l2_energy storage elements

8/10/2019 L2_Energy Storage Elements

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Energy Storage Elements

ELCITWO


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Contents Introduction Capacitors Energy Storage in a Capacitor Series and Parallel Capacitors Inductors Energy Storage in an Inductor Series and Parallel Inductors Plotting Capacitor and Inductor Voltage and

Current (in MatLab)


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Introduction

This lecture aims to discuss the nature ofcapacitors and inductors. The analysis of theircharacteristics involves either integration or

differentiation.

Electric circuits that contain capacitors and/orinductors are represented by differentialequations. Circuits that do not contain themare represented by algebraic equations.


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Introduction

Circuits that contains capacitors and/orinductors are dynamic circuits , while circuitsthat do not contain them are static circuits .

Circuits that contain capacitors and/orinductors are able to store energy.

Circuits that contain capacitors and/orinductors have memory.


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Introduction

In a dc circuit, capacitors act like open circuitsand inductors act like short circuits.

A set of series or parallel capacitor/inductorcan be reduced to an equivalent one.

Integrating and differentiating circuits can beaccomplished by these components working withan operational amplifier.


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Capacitors

A capacitor is a two-terminal element that is amodel of a device consisting of two conductingplates separated by a non-conducting material.

Electric energy is stored on the plates.


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http://sdsu-physics.org/physics180/physics196/Topics/capacitance.html
http://sdsu-physics.org/physics180/physics196/Topics/capacitance.htmlhttp://sdsu-physics.org/physics180/physics196/Topics/capacitance.htmlhttp://sdsu-physics.org/physics180/physics196/Topics/capacitance.htmlhttp://sdsu-physics.org/physics180/physics196/Topics/capacitance.html


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Example

1. Calculate the capacitance of the 2mm 2 platesseparated by 0.2mm.

2. For a charge of 30C, what would be thepotential difference for the capacitor inproblem 1?

3. What is the electric field that exist betweenthe plates?


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Capacitance is a measure of the ability of adevice to store energy in the form of

separated charge or an electric field. It has the unit C/V and in called Farad (in

honor of Michael Faraday)

Current flows while thecharge flow from oneplate to the other.

i = dq / dt

If differentiated,

i = C (dv/dt)


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Capacitors use various dielectrics and are built in several forms.Some common capacitors use impregnated paper for a

dielectric, whereas others use mica sheets, ceramics, and

organic and metal films.


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The voltage across a capacitor cannot changeinstantaneously.


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Examples

The plots represent the current and voltage ofthe capacitor in the circuit. Determine thevalues of the constants a and b, used to labelthe plot of the capacitor current.


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The current i(t) = 3.75e -1.2t A for t>0 and theoutput voltage of the capacitor is v(t) = 4 1.25e -1.2t V for t>0. Find the value of thecapacitance C.


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Energy Storage in a Capacitor

A current flows and a charge is stored on the plates of thecapacitor. Eventually, the voltage across the capacitor is aconstant, and the current through the capacitor is zero.

The capacitor has stored energy by virtue of theseparation of charges between the capacitor plates.

These charges have an electrical force acting on them.


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The forces acting on the charges stored in a

capacitor are said to result from an electricfield. An electric field is defined as the forceacting on a unit positive charge in a specifiedregion.

The energy required originally to separate thecharges is now stored by the capacitor in theelectric field. The energy stored in thecapacitor is w c(t) = (1/2) Cv 2(t) Joules


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Example

A 10 mF is charged to 100V. Find the energystored by the capacitor and the voltage of thecapacitor at t = 0 + after the switched isopened.


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Series and Parallel Capacitors Parallel: Use KCL and analyze the current in each

branch.Cp = C1 + C2 + C3 + + CN

Series: Use KVL and analyze the voltage drops ineach capacitor.

1/C s = 1/C 1 + 1/C 2 + 1/C 3 + + 1/CN


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Inductors

A wire maybe shaped as a multi-turn coil.

If we use a current source, we find that the voltageacross the coil is proportional to the rate of change ofthe current. This proportionality may be expressed by

the equation v = L (di/dt).


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Inductor a two-terminal element consisting of a winding of N-turns forintroducing inductance into an electric circuit.

Inductance the property of an electric device by which a time-varyingcurrent through the device produces a voltage across it. It is a measure of

the ability of a device to store energy in the form of a magnetic field.


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Single-layer coils wound in a helix are oftencalled solenoids. If the length of the coil isgreater than half of the diameter and the coreis of non-ferromagnetic material, the

inductance of the coil is given by

L = [(uoN2A) / (l + 0.45d)] Henry


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Current equation for inductors.

t

t

t iv dt L

i

0

)(1

0


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Example

The input to the circuit shown is v(t)=4e -20t Vfor t>0. The output current is i(t) = -1.2e -20t 1.5 A for t>0. The initial inductor current isiL(0) = -3.5A. Find the value of L and R.


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Energy Storage in an Inductor

The power in an inductor is p = vi = [L(di/dt)]i.

The energy stored in the inductor is stored inits magnetic field. The energy stored in theinductor during the interval t 0 to t is given by

W = Li2


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Example

Find the power and the energy stored in a0.1H inductor when i=20te -2t A and v = 2e -2t (1-2t) V for t>0 and i = 0 for t < 0.


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Series and Parallel Inductors Series: Use KVL and analyze the voltage in each

inductor.Ls = L1 + L2 + L3 + + LN

Parallel: Use KCL and analyze the current in eachbranch.

1/LP = 1/L1 + 1/L2 + 1/L3 + + 1/LN


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