En CURS 1 Two Pole Circuit Elements

8/8/2019 En CURS 1 Two Pole Circuit Elements

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COURSE ONE

TWO POLE CIRCUIT ELEMENTS

Circuit elements are systems characterized by input and output magnitudes and by the

relationship between them. We denote the instantaneous input or excitation magnitude by x(t)

and the instantaneous output or response magnitude by y(t). The relationship is generally non-

linear.

Curves y(x) represented in plane (y,x) for the different values of t are called operational

characteristics. A point )( 000 y x M on the curve at the moment 0t is called operational point.

The value of the derivative in a point at moment 0t is called dynamical parametre d P .

0/ t t d

dx

dyP ==

The circuit elements are classified according to the fact that the characteristic is linear or

non-linear, variable or invariable with the time:

- Linear elements invariable with the time Cx y =

- Linear elements variable with the time or parametric )()( t xt C y ⋅=

- Non-linear elements invariable with the time )( x y y =

- Non-linear elements variable with the time ),( t x y y = or ( )[ ]t t x y y ,=

Irrespective of the nature of magnitude pair (x,y) the hub voltage )(t u and the current intensity

)(t i are uniquely determined at the hubs of the circuit elements and their product is denoted:

iu p ⋅=

called instantaneous power ; the power integral in ratio with the time during the interval 12 t t −

is the energy 12W .

0 x

2t

x

)0(t 1t

0 y

y

0 M

)(t y )(t x


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2

∫=2

1

12

t

t

pdt W

If at least in one point of the operation characteristic, the instantaneous power is negative

p0 the element

receives power from the hubs and it is called a passive element, receiver or load . Their

operational characteristics are only placed within scales 1and 3.

Passive elements able to amass energy in an electric or magnetic field are called reactive

(the coil, the condenser). The resistor irreversibly transforms the energy into heat.

THE RESISTOR

The characeristic equation of the resistor is:

( )[ ]t t iuu ,= or ( )[ ]t t uii ,=

The charcteristic curve in plane [ ]iu, at moment t is called characteristic voltage-

current, in plane i-u characteristic current-power, respectively.

The linear resistor variable with the time

The characteristic equation is:

)()( t Rit u = or )()( t Gut i = G

R1

= (1)

i (t)

u t


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namely Ohm’s Law, where R>0 ( )Ω and G>0 (siemens conductance)

in plane (u,i) the characteristic curve is a straight line which passes through the origin, the

voltage and the current having the same variation with the time.

If we multiply both terms of the equations (1) by i(t) and by u(t) respectively, we get the

instantaneous power p

).2(22 Gu Riui p ===

Irrespective of the reference sense of the voltage or current , power p is positive and

corresponds to the electro-calorific effect of irreversible transforming of electric energy into heat

- If ∞=⇒= G R 0 and 0)( =t u , the equation (1) becomes:

0=u - SHORT CIRCUIT

if ∞= R 0)( =G equation (1) becomes 0=i

and the element is called open or broken circuit.

Linear resistor variable with the time (parametric)

has the characteristic equation )()()( t it Rt u ⋅=

=)(t R parametric resistance with the symbol

u u

i t

u

i

i (t)

u t

R(t)


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An example of parametric resistor is the potentiometer whose mobile contact alternately

swings in the two senses with frequency f . Denoting =0 R the resistance at 0=t and max R the

peak amplitude of the part variable with the time. The parametric resistance has the expression:

ft R Rt R π 2sin)( max0 +=

The voltage variation with the time differs from that of the current .

If the resistor is passed by a sinusoidal current, ( )α π += t f I t i 1max 2sin)( , the hub voltage

)(t u has the expression:

( ) ( )[ ] ( )[ ]α π α π α π ++−−−++= t f f I R

t f f I R

t f I Rt u 1maxmax

1maxmax

1max0 2cos2

2cos2

2sin)(

The expression has two frequency terms 1 f f − and 1 f f + and a frequency term 1 f .

A switch is modelled as a circuit element by a parametric resistor which modifies the

resistance from a very low value when the ciruit is closed, to a very high value when it is open.

An ideal switch is represented at the closing of the circuit by the short circuit element and at the

opening of the circuit by the open circuit element.

