Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 43 Capacitors &Inductors Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Capacitance & Inductance • Introduce Two Energy STORING Devices • Capacitors • Inductors • Outline • Capacitors • Store energy in their ELECTRIC field (electroStatic energy) • Model as circuit element • Inductors • Store energy in their MAGNETIC field (electroMagnetic energy) • Model as circuit element • Capacitor And Inductor Combinations • Series/Parallel Combinations Of Elements

3. First of the Energy-Storage Devices Basic Physical Model The Capacitor • Circuit Representation • Note use of the PASSIVE SIGN Convention • Details of Physical Operation Described in PHYS4B & ENGR45

4. Consider the Basic Physical Model Capacitance Defined • Where • A The Horizontal Plate-Area, m2 • d  The Vertical Plate Separation Distance, m • 0  “Permittivity” of Free Space; i.e., a vacuum • A Physical CONSTANT • Value = 8.85x10-12 Farad/m • Then What are the UNITS of Capacitance, C • Typical Cap Values →“micro” or “nano” • The Capacitance, C, of the Parallel-Plate Structure w/o Dielectric

5. Recall the Circuit Representation Capacitor Circuit Operation • LINEAR Caps Follow the Capacitance Law; in DC • Where • Q  The CHARGE STORED in the Cap, Coulombs • C  Capacitance, Farad • Vc DC-Voltage Across the Capacitor • Discern the Base Units for Capacitance • The Basic Circuit-Capacitance Equation

6. Pick a Cap, Say 12 µF “Feel” for Capacitance • Recall Capacitor Law • Solving for Vc • Caps can RETAIN Charge for a Long Time after Removing the Charging Voltage • Caps can Be DANGEROUS! • Now Assume That The Cap is Charged to hold 15 mC • Find V c

7. The time-Invariant Cap Law Forms of the Capacitor Law • If vC at − = 0, then the traditional statement of the Integral Law • Leads to DIFFERENTIAL Cap Law • The Differential Suggests SEPARATING Variables • If at t0, vC = vC(t0) (a KNOWN value), then the Integral Law becomes • Leads to The INTEGRAL form of the Capacitance Law

8. Express the VOLTAGE Across the Cap Using the INTEGRAL Law Capacitor Integral Law • Thus a Major Mathematical Implication of the Integral law • The Voltage Across a Capacitor MUST be Continuous • An Alternative View • The Differential Reln • If i(t) has NO Gaps in its i(t) curve then • Even if i(y) has VERTICAL Jumps: • If vC is NOT Continous then dvC/dt → , and So iC → . This is NOT PHYSICALLY possible

9. Express the CURRENT “Thru” the Cap Using the Differential Law Capacitor Differential Law • Thus a Major Mathematical Implication of the Differential Law • A Cap with CONSTANT Voltage Across it Behaves as an OPEN Circuit • If vC = Constant Then • Cap Current • Charges do NOT flow THRU a Cap • Charge ENTER or EXITS The Cap in Response to Voltage CHANGES • This is the DC Steady-State Behavior for a Capacitor

10. Capacitor Current • Charges do NOT flow THRU a Cap • Charge ENTER or EXITS The Capacitor in Response to the Voltage Across it • That is, the Voltage-Change DISPLACES the Charge Stored in The Cap • This displaced Charge is, to the REST OF THE CKT, Indistinguishablefrom conduction (Resistor) Current • Thus The Capacitor Current is Called the “Displacement” Current

11. The Circuit Symbol Capacitor Summary • From Calculus, Recall an Integral Property • Note The Passive Sign Convention • Now Recall the Long Form of the Integral Relation • Compare Ohm’s Law and Capactitance Law • Cap Ohm • The DEFINITE Integral is just a no.; call it vC(t0) so

12. Capacitor Summary cont • Consider Finally the Differential Reln • Some Implications • For small Displacement Current dvC/dt is small; i.e, vC changes only a little • Obtaining Large iC requires HUGE Voltage Gradients if C is small • Conclusion: A Cap RESISTS CHANGES in VOLTAGE ACROSS It

13. The fact that the Cap Current is defined through a DIFFERENTIAL has important implications... Consider the Example at Left iC Defined by Differential • Using the 1st Derivative (slopes) to find i(t) • Shows vC(t) • C = 5 µF • Find iC(t)

