1. Lecture 7, Slide 1EECS40, Fall 2004Prof. White 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

2. Lecture 7, Slide 2EECS40, Fall 2004Prof. White

3. Lecture 7, Slide 3EECS40, Fall 2004Prof. White

4. Lecture 7, Slide 4EECS40, Fall 2004Prof. White 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. Lecture 7, Slide 5EECS40, Fall 2004Prof. White Superposition Procedure: 1.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 2.Repeat for each independent source 3.Sum individual contributions to obtain desired voltage or current

6. Lecture 7, Slide 6EECS40, Fall 2004Prof. White Superposition Example Find V o –+–+ 24 V 2  4  4 A 4 V + – +Vo–+Vo– -5.3 V

7. Lecture 7, Slide 7EECS40, Fall 2004Prof. White

8. Lecture 7, Slide 8EECS40, Fall 2004Prof. White Calculating a Thévenin Equivalent 1.Calculate the open-circuit voltage, v oc 2.Calculate the short-circuit current, i sc Note that i sc is in the direction of the open-circuit voltage drop across the terminals a,b ! network of sources and resistors a b + v oc – network of sources and resistors a b i sc

9. Lecture 7, Slide 9EECS40, Fall 2004Prof. White Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

10. Lecture 7, Slide 10EECS40, Fall 2004Prof. White

11. Lecture 7, Slide 11EECS40, Fall 2004Prof. White

12. Lecture 7, Slide 12EECS40, Fall 2004Prof. White Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

13. Lecture 7, Slide 13EECS40, Fall 2004Prof. White

14. Lecture 7, Slide 14EECS40, Fall 2004Prof. White Alternative Method of Calculating R Th For a network containing only independent sources and linear resistors: 1.Set all independent sources to zero voltage source  short circuit current source  open circuit 2.Find equivalent resistance R eq between the terminals by inspection Or, set all independent sources to zero 1.Apply a test voltage source V TEST 2.Calculate I TEST network of independent sources and resistors, with each source set to zero R eq network of independent sources and resistors, with each source set to zero I TEST –+–+ V TEST

15. Lecture 7, Slide 15EECS40, Fall 2004Prof. White R Th Calculation Example #1 Set all independent sources to 0:

16. Lecture 7, Slide 16EECS40, Fall 2004Prof. White Potential Plots for a Single Resistor and Two Resistors in Series (Potential is Plotted Vertically) Arrows represent voltage drops

17. Lecture 7, Slide 17EECS40, Fall 2004Prof. White Potential Plot for Two Resistors in Parallel Arrows represent voltage drops

18. Lecture 7, Slide 18EECS40, Fall 2004Prof. White Schematic Symbol and Water Model of DC Voltage Source (assumes gravity acting downward)

19. Lecture 7, Slide 19EECS40, Fall 2004Prof. White Resistor (top left), its Schematic Symbol (top right), and Two Water Models of a Resistor

20. Lecture 7, Slide 20EECS40, Fall 2004Prof. White The Capacitor Two conductors (a,b) separated by an insulator: difference in potential = V ab => equal & opposite charge Q on conductors Q = CV ab where C is the capacitance of the structure,  positive (+) charge is on the conductor at higher potential Parallel-plate capacitor: area of the plates = A (m 2 ) separation between plates = d (m) dielectric permittivity of insulator =  (F/m) => capacitance (stored charge in terms of voltage) F(F)

21. Lecture 7, Slide 21EECS40, Fall 2004Prof. White Symbol: Units: Farads (Coulombs/Volt) Current-Voltage relationship: or Note: Q (v c ) must be a continuous function of time +vc–+vc– icic C C (typical range of values: 1 pF to 1  F; for “supercapa- citors” up to a few F!) Electrolytic (polarized) capacitor C If C (geometry) is unchanging, i C = dv C /dt

22. Lecture 7, Slide 22EECS40, Fall 2004Prof. White 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

23. Lecture 7, Slide 23EECS40, Fall 2004Prof. White

24. Lecture 7, Slide 24EECS40, Fall 2004Prof. White Schematic Symbol and Water Model for a Capacitor

25. Lecture 7, Slide 25EECS40, Fall 2004Prof. White You might think the energy stored on a capacitor is QV = CV 2, 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. Thus, energy is. 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!) Stored Energy CAPACITORS STORE ELECTRIC ENERGY