The real switch can be modelled by an ideal switch with two linear resistors invariable

with the time, the first having very high 1 R and the second very low resistance 2 R , as shown in

the figure below.

u

i

max

max

R R

R R

o

o

+

〉

ft R R π

2sinmax0 +

max R Ro −


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The coil

When not magnetically coupled, the coil is a passive cirucit element with the following

characteristic equation:

( )[ ]t t i ,Φ=Φ

The graphic symbol is:

The characteristic curve in plane ( )i,Φ is called characteristic flux-current. The link

equation between the flux and the hub voltage or the voltage inductive drop is the evolution

equation:

dt

d u

Φ= (3)

By integrating in the interval t →0 , we get:

t d t ut

t

o

′′+Φ=Φ ∫ )()0()(

t d t u

O

′′=Φ ∫∞−

)()0(

Since the magnetic flux is conditioned by the previous values of voltage, the coil is a

memory element.

(t)

u (t)


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The absolute continuity of the magnetic flux

By writing t d t ut

t

′′=Φ ∫∞−)()( ( 3′ ) where )(t u ′ is an integrable function, the magnetic flux

defined by the equation (3) in interval (-∞, ∞) is an absolute continuous function of time.

Irrespective of the means of switching the coil, in the electric circuit, the magnetic flux

does not vary discontinuously (the magnetic flux in the coil is preserved).

The linear coil, invariable with the time and not magnetically coupled

has the equation:

1

)()(

)()(

−Γ=

ΦΓ=

=Φ

L

t t i

t i Lt

= L independent inductance from )(,, H t iΦ

Γ >0 mutual inductance

In plane ),( iΦ the characteristic curve is a straight line through the origin and,

consequently, the magnetic flux and the current have the same form of time variation. Taking

into account (4), equation (3) becomes:

dt

di Lt u =)( (5)

Integrating on interval 0-t we get:

)6()(1

)0(

)()0()(1

)0()(

0

0

00

t d t u L

iunde

t d t uit d t u L

it i

t

t t

′′=

′′Γ+=′′+=

∫

∫∫

∞−

The current strength )(t i at the moment )(t is conditioned by the current strength at the

initial moment (0) and previous values of voltage 0)(t u ′ < t t


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By multiplying the equation (5) by t id ′ and integrating we get energym

W amassed in the

magnetic field of the coil.

( ) L

ii Lid i Lt d t it uW

it

m

22

002

1

2

1

2

1)(

Φ⋅=Φ⋅=⋅=′′=′′′= ∫∫

where we assumed that 0)0( =i .

Irrespective of the current sense in respect with that of voltage, the magnetic energy is

positive and thus, the coil is a passive element.

The theorem of the current uniform continuity in the coil

Writing equation (6) at moment dt t + and substracting term by term from the equation,

equation (5), it results that:

t d t u L

t idt t i

dt t

t

′′=−+ ∫+

)(1

)()(

If in the time interval [ ]T −0 voltage is bordered U t u


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where =⋅dt

t dit L

)()( the inductive voltage drop by pulsating (due to the current variation with the

time);

=dt

t dL

t i

)(

)( parametric inductive voltage drop (due to inductance variation with the time).

THE CONDENSER

Is a passive circuit element, with the characteristic equation:

( )[ ] ( )[ ]t t quusaut t uqq ,, ==

The characteristic curve in plane ( )uq, at a moment t is called characteristic charge-

voltage.

The link equation between the electric charge and the current strength is:

dt

dqi = (8)

Integrating equation (8) in interval t −0 , we get:

( )

( ) t d t iq

t d t iqt q

t

′′=

′′+=

∫

∫

∞−

0

0

)0(

)0()(

(9)

The electric charge )(t q at moment ( )t is conditioned by the initial charge and the

previous values of strength )(t i ′ , t t


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The absolute continuity of the electric charge

Equation (9) can be written:

( ) t d t it qt

′′= ∫∞−)( (10)

where ( )t i ′ is an integral function, the electric charge defined on interval ( )∞∞− , is an absolute

function continuous in time. Irrespective of the condenser’s linking and switching means the

electric charge does not vary discontinuously.

The linear condenser invariable with the time

The characteristic equation:

)()( t Cut q = or )()( t Sqt u =

where C>0 [ ]F does not depend on t uq ,,

S = mutual capacity [ ] DF

In plane [ ]uq, the characteristic curve is a straight line passing through the origin and

thus the electric charge and voltage have the same variation form with the time.

