Last time energy storage elements capacitor

ESE 271 / Spring 2013 / Lecture 9

Last time : energy storage elements ‐ capacitor.

Charge on plates Energy stored in the form of electric field

Passive signPassive sign convention

V lt d l it t h b tl

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Voltage drop across real capacitor can not change abruptly because instant voltage change means instant change of accumulated charge and this, in turn, requires infinite current.


Last time : capacitor charge/discharge.

Charging capacitor by practical DC source

Energy gets stored in the capacitorthe capacitor

Discharging capacitor

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Energy previously stored in the capacitor gets dissipated in resistor


Energy can be stored in circuit element in the form of magnetic field.

Ampere’s law – magnetic field created by electric current in vacuum

Coil of wire can be used to store i h f f i fi ld

magnetic flux densityenergy in the form of magnetic field

Biot‐Savart law:

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Magnetic flux density generated by current in the coil of wire.

Magnetic field in center generated by

Magnetic field in center generated by full coil of wire

Biot‐Savart law:

F i it l t t i i NFor circuit element containing N coils and in media with magnetic permeability

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In every case depends on geometry but always:


Inductance.

Closed path for Ampere’s law calculation

Number of turns inside the closed path

Total number of turns in solenoidf

Length of solenoid

Magnetic flux density inside solenoid core:

Magnetic flux inside solenoid core:

Inductance (self‐inductance) relates the magnetic flux to the current that created it.

Depends on geometry of the circuit element.

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Linear inductor.

Energy stored in inductor

Recall that in parallel plate capacitor with plate area A and spacing d:

, where volume energy density

Volume energy density of magnetic field: Energy stored in solenoid of length l

In solenoid we just considered:

and cross‐section of the core A:

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Voltage drop across ideal linear inductor.

Magnetic flux:

When DC current is flowing through ideal inductor the voltage drop across it is zero.

What happens if the current is changing with time?

Magnetic flux:

Magnetic flux change due to change of current:

F d ’ l l t ti fFaraday’s law – electromotive force:

Recall EMF in battery:

In inductor:

Change of current in inductor generates

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inductor generates the voltage drop.


Voltage and current in inductor.

Passive sign convention

Voltage drop appears because the induced EMF force opposes the change of current

Assume:

moved charges in the direction opposite to the direction of current changemoved charges in the direction opposite to the direction of current change.

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Inductor power and energy.

Initial current

Current in inductor implies presence of magnetic field, hence, current is associated with energy and energy can not be changed abruptly without infinite power.

Hence, current through practical inductor can not be changed abruptly since it would imply infinite voltagewould imply infinite voltage.

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Let’s put the energy into inductor.

1. Simple ideal case:

2. More realistic case with practical power supply:

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‐ Time constant of the RL‐circuit


Increase of current through inductor.

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Increase of current through inductor by practical voltage source.

Obviously, everything is just the same since we used Thevenin form of the practicalThevenin form of the practical source and it is equivalent to Norton one used on slide 11.

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Decrease of current through inductor – removal of energy.

1. Simple ideal case:

See ignition coil in cars

2. Realistic case:

coil in cars

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Series connection of inductors

Physical sense:

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Parallel connection of inductors

Physical sense – current divider.

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Example 1.

A DCAssume DC steady‐state

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Example 2.

‐ Current trough inductor in DC steady‐state

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‐ Energy stored in inductor


Example 2 – cont.

Energy stored in inductor:

Energy dissipated in the circuit during

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Energy dissipated in the circuit during transition from initial DC steady‐state to final DC steady‐state condition:


Example 2 – cont.

KVL:

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Example 2 – cont.

KVL:

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Example 2 – cont.

Find B from initial conditions:

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Example 2 – cont.

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Superposition.

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Superposition ‐ cont.

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Example.

DC steady‐statey

Energy stored in inductor will get dissipated in resistor.

Eventually all voltages and currents will become zero since circuit will contain no sources and there is resistor that does not store but dissipates energy.

To find the transient values of voltage and currents we will need to solve integro‐differential equation – not an easy task, in general.

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Last time energy storage elements capacitor