14. UNlike an I-src or V-src a Cap Does NOT Produce Energy A Cap is a PASSIVE Device that Can STORE Energy Recall from Chp.1 The Relation for POWER Capacitor Energy Storage • For a Cap • Recall also • Subbing into Pwr Reln • By the Derivative CHAIN RULE • Then the INSTANTANEOUS Power

15. Again From Chp.1 Recall that Energy (or Work) is the time integral of Power Mathematically Capacitor Energy Storage cont • Integrating the “Chain Rule” Relation • Recall also • Subbing into Pwr Reln • Comment on the Bounds • If the Lower Bound is − we talk about “energy stored at time t2” • If the Bounds are −  to + then we talk about the “total energy stored” • Again by Chain Rule

16. Then Energy in Terms of Capacitor Stored-Charge VC(t)C = 5 µF Capacitor Energy Storage cont.2 • The Total Energy Stored during t = 0-6 ms • Short Example • wC Units? • Charge Stored at 3 mS

17. For t > 8 mS, What is the Total Stored CHARGED? vC(t)C = 5 µF Some Questions About Example • For t > 8 mS, What is the Total Stored ENERGY? CHARGING Current DIScharging Current

18. Consider A Cap Driven by A SINUSOIDAL V-Src i(t) Capacitor Summary: Q, V, I, P, W • Charge stored at a Given Time • Current “thru” the Cap • Energy stored at a given time • Find All typical Quantities • Note • 120 = 60∙(2) → 60 Hz

19. Consider A Cap Driven by A SINUSOIDAL V-Src Capacitor Summary cont. • At 135° = (3/4) i(t) • The Cap is SUPPLYING Power at At 135° = (3/4) = 6.25 mS • That is, The Cap is RELEASING (previously) STORED Energy at Rate of 6.371 J/s • Electric power absorbed by Cap at a given time

20. WhiteBoard Work • Let’s Work this Problem • The VOLTAGE across a 0.1-F capacitor is given by the waveform in the Figure Below. Find the WaveForm Eqn for the Capacitor CURRENT ANOTHER PROB 0.5 μF See ENGR-43_Lec-06-1_Capacitors_WhtBd.ppt + vC(t) -

21. Second of the Energy-Storage Devices Basic Physical Model: The Inductor Ckt Symbol • Details of Physical Operation Described in PHYS 4B or ENGR45 • Note the Use of the PASSIVE Sign Convention

22. Inductors are Typically Fabricated by Winding Around a Magnetic (e.g., Iron) Core a LOW Resistance Wire Applying to the Terminals a TIME VARYING Current Results in a “Back EMF” voltage at the connection terminals Physical Inductor • Some Real Inductors

23. From Physics, recall that a time varying magnetic flux, , Induces a voltage Thru the Induction Law Inductance Defined • Where the Constant of Proportionality, L, is called the INDUCTANCE • L is Measured in Units of “Henrys”, H • 1H = 1 V•s/Amp • Inductors STORE electromagnetic energy • They May Supply Stored Energy Back To The Circuit, But They CANNOT CREATE Energy • For a Linear Inductor The Flux Is Proportional To The Current Thru it

24. Inductors Cannot Create Energy; They are PASSIVE Devices All Passive Devices Must Obey the Passive Sign Convention Inductance Sign Convention

25. Recall the Circuit Representation Inductor Circuit Operation • Separating the Variables and Integrating Yields the INTEGRAL form • In a development Similar to that used with caps, Integrate − to t0 for an Alternative integral Law • Previously Defined the Differential Form of the Induction Law

26. From the Differential Law Observe That if iL is not Continuous, diL/dt → , and vL must also →  This is NOT physically Possible Thus iL must be continuous Inductor Model Implications • Consider Now the Alternative Integral law • If iL is constant, say iL(t0), then The Integral MUST be ZERO, and hence vL MUST be ZERO • This is DC Steady-State Inductor Behavior • vL = 0 at DC • i.e; the Inductor looks like a SHORT CIRCUITto DC Potentials

27. From the Definition of Instantaneous Power Inductor: Power and Energy • Time Integrate Power to Find the Energy (Work) • Subbing for the Voltage by the Differential Law • Units Analysis • J = H x A2 • Energy Stored on Time Interval • Again By the Chain Rule for Math Differentiation • Energy Stored on an Interval Can be POSITIVE or NEGATIVE

