1 ECE 3336 Introduction to Circuits & Electronics Note Set #7a Capacitors Fall 2012, TUE&TH 4:00-5:30pm Dr. Wanda Wosik

2 Capacitors Capacitance C[F] is important in circuits (ex. filters, tuning circuits, electron devices etc.) but also in electromechanical applications (sensors, transducers, motors etc). Capacitance may be also parasitic (between electronic elements, connectors etc) A capacitor is a circuit element that stores electric charges and energy in electric fields. For DC conditions it acts as an open circuit For AC conditions a current will flow through C and it will be proportional to the voltage derivative. Charge stored when current flows

Analogy between fluid capacitance and electrical capacitance q f q f P 1 p 1 gas p 2 p 2 + _ C f p i v 2 v 1 + _ Cv 3

Parallel Plate Capacitor – Equations without considering fringe electric field. – A note on fringe electric field: The fringe field is frequently ignored in first-order analysis. It is nonetheless important Fringe electric field (ignored in first order analysis) Electric field (Gauss law) Energy stored in the capacitor Capacitance Differential form Integral form Dielectric=air But can be low or high e r materials solid or liquid 4 Inserting dielectric:  E  V  C  (same Q) better charge storage efficiency

5 The defining equation for the capacitor, If we want to express the voltage in terms of the current, we can integrate both sides. Defining Equation, Integral Form, Derivation We pick t 0 (initial time, ex. 0) and t (variable) for limits of the integral - The capacitance, C X, is constant – does not change with current or voltage i c dt=dQ dQ=C x dv c Meaning: Q=C x V [A][s]=[C][C]=[F][V] Charge storage

6 Series Capacitors - Equivalent Circuits Series capacitors, C 1 and C 2 and …C n, can be replaced with an equivalent circuit with a single capacitor C EQ Use KCL i C1 (t)=i C2 (t)=…=i CEQ (t) v C1 (t)+v C2 (t)+…+v Cn (t)=v CEQ (t) i C =Cdv c (t)/dt v C1 (t) v C2 (t) v C3 (t) v CEQ (t) EQ 

7 Parallel Capacitors Equivalent Circuits Parallel capacitors, C 1 and C 2 and …C n, can be replaced with an equivalent circuit with a single capacitor C EQ Use KCL v C1 (t)=v C2 (t)=…=v CEQ (t) i C1 (t)+i C2 (t)+…+i Cn (t)=i CEQ (t) i C =Cdv c (t)/dt i CEQ (t) i C1 (t) i C2 (t)i Cn (t) + v C (t) - EQ 

8 Power and Energy in Capacitors Power: d(energy)/dt Energy:

Applications (others than EE) Basic Principles Sensing – capacitance between moving and fixed plates change as distance and position is changed media is replaced Actuation – electrostatic force (attraction) between moving and fixed plates as a voltage is applied between them. Two major configurations Parallel plate configuration Interdigitated finger configuration 9 C.Liu

Forces of Capacitor Actuators Stored energy Force is derivative of energy with respect to pertinent dimensional variable Plug in the expression for capacitor We arrive at the expression for force 10

An Equivalent Electromechanical Model d No forces applied to the structure Note: direction definition of variables Mechanical force develops to balance the electrical force “d” will change with increasing voltage 11 C.Liu

Micromotors Slightly smaller pitch and misaligned electrodes Power AA’ so that they would align That would misalign BB’ so now apply bias to BB’ Then repeat the steps for CC’ Similar idea in rotary motors but the main problem comes from the bearings in the rotors. Electrodes are in the rotor and stator are misaligned. That generates electrostatic force that moves the rotor. There is more poles in the stator than in the rotor 3:2. The motor rotates with 10,000 rmp so there is a problem with wearing and lubrication; tribology concentrate on these issues. µm2µm 20-25µm Hsu 12

Actuators that Use Fringe Electric Field - Rotary Motor Three phase electrostatic actuator. Arrows indicate electric field and electrostatic force. The tangential components cause the motor to rotate. 13 C.Liu

Electrostatic Micromotor rotor has movednew electrodes biased ROTOR 14 C.Liu

Three Phase Motor Operation Principle 15 C.Liu

Starting Position -> Apply voltage to group A electrodes 16 C.Liu

Motor tooth aligned to A -> Apply voltage to Group C electrodes 17 C.Liu

Motor tooth aligned to C -> Apply voltage to Group B electrodes 18 C.Liu

Motor tooth aligned to B -> Apply voltage to Group A electrodes 19 C.Liu

Motor tooth aligned to A -> Apply voltage to Group C electrodes 20 C.Liu

21 Polarity is important for current sources and voltages sources. Resistors, inductors and capacitors (except for electrolytic) do not have this restriction. We will use the same passive and active sign conventions as for resistors to find relations between currents and voltages in capacitors Capacitors in Circuits Polarities Passive Sign ConventionActive Sign Convention

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Circuit Symbols of Capacitor 24

“How an Ultracapacitor Works An ultracapacitor, also known as a double-layer capacitor, polarizes an electrolytic solution to store energy electrostatically. Though it is an electrochemical device, no chemical reactions are involved in its energy storage mechanism – it is electrostatic. This mechanism is highly reversible, and allows the ultracapacitor to be charged and discharged hundreds of thousands of times.” source: NREL e/ultracapacitors.html 25

26 Time dependence in i c (t) and v c (t) AC: If there is no voltage change there is no current flowing through the capacitor. DC: If a charge is stored but it does not change (no current) there is a constant voltage that can be measured (be careful – it can be large). A capacitor will still has no current (unless it is leaky). and

27 Change in voltage is limited Current is not infinitive The voltage change across a capacitor cannot be infinitively fast. The current cannot be infinite but it can be very large and

28 Rules for Capacitors Passive sign convention For capacitors, we had the following rules and equations:

29 Charging and Discharging the Capacitor Wikipedia

30 One of the Applications High pass filter

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