Since

dt

dqi = and

dt

duC t it Cut q =⇒= )()()(

Integrating, it results that:

( ) ( )

( ) ( ) t d t iC

u

t d t iS ut d t iC

ut u

t t

′′=

′′+=′′+=

∫

∫∫

∞−

0

00

10

)0(1

)0()(

Hence, voltage )(t u at moment t is conditioned by the initial voltage ( )0u and previous

values of the current t t


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( ) ( ) qU qC

CU duuC t d t it uW

t t

e2

1

2

1

2

1 22

00

====′′′= ∫∫

where we considered 0)0( =u .

Irrespective of reference senses, the condenser’s energy is positive, which makes thecondenser a passive element.

The theory of the voltage uniform continuity at the condenser’s hubs

In a similar manner with the magnetic flux, the electric voltage at the condenser’s hubs

continuously vary in the open interval (0,T) if the current strength is bordered in the closed

interval [ ]T ,0 .

THEREFORE: Voltage at the condenser’s hub cannot pass from a finite value to anotherfinite value if the current strength is bordered.

The linear condenser variable with the time (parametric)

The characteristic equation:

)()()( t ut C t q ⋅=

where =)(t C parametric capacity.

Graphically, the parametric condenser is represented below:

The current strength is derived form the relation:

dt

t dC t u

dt

dut C t i

si

t ut C t qundedt

dqt i

)()()()(

)()()()(

+=

⋅==

The first termdt

dut C )( represents the pulsating component

u t C t

i t


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dt

t dC t u

)()( represents the parametric component.

THE INDEPENDENT GENERATOR

This is an active circuit element, whose hubs voltage does not depend on the current

strength, its characteristic equation being: ( )t eu = .

In plane (u,i) the operating characteristic is a straight line parallel to axis (0,i).

Since a voltage value uniquely corresponds to a current strength, the generator can be

considered a nonlinear active resistor with current control .

As a circuit element, the independent generator is characterised by the manner in which

the electromotive voltage ( )t e varies with the time.

The generator is of direct voltage if the electromotive voltage E t e =)( is constant.

In the direct current the dependence of hub voltage bU to the current strength I is due to

the internal resistance of generatorg R .

The real generator of direct voltage is characterised by the electromotive voltage E and

internal resistanceg

R and Joubert equation has the form:

ggb I RU E =−

The operation characteristic is a straight line which does not pass through the origin .


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THE INDEPENDENT CURRENT GENERATOR

This is the active element which has the current strength independent of voltage and its

characteristic equation is :

( )t ii g=

In plane (u,i) the operation characteristic is a straight line parallel to the axis 0-u.

The current generator is completely characterised by the manner in which the injected

current ( )t ig varies with the time. If the current strength is constant with the time, ( ) gg I t i = the

source is of direct current. The current strength of real injectors depend on voltage. In direct

current this is due to the internal conductance G.


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The real injector is characterised by the current injection g I and conductance gG .

On considering the equation:

ggb I RU E =−

and dividing both terms byg

gG

R 1= , we get:

bgg U G I I =− , where ggg U Gl = is the short circuit current of the generator

The equations of voltage drop in the generator and current decrease in the current

generator are dual and the correspondence of dual magnitude is the following:

gggg G R I U I E ↔↔↔ ;; 0 ;

As circuit elements sources admit dual models of the voltage generator and current

generator.

WAYS OF VOLTAGE AND CURRENT VARIATION WITH THE TIME

• CONTINUOUS – as defined by the equation: .)( const C t C t Y =∞


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The function delayed by 0t can be written as:

( )⎩⎨⎧

≥−

<=−−

00

0

00)(

0)(

t t t t f

t t t t ht t f

• SLOPE UNIT - ( )⎩⎨⎧

≥

<=

0

00

t t

t r t

The slope unit function is equal to the non-definite integral of the unit function.

1

1 t

г

г t

1

1

г t-t0

г

t0

t

h

1

h t

t

h

h t-t0

t0

t

f(t).h(t)

f t

f(t-t0).h(t-t0)

f t-t0


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)()()(0

t tht d t ht r

t

=′′∫

The delayed slope unit function can be written as:

( ) ( ) ( )000 t t ht t t t r −−=−

RECTANGULAR UNIT IMPULSE (of unit area)

⎪⎪

⎩

⎪⎪

⎨

⎧

∞


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Since ( )10 ,lim)( 1 t t pt t →=δ

integrating, ( ) ( ) 1,lim 101

=′=′∂′ ∫∫ →∞

∞−

t d t t pt t t

δ

Multiplying both terms of the equation

( ) ( )t hdt

t dht

1)()( ==δ with a function )(t f continuous in its origin, bordered and integrable

and integrating between ∞∞− si we get:

( ) ( ) ( ) ( ) ( ) ( ) ( )000

0

0

0

f t d t f t d t t f t d t t f =′′=′′′=′′′ ∫∫∫+

−−

∞

∞−

δ δ δ

This property is called filtering.

