1 / 11

Capacitance Mathematical

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

velvet
Download Presentation

Capacitance Mathematical

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Capacitance Mathematical

  2. 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)

  3. 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 Power Application By the Derivative CHAIN RULE Then the INSTANTANEOUS Power

  4. 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 Power Application • 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

  5. 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?

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

  7. Given iC, Find vC Numerical Example The Piecewise Function for iC C= 4µFvC(0) = 0 > Integrating & Graphing Linear Parabolic

  8. 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

  9. 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

  10. Summary of Formulas

  11. Capacitor Summary From Calculus, Recall an Integral Property The Circuit Symbol Note The Passive Sign Convention Now Recall the Long Form of the Integral Relation • Compare Ohm’s Law and Capacitance Law • Capacitor Ohm The DEFINITE Integral is just a number; call it vC(t0) so

More Related