26. Lecture 7, Slide 26EECS40, Fall 2004Prof. White +vc–+vc– icic A more rigorous derivation

27. Lecture 7, Slide 27EECS40, Fall 2004Prof. White Example: Current, Power & Energy for a Capacitor –+–+ v(t)v(t) 10  F i(t)i(t) t (  s) v (V) 023451 t (  s) 0 23451 1 i (  A) v c and q must be continuous functions of time; however, i c can be discontinuous. Note: In “steady state” (dc operation), time derivatives are zero  C is an open circuit

28. Lecture 7, Slide 28EECS40, Fall 2004Prof. White 023451 w (J) –+–+ v(t)v(t) 10  F i(t)i(t) t (  s) 0 23451 p (W) t (  s)

29. Lecture 7, Slide 29EECS40, Fall 2004Prof. White Capacitors in Parallel i(t)i(t) +v(t)–+v(t)– C1C1 C2C2 i1(t)i1(t)i2(t)i2(t) i(t)i(t) +v(t)–+v(t)– C eq Equivalent capacitance of capacitors in parallel is the sum

30. Lecture 7, Slide 30EECS40, Fall 2004Prof. White Capacitors in Series i(t)i(t) C1C1 + v 1 (t) – i(t)i(t) + v(t)=v 1 (t)+v 2 (t) – C eq C2C2 + v 2 (t) – 21 111 CCC eq 

31. Lecture 7, Slide 31EECS40, Fall 2004Prof. White Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors is changed by  v. How will this affect the voltage across each individual capacitor? v+vv+v C1C1 C2C2 + v 2 (t)+  v 2 – + v 1 +  v 1 – +–+– Note that no net charge can can be introduced to this node. Therefore,  Q 1 +  Q 2 =0 Q1+Q1Q1+Q1 - Q 1  Q 1 Q2+Q2Q2+Q2  Q 2  Q 2 Q1=C1v1Q1=C1v1 Q2=C2v2Q2=C2v2 Note: Capacitors in series have the same incremental charge.

32. Lecture 7, Slide 32EECS40, Fall 2004Prof. White Application Example: MEMS Accelerometer to 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 FIXED OUTER PLATES g1g1 g2g2

33. Lecture 7, Slide 33EECS40, Fall 2004Prof. White Sensing the Differential Capacitance –Begin with capacitances electrically discharged –Fixed electrodes are then charged to +V s and –V s –Movable electrode (proof mass) is then charged to V o C1C1 C2C2 VsVs –Vs–Vs VoVo Circuit model

34. Lecture 7, Slide 34EECS40, Fall 2004Prof. White 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 Practical Capacitors

35. Lecture 7, Slide 35EECS40, Fall 2004Prof. White 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: Li L When the current changes, the magnetic flux changes  a voltage across the coil is induced: iLiL vL(t)vL(t) + _ Note: In “steady state” (dc operation), time derivatives are zero  L is a short circuit

36. Lecture 7, Slide 36EECS40, Fall 2004Prof. White Symbol: Units: Henrys (Volts second / Ampere) Current in terms of voltage: Note: i L must be a continuous function of time +vL–+vL– iLiL L (typical range of values:  H to 10 H)

37. Lecture 7, Slide 37EECS40, Fall 2004Prof. White Schematic Symbol and Water Model of an Inductor

38. Lecture 7, Slide 38EECS40, Fall 2004Prof. White Stored Energy Consider an inductor having an initial current i(t 0 ) = i 0 2 0 2 2 1 2 1 )( )()( )()()( 0 Li tw dptw titvtp t t      INDUCTORS STORE MAGNETIC ENERGY

39. Lecture 7, Slide 39EECS40, Fall 2004Prof. White Inductors in Series v(t)v(t) L1L1 + v 1 (t) – v(t)v(t) + v(t)=v 1 (t)+v 2 (t) – L eq L2L2 + v 2 (t) – +–+– +–+– i(t)i(t) i(t)i(t) Equivalent inductance of inductors in series is the sum

40. Lecture 7, Slide 40EECS40, Fall 2004Prof. White L1L1 i(t)i(t) i2i2 i1i1 Inductors in Parallel L2L2 +v(t)–+v(t)– L eq i(t)i(t) +v(t)–+v(t)–

41. Lecture 7, Slide 41EECS40, Fall 2004Prof. White 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: Summary