EQUIVALENT THEOREMS OF REACTIVE ELEMENTS

THEOREM 1 The linear coil invariable with the time of inductance L and initial current

)0( Li is equivalent either with current generator ( )t hit i Lg )0()( = connected in parallel with

coil L, with the initial current nil, or with electromotive generator )()0()( t Lit e L δ =

connected in series with coil L having the initial current nil.

δ

δ

δ t

f t

δ t-tδ t-t0

t

tt

t


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Using the equation determined for the linear coil invariable with the time

( ) ( ) ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ′′=′′+= ∫∫

∞−

t d t u L

it d t u L

it i

t 0

0

)(1

0)(1

0)(

According to Kirchhoff’s first law by using function )(t h

t d t u L

t hit i

t

L ′′+⋅= ∫

0

)(1

)()0()(

to the first term in the second member corresponds the current generator

)()0()( t hit i Lg = , and to the second term the inductance coil L and initial nil current.

If both terms of previous relations are multiplied by L and are derived in respect with the

time we get:

( ) )()0( t ut i Ldt di L L +⋅= δ for scheme 2

THEOREM 2 A circuit made up of a linear coil of inductance L and initial nil current,

connected in parallel with a current injector )(t ig is equivalent to a circuit having an

electromotive generatordt

di Lt e

g=)( in series with coil L


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If we write Kirchhoff’s first law in a node, t d t u L

t it i

t

g ′′+= ∫0

)(1

)()( .

By deriving in respect with the time and multiplying by L, we get:

)()()( t udt

t di Ldt

t di L g +=

şi ( ) ( )dt

di Lge

dt

t di L

dt

t di Lt u

g +−=+−=)()(

THEOREM 3 The linear condenser, invariable with the time of capacitance C and initial

voltage )0(cu is equivalent either to the electromotive generator ( )t hut e c )0()( = in series

with C having the initial voltage nil, or to the current generator

( )t Cut i

cg δ )0()( = in parallel

with the condenser of initial voltage nil.

Considering the relation:

t d t iC ucudt t iC ut u

t

′′=′+= ∫∫ ∞−

0

0)(

1

)0()(

1

)0()(

and using function )(t h it becomes:

( ) t d t iC

t hut u

t

′′+⋅= ∫0

0 )(1

)(

Multiplying by C and deriving we get:

( )t it uC dt

duC c += )()0( δ


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According to Kirchhoff’s first law this corresponds to figure 2.

THEOREM 4 A circuit made up of a linear condenser invariable with the time of capacitance C

and initial nil voltage, connected in series to an electromotive generator is equivalent to the

circuit having in parallel to condenser C a current generatordt

deC ig = .

From Kirchhoff ' 2nd equation written as ( )t uet d t iC

t

=−′′∫0

)(1

derived and multiplied by C we

get:

dt duC

dt deC t i +=)( , namely scheme (b)

SERIES AND PARALLEL CONNECTIONS OF TWO POLE ELEMENTS

Let us consider m two-pole elements passed by currents k i , with hub voltage k u .

The series connection is obtained by connecting hubs ,32,21 cucu ′′ etc.

T1 Kirchoff at node 1k it results that: ek ii =

and voltage at hubs (1), ( )m′ has the expression:

∑=

=m

k

k e uu1

The parallel connection is obtained by connecting together hubs 1,2,...m

m′′′ ...2,1 ,respectively. Applying T2 Kirchhoff to the loop made up of elements 1+k sik , it

results that:

ck k uuu == +1 mk ,...2,1= .


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Applying T1 Kirchhoff to one of the nodes, we get the total strength of the current ei .

∑=

=m

k

k e ii1

CIRCUITS WITH RESISTORS

Series

( )

∑

∑

=

=

=

=

m

k

K e

m

k

eK e

R R

iuu

1

1

Parallel the equivalent resistor has the characteristic equation ( )∑=

=m

k

ek e uii1

and

∑=

=m

k K

e

R

R

1

1

1

Mixed it contains dipole elements connected in series and parallel .