28. 1 = 2 Li ( t ) 0 1 L 2 1 = 2 w ( t ) Li ( t ) L 2 Inductor: P & W cont. • In the Interval Energy Eqn Let at time t1 • Then To Arrive At The Stored Energy at a later given time, t • Thus Observe That the Stored Energy is ALWAYS Positive In ABSOLUTE Terms as iL is SQUARED • ABSOLUTE-POSITIVE-ONLY Energy-Storage is Characteristic of a PASSIVE ELEMENT

29. Given The iL Current WaveForm, Find vL for L = 10 mH Example • The Derivative of a Line is its SLOPE, m • Then the Slopes • And the vL Voltage • The Differential Reln

30. The Energy Stored between 2 & 4 mS Example  Power & Energy • The Value Is Negative Because The Inductor SUPPLIES Energy PREVIOUSLY STORED with a Magnitude of 2 μJ • The Energy Eqn • Running the No.s

31. Given The Voltage Wave Form Across L , Find iL if L = 0.1 H i(0) = 2A Numerical Example • The PieceWise Function • A Line Followed by A Constant; Plotting • The Integral Reln

32. The Current Characteristic Numerical Example - Energy • The Initial Stored Energy • The “Total Stored Energy” • Energy Stored between 0-2 • The Energy Eqn • → Consistent with Previous Calculation • Energy Stored on Interval Can be POS or NEG

33. Example • Find The Voltage Across And The Energy Stored (As Function Of Time) • Notice That the ABSOLUTE Energy Stored At Any Given Time Is Non Negative (by sin2) • i.e., The Inductor Is A PASSIVE Element • For The Energy Stored

34. L = 5 mH; Find the Voltage

35. Example  Total Energy Stored • The Ckt Below is in the DC-SteadyState • Find the Total Energy Stored by the 2-Caps & 2-Inds • Recall that at DC • Cap → Short-Ckt • Ind → Open-Ckt • Shorting-Caps; Opening-Inds • KCL at node-A • Solving for VA

36. Example  Total Energy Stored • Continue Analysis of Shorted/Opened ckt • Using Ohm and VA = 16.2V • By KCL at Node-A • VC2 by Ohm • VC1 by Ohm & KVL

37. Example  Total Energy Stored • Have all I’s & V’s: • Next using the E-Storage Eqns • Then the E- Storage Calculations −1.2 A 1.8 A 10.8 V 16.2 V

38. Caps & Inds Ideal vs. Real • A Real CAP has Parasitic Resistances & Inductance: • A Real IND has Parasitic Resistances & Capacitance: GenerallySMALLEffect

39. Ideal C & L Ideal vs Real • Practical Elements “Leak” Thru Unwanted Resistance

40. Capacitors in Series • By KVL for 1-LOOP ckt • If the vi(t0) = 0, Then Discern the Equivalent Series Capacitance, CS • CAPS in SERIES Combine as Resistorsin PARALLEL

41. Find Equivalent C Initial Voltage Example • Spot Caps in Series • Or Can Reduce Two at a Time • Use KVL for Initial Voltage • This is the Algebraic Sum of the Initial Voltages • Polarity is Set by the Reference Directions noted in the Diagram

42. Two charged Capacitors Are Connected As Shown. Find the Unknown Capacitance Numerical Example • Recognize SINGLE Loop Ckt → Same Current in Both Caps • Thus Both Caps Accumulate the SAME Charge • And Find VC by KVL • VC = 12V-8V = 4V • Finally Find C by Charge Eqn

43. Capacitors in Parallel • By KCL for 1-NodePair ckt • Thus The Equivalent Parallel Capacitance • CAPS in Parallel Combine as Resistors in SERIES

44. Complex Example → Find Ceq

45. Inductors in Series • By KVL For 1-LOOP ckt • Thus • Use The Inductance Law • INDUCTORS in Series add as Resistors in SERIES

46. Inductors in Parallel • By KCL for 1-NodePair ckt • And • Thus • INDUCTORS in Parallel combine as Resistorsin PARALLEL

47. Example – Find: Leq, i0 • Series↔Parallel Summary • INDUCTORS Combine as do RESISTORS • CAPACITORS Combine as do CONDUCTORS

48. Find Leq for Li = 4 mH Inductor Ladder Network • Place Nodes In Chosen Locations • Connect Between Nodes • When in Doubt, ReDraw • Select Nodes

49. Find Leq for Li = 6mH • ReDraw The Ckt for Enhanced Clarity • Nodes Can have Complex Shapes • The Electrical Diagram Does NOT have to Follow the Physical Layout • It’s Simple Now

50. C&L Summary