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Passive components and circuitsLecture 6

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Index

Inductance Inductance as circuit element DC regime behavior AC regime behavior Transient regime behavior

RLC series circuit RLC parallel circuit

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Web addresses: http://en.wikipedia.org/wiki/RL_circuit http://www.play-hookey.com/ac_theory/ac_rl_series.html http://en.wikibooks.org/wiki/Circuit_Theory/RLC_Circuits

#Series_RLC_Circuit http://members.aol.com/_ht_a/RAdelkopf/rl.html http://www.tpub.com/neets/book2/4l.htm http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcpar

.html

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Inductance as circuit element The main electrical property

of the inductance – to generate magnetic field when a current is flowing through it.

Measurement unit - Henri [H].

Practical units: starting nH

and H up to mH and H.

LI

dt

dILV

dtdIV

dtdIdtd

L LL

I

L

V L

C o n d u c to r

C o n d u c to r

M e d iu c a rc a te r iza t

d e p e rm e b il ita te am a g n e tic ă re la t iv ă r

I

L in ii le d e f lu x m a g n e tic Magnetic flux

lines

Conductive element

Conductive element

Magnetic substrate

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Inductance as circuit element The electronic component characterized

by the inductance is called inductor (coil).

Inductance L of a particular coil depends on its geometrical dimensions (A – turn area, L – length or l – width of the winding on the support), number of turns N, relative permeability r of the core, working temperature, etc. The theoretical formula for a coil with one turn after another, disposed on one layer, linear shape, is: H/m104 7

0

20

l

ANL r

l

D

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The energy stored in inductance The inductance doesn’t dissipate power, but it stores

electric energy:

2

000 2

1LIdt

dt

dIILVIdtPdtW

TTT

m

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Series connection The equivalent inductance of a

series connection is equal with the sum of all inductances:

n

iiech

n

iiABiAB

i

ii

AB

ABech

LL

iivv

dt

div

L

dtdiv

L

1

1

;

;

A

A

B

B

LnL2

Lech

L1

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Parallel connection: The equivalent inductance of a

parallel connection is given by the following relation:

n

i iech

n

iiABiAB

i

ii

AB

ABech

LL

vvii

dtdiv

L

dtdiv

L

1

1

11

;

;

B

A

B

A

L1 L2 Ln Lech

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DC regime behavior In DC regime, the inductances

are equivalent with short circuit:

Circuitelectronic

A B

V1 V2=V1

V =0AB

DC

L

Circuitelectronic

A B

V1 V2=V1

V =0AB

0. dt

diLvcsti AB

ABAB

Electronic circuit

Electronic circuit

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AC regime behavior In AC regime, inductances are equivalent with

impedances ZL.

LZX

LjZI

VILj

eeILjdt

eeVIdLV

eeIIdt

dILV

LL

LL

LL

jtjjtj

L

jtjL

LL

;

Inductive reactance

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AC regime behavior

Inductive reactance (impedance) depends on the frequency.

In AC, the imittances of the circuits with inductances depend on the signal frequency.

In consequence, the circuits with inductances have the signal filtering property.

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RL high-pass filter

RL

j

RL

j

jv

jvjH

vLjR

Ljv

ZR

Zv

i

o

iiL

Lo

1)(

)()(

For R=1K and L=160H is obtained: 6

6

3

6

3

6

101

10

1010160

21

1010160

2)(

fj

fj

fj

fjjfH

vo

R

vi L

Exercises::

Calculate vo=f(vi).

Identify, from the previous course, the circuit with the same transfer function.

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RL high-pass filter - frequency characteristics

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RL low-pass filter

RL

jjv

jvjH

vLjR

Rv

ZR

Rv

i

o

iiL

o

1

1

)(

)()(

6

3

6 101

1

1010160

21

1)(

fj

fjjfH

For R=1K and L=160H is obtained:

voR

L

Exercises::Calculate vo=f(vi).Identify, from the previous course, the circuit with the same transfer

function.

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RL low-pass filter - frequency characteristics

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High frequency inductance behavior At a very high frequency, the inductive reactance becomes much higher than the resistances from the circuits. In this case, the inductance is equivalent with open-circuit.

vi

R

vi

vo

vi

voR

=

0=

V H Fvo

R

vi L

voR

L

vi

V H F

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High frequency shocks

In some circuits, the inductances are used to separate AC high frequency components between two circuits (AC high frequency -> open-circuits), without affecting the DC components (DC -> short-circuits). In these situations, the inductances are called high frequency shocks.

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Transient regime behavior

In this case, the transient regime consists in the modification of a DC circuit state in a new DC state.

During these modifications, the inductance cannot be considered open-circuit or short-circuit.

The transient regime analysis presume determining the way the inductances charge and discharge.

In transient regime, the circuit operations are described by differential equations.

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Determining the current when a constant voltage is applied

Considering the switch K on position 1. The current through the inductance is zero.

At the time t=t0, the switch is moved on position 2.

After enough time, t, the current through inductor will be E/R.

The transient regime is taking place between these two DC states.

R

E vL

1

2K

iL

vR

L

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Determining the current when a constant voltage is applied

dt

dii

R

ER

L

dt

diLRiE

dt

diLvviRE

vvE

LL

LL

LLLL

LR

;

;

:TKV

t

LLLL eiiiti

)]()0([)()(

)()(

)()0( 0

tii

ttii

LL

LL

Solution of differential equations

R

L Circuit time

constant

R

E vL

1

K

iL

2

vR

L

21/43

Current variation

)1()(

)(;0)0(

t

L

LL

eR

Eti

R

Eii

22/43

Voltage variation

t

RL

t

LR

eEtvEtv

eERtitv

)()(

)1()()(

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Significance of time constant

If the transient process has the same slope like in the origin (initial moment), the final values of voltages and currents will be obtain after a time equal with the circuit time constant.