CIRCUITS WITH COILSSERIES We consider m linear coils characterised by inductivity LK or reciprocal

inductivity 1−=Γ K K L and currents’ strengths at the initial moment )0(k i , connected in series.

)()( t it i ek =

Voltage at the coil’s hubs k has the expression:

dt

di Lt u eK k =)(

Voltage at the circuit’s hubs is computed in the following manner:

( )dt

di

dt

di Lt ut u e

m

k K

em

k

K

m

k

k e ⎟⎟ ⎠

⎞⎜⎜⎝

⎛

Γ=⎟

⎠

⎞⎜⎝

⎛ == ∑∑∑

=== 111

1)(

therefore

∑

∑

Γ

=Γ

=

K

e

K e L L

1

1


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PARALLEL Current ( ) ( ) ( ) t d t u L

it i

t

e

K

K k ′′+= ∫

0|

10

Applying T1 Kirchhoff we derive that:

t d t u L

it it i

t

e

m

k K

m

k

k

m

k

k e ′′⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛ +== ∫∑∑∑

=== 0111

)(1)0()()(

Therefore,

∑=

=m

k K

e

L

L

1

1

1

CIRCUITS WITH CONDENSERS

We shall consider m linear condensers having capacity K C and hub voltages at the initial

moments )0(K u .When connecting in parallel, conditions are met by equalling the currents )()( t it i eK = .

The characteristic equation of condenser k is written as:

( ) t d t iC

ut u

t

e

K

k K ′′+= ∫

0

1)0()(

Hence, voltageeu has the expression:

t d t iC ut ut u

t

e

m

k

m

k K K

m

k K e

′′⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

+== ∫∑ ∑∑ = == 01 11 )(1

)0()()(

Thus:

∑=

=n

k K

e

C

C

1

1

1 ∑

=

=m

k

K e uu1

)0()0(

PARALLEL The equality of voltages is implied

)()( t ut u ek =

The strength of currentk i through the capacity condenser K C is given by the relation

below:

dt

duC t i eK k =)(

The total currentei is obtained by:

∑∑ == ==

m

k

e

K

m

k k e dt

du

C t ii 11 )(


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and ∑=

=m

k

K e C C 1

CIRCUITS WITH INDEPENDENT GENERATORS

SERIES The electromotive voltage of m generators series connected computed with the

formula below are:

∑=

=m

k

K e t et e1

)()(

Due to the restriction in the series connection, the generators which are independent from

current can be series connected only if they have the same current.

mk t it i gegk ...2,1)()( ==

The series circuit with m generators of continuous voltage k E and internal resistors gk R

is equivalent to the voltage generator e E and internal resistor ge R .

∑∑==

==m

k

gk ge

m

k

k e R R E E 11

;

The circuit with m current generators gk I and internal resistors gk R series connected is

equivalent to m voltage generators gk gk K I R E = and resistors gk R series connected.

The electromotive voltagese E and internal resistor ge R of the equivalent voltage

generator has the expressions:

∑∑==

==m

k

gk gegk

m

k

gk e R R I R E 11

;

Going back to the scheme of the current generator we get:

∑

∑

=

==m

k

gk

m

k

gk gk

ge

R

I R

I

1

1

PARALLEL The current generator is an active element, of resistive type with voltage

control; then, it results that the current strength )(t ige of the equivalent generator with m current

generators parallel connected is computed with the relation below:


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)()(1

t it im

k

gk ge ∑=

=

Due to the restriction of parallel connexion, the independent voltage generators can be

connected in parallel only if they have the same electromotive voltage

mk t et e ek ...2,1)()( ==

The parallel circuit with m generators of direct current gk I and internal conductance gk G

is equivalent to the current generatorge I and internal conductance geG :

∑∑==

==m

k

gk ge

m

k

gk ge I I GG11

;

The circuit with m voltage generators k E and internal resistors gk R parallel connected is

equivalent to m current generatorsgk

K

gk R

E I = connected in parallel.

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ==

∑∑

=

= m

k gk

ge

m

k gK

K

ge

R

R R

E I

1

1 1

1

Therefore, the current generator will have:

∑

∑=

gk

gk

K

e

R

R

E

E 1

.

En CURS 1 Two Pole Circuit Elements

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