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

1. Engineering 43 Chp 6.1Capacitors 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 • Model as circuit element • Capacitor And Inductor Combinations • Series/Parallel Combinations Of Elements • RC OP-AMP Circuits • Electronic Integration & Differentiation

3. First of the Energy-Storage Devices Basic Physical Model (R  ) 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 • The Capacitance, C, of the Parallel-Plate Structure w/o Dielectric • Then What are the UNITS of Capacitance, C • Typical Cap Values →“micro” or “nano”

5. Recall the Circuit Representation Capacitor Circuit Operation • LINEAR Caps Follow the Capacitance Law • Where • Q  The CHARGE STORED in the Cap, Coulombs • C  Capacitance, Farad • VC  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 Vc

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 voltage is defined through an INTEGRAL has important implications... Consider the Example at Left vC 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. Given iC, Find vC Numerical Example • The Piecewise Fcn for iC C= 4µFvC(0) = 0 • Integrating & Graphing Linear Parabolic

19. For The Previous Conditions, Find The POWER Characteristic C = 4 µF iC by Piecewise curve • From Before the vC Power Example • Using the Pwr Reln

20. Finally the Power Characteristic Power Example cont • Absorbing or Supplying Power? • During the CHARGING Period of 0-2 mS, the Cap ABSORBS Power • During DIScharge the Cap SUPPLIES power • But only until the stored charge is fully depleted

21. For The Previous Conditions, Find The ENERGY Characteristic C = 4 µF pC by Piecewise curve Energy Example • Now The Work (or Energy) is the Time Integral of Power • For 0  t  2 mS

22. For 2 < t  4 mS Energy Example cont • Taking The Time Integral and adding w(2 mS) • Then the Energy Characteristic

23. 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

24. 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 STORED Energy at Rate of 6.371 J/s • Electric power supplied to Cap at a given time

25. 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 for the current in the capacitor ANOTHER PROB 0.5 μF See ENGR-43_Lec-06-1_Capacitors_WhtBd.ppt + vC(t) -

26. Engineering 43 Appendix Complex Cap Example Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

27. If the current is known ... SAMPLE PROBLEM Current through capacitor Voltage at a given time t Voltage at a given time t when voltage at time to<t is also known V C Charge at a given time Voltage as a function of time W Electric power supplied to capacitor V J Energy stored in capacitor at a given time “Total” energy stored in the capacitor J

28. SAMPLE PROBLEM Given current and capacitance Compute voltage as a function of time At minus infinity everything is zero. Since current is zero for t<0 we have In particular Charge stored at 5ms Total energy stored Before looking into a formal way to describe the current we will look at additional questions that can be answered. Total means at infinity. Hence Now, for a formal way to represent piecewise functions....

29. Formal description of a piecewise analytical signal