As can be seen in the previous figures, the charging process continues to infinite.

Practically, the transient regime is considered to be finished after 3 (95% from the final values) or 5 (99% from the final values).

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Example (E=1V, R=1K, L=1mH)

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Inductance discharging

At the initial time, consider the switch on position 2. The current through the inductor is E/R.

At a reference time moment t=t0, The switch K is moved on position 1.

After enough time, t, the current becomes zero.

The transient regime is the time between these two DC states.

R

E vL

1

K

iL

2

vR

L

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Inductance discharging

dt

dii

R

L

dt

diLiR

dt

diLvviR

vv

LL

LL

LLLL

LR

0

;0

;0

0:TKV

Solution of differential equation

;)()(

;)()(;)(

t

RL

t

LR

t

L

eEtvtv

eEtiRtveR

Eti

t

LLLL eiiiti

)]()0([)()(

0)()(

)()0( 0

tiiR

Ettii

LL

LL

R

E vL

1

2K

iL

vR

L

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Example (E=1V, R=1K, L=1mH)

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Observation At the switching between 2 and

1 positions, we can have an open circuit and the current becomes zero instantaneously. That means di/dt. This phenomenon determines an over voltage across the inductance which can be dangerous for other circuits.

Over voltage protection – introducing a diode in circuit.

R

E vL

1

2K

iL

vR

L

R

E vL

1

2K

iL

vR

L

29/43

The RL circuits behavior when pulses are applied Consider a pulses signal source

applied to a series RL circuit. In analyzing of circuit behavior, we

consider both the voltage across

the inductor, vL(t), and the voltage

across the resistor, vR(t). Applying this signal source, the

phenomenon of charging and discharging described to transient regime is repetitive.

R

vL

iC

vR

vI

L

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Case A – the time constant is much lower than the pulses duration

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Case B – the time constant is much greater than the pulses duration

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Integrating circuit If the output voltage is the voltage across the resistor, the

effect under the input signal is an attenuation of edges, similarly with the integration mathematical operation.

In this situation,(when vO(t)=vR(t)), the circuit is called

integration circuit. The integration effect is higher in case B , when the time

constant is greater than the pulse duration. The integration function in transient regime corresponds

to low-pass filtering in AC regime.

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Derivative circuit

If the output voltage is the voltage across the inductance, the circuit effect under the input signal is an accentuation of edges, similarly with the derivative mathematical operation.

In this situation,(when vO(t)= vL(t)), the circuit is called

derivative circuit. The derivative effect is higher in case A , when the time

constant is lower than the pulse duration. The derivative function in transient regime corresponds

to high-pass filtering in AC regime.

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RLC series circuit – AC regime behavior The equivalent impedance

between AB terminals is:

CR L

A Bi

vAB

CLjR

CjLjRZZ SechAB

11

Modulus of this impedance is:

C

LCCR

CLRZSech

222222

2 11

35/43

RLC series circuit – AC regime behavior We can notice that when the frequency is 0 or , the modulus is .

In DC, the capacitance is equivalent with open-circuit. At very high frequency, the inductance is equivalent with open-circuit.

The imaginary part of impedance becomes zero at the frequency:

0

1

2f

LC

This frequency is called resonance frequency. From the energetic point of view, at this frequency the energy is transferred between inductor and capacitor.

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RLC series circuit – AC regime behavior The derivate of impedance modulus becomes zero at the resonance

frequency.

The resonance frequency is an extreme point for impedance modulus (minimum, for this case).

At the resonance frequency, the impedance is pure resistive.

RZSech )( 0

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Modulus of ZSech for R=10, L=10H, C=100nF

Exercise: Represent |ZSech| versus frequency at the logarithmic scale.

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RLC parallel circuit – AC regime behavior

The equivalent impedance between AB terminals is:

RL

jLC

Lj

LC

LjRZZRZZ CLPechAB

22

11||||||

Modulus of this impedance is:

222

2 1 LCRL

LZPech

R

C

L

i

A B

vAB

39/43

RLC parallel circuit – AC regime behavior

We can notice that when the frequency is 0 or , the modulus is 0. In DC, the inductance is equivalent with short-circuit. At very high frequency, the capacitance is equivalent with short-circuit.

The imaginary part of impedance becomes zero at the frequency:

This frequency is called resonance frequency. From the energetic point of view, at this frequency the energy is transferred between inductor and capacitor.

0

1

2f

LC

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RLC parallel circuit – AC regime behavior

The derivate of impedance modulus becomes zero at the resonance frequency.

The resonance frequency is an extreme point for impedance modulus (maximum, for this case).

At the resonance frequency, the impedance is pure resistive.

RZPech )( 0

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Modulus of ZPech for R=100, L=10H, C=100nF

Exercise:Represent |ZPech | versus frequency at the logarithmic scale.

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Quality factor - Q These two structures (series and parallel RLC circuits) are used in

order to obtain Band-pass and Band-reject filters.

The selectivity of these circuits respective to some frequencies is characterized by quality factor. The quality factor is defined as the ratio between resonance frequency and 3dB frequency band.

L

RQ

R

LQ

P

S

0

0

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Homework For the following circuit and

each situation (from table), determine the circuit function.

Make an essay: “Complementarity's between inductance and capacitance behavior in electronic circuits”

Circuit 1 Circuit 2 FunctionR series RLCR parallel RLC series RLC Rparallel RLC Rseries RLC parallel RLC parallel RLC series RLC

Circuit 1

Circuit 2vi

vo