C2Dv2 Note: Capacitors in series have the same incremental charge.

32. Application Example: MEMS Accelerometerto deploy the airbag in a vehicle collision • Capacitive MEMS position sensor used to measure acceleration (by measuring force on a proof mass) MEMS = micro- • electro-mechanical systems g1 g2 FIXED OUTER PLATES

33. Sensing the Differential Capacitance • Begin with capacitances electrically discharged • Fixed electrodes are then charged to +Vs and –Vs • Movable electrode (proof mass) is then charged to Vo Circuit model Vs C1 Vo C2 –Vs

34. Practical Capacitors • A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size. • To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high. • Real capacitors have maximum voltage ratings • An engineering trade-off exists between compact size and high voltage rating

35. The Inductor • An inductor is constructed by coiling a wire around some type of form. • Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: LiL • When the current changes, the magnetic flux changes  a voltage across the coil is induced: + vL(t) iL _ Note: In “steady state” (dc operation), time derivatives are zero  L is a short circuit

36. Symbol: Units: Henrys (Volts • second / Ampere) Current in terms of voltage: L (typical range of values: mH to 10 H) iL + vL – Note: iL must be a continuous function of time

37. Schematic Symbol and Water Model of an Inductor

38. Stored Energy INDUCTORS STORE MAGNETIC ENERGY Consider an inductor having an initial current i(t0) = i0 = = p ( t ) v ( t ) i ( t ) t ò = t t = w ( t ) p ( ) d t 0 1 1 2 = - 2 w ( t ) Li Li 0 2 2

39. Inductors in Series + v1(t) – + v2(t) – + v(t)=v1(t)+v2(t) – L1 L2 i(t) i(t) + – + – v(t) v(t) Leq Equivalent inductance of inductors in series is the sum

40. Inductors in Parallel + v(t) – + v(t) – i1 i2 i(t) i(t) Leq L1 L2

41. Summary • Capacitor • v cannot change instantaneously • i can change instantaneously • Do not short-circuit a charged • capacitor (-> infinite current!) • n cap.’s in series: • n cap.’s in parallel: • Inductor • i cannot change instantaneously • v can change instantaneously • Do not open-circuit an inductor with current (-> infinite voltage!) • n ind.’s in series: • n ind.’s in parallel: