Download Ch01 - lmn.pub.ro

8. DYNAMIC CRCUITS
133
_____________________________________________________________________________________________________________________________________________________________
_________
8. DYNAMIC CIRCUITS
8.1. Circuit elements and fundamental relations
A dinamic circuit is a set of interconnected ideal circuit elements operating under
quasi–steady state conditions. The ideal circuit elements have been introduced already in
previuos chapters and are only briefly presented here.
1. An active circuit element is an ideal circuit element where electromagnetic
energy is generated – more precisely transformed from some energy of another type. The
active circuit elements are the ideal generators.
The ideal voltage generator is the active dipole element where the
electromagnetic energy is generated under imposed voltage conditions. It is characterised
by the (time–dependent) electromotive force (e.m.f.) e(t) and its direction (fig. 8.1). The
electromotive force e is determined by the internal energy conversion processes and
depends on different physical parameters and, generaly, on time – this dependence on
time is considered here. The state/ operating equation of the ideal voltage generator is
u  et 
,
for any i
,
where the reference directions of current and voltage are associated according with the
source rule. The energy conversion processes are simply characterised by the fact that the
power delivered at the terminals is just the generated power,
pt  u i  e i  pG
.
Fig. 8.1.
Fig. 8.2.
The ideal current generator is the active dipole element where the
electromagnetic energy is generated under imposed current conditions and delivered at
terminals. It is characterised by the (time–dependent) generated current a(t) and its
direction (fig. 8.2). The generated curent a is determined by the internal energy
conversion processes and depends on different physical parameters and, generaly, on time
– this dependence on time is considered here. The state/ operating equation of the ideal
current generator is
i  at 
,
for any u
,
where the reference directions of current and voltage are associated according with the
source rule. The energy conversion processes are simply characterised by the fact that the
power delivered at the terminals is just the generated power,
8. DYNAMIC CRCUITS
134
_____________________________________________________________________________________________________________________________________________________________
_________
pt  u i  u a  pG
.
2. A passive circuit element is an ideal circuit element where the electromagnetic
energy received at terminals is dissipated or stored – the element is called dissipative in
the former case and reactive in the latter case.
The ideal resistor is the dissipative dipole element where the electromagnetic
energy received at terminals is irreversibly transformed into internal energy (heat) of the
internal conducting substance. It is characterised by the resistance R or its reciprocal, the
conductance G . Correspondingly, the state/operating equation is,
u  Ri
 i  Gu ,
where G  1 R and the reference directions of current and voltage are associated
according with the load rule (fig. 8.3). The energy conversion processes are simply
characterised by the fact that the power received at the terminals is dissipated,
pt  u i  R i 2  G u 2  p J
.
Fig. 8.3.
Fig. 8.4.
The ideal capacitor is the reactive dipole element where the electromagnetic
energy received at terminals is stored as electric energy, associated with the opposite
electric charges accumulated within the element. The ideal capacitor is characterised by
the capacitance C and its state equation is
q  Cu
,
where the reference direction of the voltage goes from the positive to the negative charge
(fig. 8.4). Supposing that the capacitance is time–invariant, the operating equation is
dq
du
i
C
,
dt
dt
where the reference directions of current and voltage are associated according with the
load rule. The energy conversion processes are simply characterised by the fact that the
power received at the terminals,
d q u d Wel .
pt  u i 

,
dt 2
dt
is the rate of increase of the accumulated electric energy,
Wel . 
q u C u 2 q2


2
2
2C
.
The ideal coil is the reactive dipole element where the electromagnetic energy
received at terminals is stored as magnetic energy, associated with the magnetic flux
8. DYNAMIC CRCUITS
135
_____________________________________________________________________________________________________________________________________________________________
_________
developed within the element. The ideal coil is characterised by the inductance L and its
state equation is
  Li ,
where the reference directions of the current and magnetic flux are associated according
with the right-hand rule (fig. 8.5). Supposing that the inductance is time–invariant, the
operating equation is
d
di
u
L
,
dt
dt
where the reference directions of current and voltage are associated according with the
load rule. The energy conversion processes are simply characterised by the fact that the
power received at the terminals,
d  i d Wmg .
pt  u i 

,
dt 2
dt
is the rate of increase of the accumulated magnetic energy,
Wmg .
i
Li2  2



2
2
2L
.
Fig. 8.5.
Fig. 8.6.
Additionaly, the system of (ideal) coupled coils is the reactive multipole element
where the electromagnetic energy received at terminals is stored as magnetic energy,
associated with the magnetic fluxes developed within the element. The system of (ideal)
coupled coils is characterised by the set of self and mutual inductances Ljj , Ljk = Lkj , j,k
= 1,…,n and its state equations are
k 
n
 Lkj i j
,
k  1,..., n
,
j 1
where the reference directions of the currents and magnetic fluxes are determined, in
relation with the specified marked terminals of the coils, according with the right-hand
rule (fig. 8.6). Assuming that the inductances are time–independent, the operating
equations are
n
d k
di j
uk 
  Lkj
, k  1,..., n ,
dt
dt
j 1
8. DYNAMIC CRCUITS
136
_____________________________________________________________________________________________________________________________________________________________
_________
where the reference directions of current and voltage are associated according with the
load rule for each coil. The energy conversion processes are simply characterised by the
fact that the power received at the terminals,
n
p t   u k ik 
k 1
d n  k ik dWmg .


d t k 1 2
dt
,
is the rate of increase of the accumulated magnetic energy,
Wmg . 
n

 k ik
k 1
2

n
n

k 1 j  1
Lkj ik i j
2
.
3. The state of a dynamic (linear) circuit is characterised by the set of appropriate
state quantities associated with each couple of terminals of constituting elements, as
follows.
For a given electromotive force, the state of a voltage generator is characterised by
the current i , while for a given generated current, the state of a current generator is
characterised by the voltage u .
The state of a resistor can be characterised either by the curent or the voltage –
usualy the current i is taken as the state variable of this circuit element. The state of a
capcitor is characterised by the electric charge q ; usualy, however, the voltage u is
equivalently taken as the state variable. The state of a coil is characterised by its magnetic
flux  ; usualy, however, the current i is equivalently taken as the state variable.
Similarly, the state of a system of coupled coils is characterised by the set of magnetic
fluxes  k ; usualy, however, the set of currents ik is equivalently taken as the set of state
variables.
A very important remark is that, according with the manner of defining the ideal
circuit elements, electric charges are present in capacitors only, while magnetic fluxes
are present in coils only – there is no electric charge outside capacitors and there is no
magnetic flux outside coils. This remark is fundamental in deriving the equations that
describe the behaviour of dynamic circuits.
4. The evolution of the state variables of a dynamic circuit is described by
Kirchhoff’s theorems (equations) for dynamic circuits.
Kirchhoff’s current theorem is formulated for a node; let then (a) be a node – that
is, a ramification point in the circuit – as in fig. 8.7. Let the node be included into a closed
surface  , which can always be so chosen that it does not separate two plates of a
capacitor, and let the law of electric charge conservation be invoked for this surface,
d
i   q D  .
dt
On the left-hand side, the total electric conduction current getting out across the surface 
is obviously the algebraic sum of currents carried by tha brances concurring in the node,
i   ik   ik ;
k 
k  a 
on the right-hand side, the electric charge inside the domain bounded by the closed
surface  is identicaly zero,
8. DYNAMIC CRCUITS
137
_____________________________________________________________________________________________________________________________________________________________
_________
q D  0
since the surface includes no unbalanced electric charge on a single capacitor plate. It
follows that

ik
k  a 
0
at any t
:
the algebraic sum of currents concurring in a node equals zero at any moment.
Fig. 8.7.
Fig. 8.8.
Kirchhoff’s voltage theorem is formulated for a loop; let then (p) be a loop – that
is, a sequence of branches assembling into a contour in the circuit – as in fig. 8.8. Let 
be the closed line traced along the voltage lines at the terminals of the branches in the
loop, and let Faraday’s law (on electromagnetic induction) be invoked for this contour,
d
e    S  .
dt
On the left-hand side, the induced e.m.f. is just the total electric voltage along the contour
 , which is obviously the algebraic sum of voltages at the terminals of the branches in the
loop,
e  u    u t k   u t k ;
k 
k  p 
on the right-hand side, the magnetic flux over the surface bounded by the closed line  is
identicaly zero,
 S  0
since the surface is traced outside the coils, where the magnetic flux is present. It follows
that
 ut k  0
k  p 
at any t
:
the algebraic sum of voltages at the terminals of branches in a loop equals zero at any
moment.
The link between the variables occuring in Kirchhoff’s equations above is offered
by Joubert’s theorem. Let the most general branch (k) be considered (fig. 8.9), where the
same reference direction is considered for the current ik , the voltage between terminals
ut k , the generated current ak , the e.m.f. ek , and the voltages across the circuit element in
the branch. The sequence of the voltages across the circuit elements in the branch and the
8. DYNAMIC CRCUITS
138
_____________________________________________________________________________________________________________________________________________________________
_________
voltage between the terminals of the branch constitute a loop, so that Kirchhoff’s voltage
theorem gives
uR k  uC k  uL k  uk  uE k  ut k  0 ,
where, in particular, u E k  ek . It readily follows Joubert’s theorem,
ut k  uR k  uC k  uL k  uk  ek
.
Fig. 8.9.
In particular, Joubert’s theorem for linear circuits takes the form
di
ut k  Rk ik  uC k  Lk k  uk  ek
,
dt
n
d is
if the coil k is included in
dt
s 1
a system of coupled coils, and the link between the current and the voltage across the
capacitor is given by the operating equation
d uC k
.
ik  C k
dt
 Lks
where the voltage across the coil is to be taken as
An extended form of Kirchhoff’s voltage theorem can be now obtained if the
expression given by Joubert’s theorem is used – it gives
 u t k  0   u R k  uC k  u L k  u k  ek  0 .
k  p 
k  p 
Transferring the e.m.f.’s to the right-hand side, this results in the desired equation,


 u R k  uC k  u L k  uk    ek
k  p 
k  p 
.
In particular, the extended form of Kirchhoff’s theorem for linear circuits is

di
 Rk ik  uC k  Lk k  uk
dt
k  p  


   ek
 k  p 
at any t
n
,
d is
if the coil k is
dt
s 1
included in a system of coupled coils, and the link between the current and the voltage
across the capacitor is given by the operating equation
where again the voltage across the coil is to be taken as
 Lks
8. DYNAMIC CRCUITS
139
_____________________________________________________________________________________________________________________________________________________________
_________
ik  C k
d uC k
.
dt
5. From the energy point of view, a consequence of Kirchhoff’s equations can be
proved, representing the theorem of power conservation in a dynamic circuit. However,
an indirect argument is invoked here, starting from the theorem of electromagnetic
energy,
dW
P 
 PS  Pmech .
dt
On the left-hand side, the electromagnetic power received by the circuit across its
surrounding surface reducess to the electromagnetic power received at the circuit
terminals (at most, at all its N nodes), if this is a unisolated circuit,
N
P  Pt   va ia
,
a 1
where va is the potential of the node (a) and ia is the electric current entering the node
from outside the circuit.
On the right-hand side, the mechanical power associated with the electromagnetic
field is identicaly zero,
Pmech  0
since there is no movement present in the system.
The electromagnetic power transferred to the substance reduces then to the power
transferred to conductors, if the absence of hysteretic phenomena is supposed, as it is
usualy the case with linear circuits. In turn, the electromagnetic power transferred to the
conductors in the circuit is given, according to Joule’s law on the power transfer
associated with electric conduction, by
PS  PC  PJ  PG ,
that is the difference between the dissipated and the generated power in the circuit.
According with the fact that different energy transfer processes are asigned to different
specific circuit elements in the B branches of the (linear) circuit, it follows that
B
B
k 1
k 1
PJ  PG   Rk ik2    ek ik  uk ak 
where the reference directions of the quantities associated with the generators eventualy
present are associated according with the source rule.
Finaly, the electromagnetic energy accumulated in the system is just the sum of
the electric and magnetic energy,
W  Wel.  Wmg.
stored in the capacitors and in the coils of the circuit, respectively. Assuming again linear
circuits, the electromagnetic energy stored in the circuit is
Wel .  Wmg . 
B

Ck uC2 k
2
in the absence of coupled coils and
k 1
Lk ik2
k 1 2
B

8. DYNAMIC CRCUITS
140
_____________________________________________________________________________________________________________________________________________________________
_________
Wel .  Wmg . 
B

Ck uC2 k
B B
Lks ik is
2
k 1s 1
 
2
in the presence of coupled coils.
The above expressions are assembled into
d Wel . d Wmg .
dW
,
P 
 PS  Pmech  Pt  PJ  PG 

dt
dt
dt
which can be rewritten in the general form of the theorem of power conservation in
dynamic circuits,
d Wel . d Wmg .
Pt  PG  PJ 

:
dt
dt
k 1
the sum between the power received at the terminals of a dynamic circuit and the power
generated by its generators equals the sum between the power dissipated in the resistors of
the circuit and the rate of increase (in time) of the electric and magnetic energy stored in
the capacitors and coils of the circuit.
In particular, in the case of linear circuits, the theorem of power conservation in
dynamic circuits takes the extended form
N
B
B
a 1
k 1
k 1
 v k i A k    ek i k  u k a k   
where the last term reduces to
2
Rk ik2
d B Ck uC k d B B Lks ik is
 
 
d t k 1 2
d t k 1s 1 2
,
d B Lk ik2
in the case where there are no coupled coils

d t k 1 2
in the circuit.
8.2. The solution to the equations of dynamic circuits
1. The evolution of (the state of) a dynamic circuit is described by the solution to
the equations of the circuit – Kirchhoff’s equations. It is extremely difficult to obtain such
a solution in the case of nonlinear circuits; therefore, only linear dynamic circuits are
considered in the following.
The standard problem of a linear dynamic circuit is the following: The given data
are: the configuration (topology) of the circuit, the characteristics of its active elements (ek
and ak ), the characteristics of its passive elements (Rk , Ck , and Lk or Lks ), and data
about the initial state of the reactive elements, as it will be analysed further on. The
unknown data are the evolution of state variables characterising the circuit elements: the
currents ik (t) through resistors, coils and voltage generators and the voltages uC k (t) and
uk (t) across capacitors and current generators, respectively, starting from an initial
moment conventionally taken as t = 0 .
2. The equations of linear dynamic circuits – Kirchhoff’s equations – are

ik
k  a 
and
0
,
8. DYNAMIC CRCUITS
141
_____________________________________________________________________________________________________________________________________________________________
_________

di
 Rk ik  uC k  Lk k  uk
dt
k  p  


   ek
 k  p 
,
n
where the voltage across a coil k is to be taken as
 Lks
s 1
d is
if that coil is included in a
dt
system of coupled coils.
Some comments are quite important.
1. Kirchhoff’s voltage equations are obviously linear differential inhomogeneous
equations: they are linear equations, include derivatives of some of the unknown
quantities, and inlcude also some given time–dependent e.m.f.’s in the right-hand side.
2. Even if it is not obvious, Kirchhoff’s current equations are generaly also linear
differential inhomogeneous equations: they are obviously linear equations, and may
d uC k
include differential terms as ik  iC k  Ck
and given time–dependent terms as
dt
ik  ak  t  , corresponding to the presence of such elements – capacitors or current
generators – in the branches concurring in the node under consideration.
3. Finaly, all the coefficients of the unknown quantities (or their derivatives) are
constant.
The problem associated with the evolution of a linear dynamic circuit is thus the
problem of solving a system of linear differential inhomogeneous equations with constant
coefficients. Such equations were extensively studied from the mathematical point of
view, and some important results are given in the following.
(1) The solution of a system of linear differential inhomogeneous equations with
constant coefficients exists and is continuous on the closed interval of continuity of
coefficients and given right-hand time–functions, and is derivable on the open interval of
continuity of coefficients and given right-hand time–functions.
(2) The solution for any unknown time–function fk (t) (that is, unknown voltage
uk or unknown current ik ) is decomposable into a so called free solution and a so called
forced solution,
f k  t   f k0  t   f kF  t 
.
(3) The free solution f k0  t  is the general solution of the homogeneous system
of equations, obtained from the original (complete) equations if all given time–functions
(in the right-hand side) are equated to zero. This corresponds with considering zero values
for all e.m.f’s of voltage generators and generated currents of current generators, i.e. the
absence of all generators in the circuit. This solution depends not only on time but also on
some undetermined integration constants,
f k0  f k0  t, A1 ,..., Am  ,
where m is called the order of the system of differential equations and is generaly equal
to the number of reactive elements in the circuit.
(4) The time–dependence of the free solution is generaly a linear combination of
products of so called functions of exponential type, namely exponential functions e r t ,
8. DYNAMIC CRCUITS
142
_____________________________________________________________________________________________________________________________________________________________
_________
powers of the variable t p , and trigonometric functions cos t or sin  t . Moreover, if
resistors are present in the circuit, then the exponential terms have a negative exponent,
e rt  e   t . The values of parameters  , p ,  depend on the characteristics of the
passive circuit elements present.
(5) The forced solution f kF  t  is a particular solution of the inhomogeneous
(complete) system of equations, and its form depends on the given time–dependence of
the functions characterising the generators – e.m.f.’s e k (t) and generated currents a k (t) .
(6) There is no general rule concerning the time–dependence of the forced
solution. However, when the given functions characterising the generators, that is, e k (t)
and a k (t) , are functions of exponential type, the forced solution is a function of the same
type.
3. The conclusion of the mathematical results briefly presented above is that the
time–functions representing the solution of a linear dynamic circuit has the form
f k  t   f k0  t, A1 ,..., Am   f kF  t 
,
and is a continuous and derivable function on the closed, respectively open, interval of
continuity of coefficients and given right-hand side functions characterising the
generators. It now remains to determine the values of the integration constants A1 , … ,
Am in the complete solution.
Let a node (a) be cosidered, and let Kirchhoff’s current equation for this node,
 ik  0 ,
k  a 
be rewritten by separating the currents carried by branches containing a capacitor from
those that do not include any capacitor,
 iC k   ik  0 .
k  a 
k   a  , no C
The first sum can be successively written as
dq
d
d
 iC k   d tk  d t  qk  d t q a  ,
k  a 
k  a 
k  a 
where q a  is the total charge of node a , that is the algebraic sum of electric charges on
the capacitor plates connected to this node. The second sum can be written as
 ik  i a  non C ,
k  a , no C
that is, as the non–capacitive total current in node a . Using these notations, Kirchhoff’s
current equation in node a becomes
d
i a  non C   q a  .
dt
For a consistent solution to exist, it is necessary that all quantities in the circuit –
including the non–capacitive total current in node a – exist at any moment, which
imposes the continuity of the total charge q a  of any node a . This is obviously true
over the interval of continuity of coefficients and given right-hand side functions of the
system of equations; it remains to impose the node charge continuity at the very first
8. DYNAMIC CRCUITS
143
_____________________________________________________________________________________________________________________________________________________________
_________
moment when the continuity is not explicitly stated, which is the initial moment t = 0 . So
a necessary condition for a consistent solution is
.
q a 
 q a 
t 0
t 0
The right-hand side is just the limit of the node charge when t  0 , t  0 , while the
left-hand side has to be given. Expressing the node charge in terms of the circuit
variables, the necessary continuity condition is
 Ck uC k 0   Ck uC k 0, A1 ,..., A2 
k  a 
where
k  a 
,
uC k 0  uC k 0  
are necessarily given initial data.
Let now a loop (p) be cosidered, and let Kirchhoff’s voltage equation for this
node,
 uk  0 ,
k  p 
for voltages across the elements in the branches of the considered loop, be rewritten by
separating the voltages across coils from those across any non–inductive circuit element,
 u L k   uk  0 .
k  p 
k  p , no L
The first sum can be successively written as
d
d
d
 u L k   d tk  d t   k  d t   p  ,
k  p 
k  p 
k  p 
where   p  is the total flux of loop p , that is the algebraic sum of magnetic fluxes of
the coils in this loop. The second sum can be written as
 uk  u p  non L ,
k  p , no L
that is, as the non–inductive total voltage of loop p . Using these notations, Kirchhoff’s
voltage equation for loop p becomes
d
u  p  non L     p  .
dt
For a consistent solution to exist, it is necessary that all quantities in the circuit –
including the non–inductive total voltage of loop p – exist at any moment, which imposes
the continuity of the total flux   p  of any loop p . This is obviously true over the
interval of continuity of coefficients and given right-hand side functions of the system of
equations; it remains to impose the loop flux continuity at the very first moment when the
continuity is not explicitly stated, which is the initial moment t = 0 . So a necessary
condition for a consistent solution is
 p
 p
.
t 0
t 0
The right-hand side is just the limit of the loop flux when t  0 , t  0 , while the lefthand side has to be given. Expressing the loop flux in terms of the circuit variables, the
8. DYNAMIC CRCUITS
144
_____________________________________________________________________________________________________________________________________________________________
_________
necessary continuity condition is
B
B
  Lks iL s 0    Lks iL s 0, A1 ,..., A2 
k   p  s 1
,
k   p  s 1
where coupled coils were generaly supposed as present and
iL s 0  iL s 0  
are necessarily given initial data.
It may be concluded that the necessary data about the initial state of the reactive
elements are the initial values (at t = 0–) of voltages across any capacitor and of
currents carried by any coil,
uC k 0  uC k 0   , iL s 0  iL s 0   ,
and the equations needed to determine the initialy undetermined integration constants are
 Ck uC k 0, A1 ,..., A2    Ck uC k 0 ,
k  a 
k  a 

B
 Lks iL s 0, A1 ,..., A2  
k   p  s 1
B
  Lks iL s 0
k   p  s 1
.
4. A simple example is illustrative for the algorithm of solving the equations of a
dynamic circuit.
Let a series connected RC circuit be considered, its terminals being switched at
the moment t = 0 from the terminals of a constant voltage source of e.m.f. E1 to the
terminals of a constant voltage source of e.m.f. E2 (fig. 8.10), starting from a steady state.
Fig. 8.10.
Fig. 8.11.
Before the initial moment t = 0 , the circuit has the configuration of fig. 8.11, and
d
0 .
is operating under steady state conditions, when all quantities are constant,
dt
Kirchhoff’s current equation is irrelevant, since there is no ramification in the circuit;
Kirchhoff’s voltage equation is
uC  R i  E1 ,
where, in view of the steady state operating conditions,
du
i C C 0 , t0 .
dt
The differential equation that describes the circuit is then
8. DYNAMIC CRCUITS
145
_____________________________________________________________________________________________________________________________________________________________
_________
uC  R  0  E1 , t  0 ,
which gives
uC  E1 , t  0 .
The needed initial data for the circuit under consideration are then
uC 0  uC 0    E1 .
After the initial moment t = 0 , the circuit
has the configuration
of fig. 8.12, and is operating as a dynamic circuit.
Kirchhoff’s current equation is again irrelevant,
and Kirchhoff’s voltage equation is now
uC  R i  E2 ,
where
du
iC C .
dt
Fig. 8.12.
The differential equation that describes the circuit is now
du
uC  R C C  E 2
, t0 ,
dt
with the solution decomposed as
uC  t   uC0  t   uCF  t  .
The free solution uC0  t  , that is the solution of the homogeneous equation
d uC0
 RC
0 ,
dt
is searched in the form
uC0  t   A e r t ,
for which
d uC0
 r Ae rt .
dt
Substituting these expressions in the circuit equation, it follows that
Ae r t  R C  r Ae r t  0  Ae r t 1  R C  r   0 ,
uC0
so that, since uC0  t   A e r t  0 , it remains that
1
1 RC  r  0  r  
.
RC
The time constant of the circuit is now defined as
  RC ,
so that the free solution is written as
t   A e

t
RC

t

 Ae
,
where A is the yet undetermined integration constant.
uC0
8. DYNAMIC CRCUITS
146
_____________________________________________________________________________________________________________________________________________________________
_________
The forced solution uCF  t  , that is the solution of the complete inhomogeneous
equation
d uCF
 E2 ,
dt
is searched in the form similar to the right-hand side functions (which is a particular case
of an exponential function with zero exponent),
uCF  t   B  const . ,
for which
d uCF
0 .
dt
Substituting these expressions in the circuit equation, it follows that
B  R C  0  E2
 B  E2 ,
so that it remains that
uCF  t   E2 .
The complete solution of the circuit equation is thus
uCF  R C

t
RC
uC  t , A   A e
 E2 ,
where the integration constant A is to be determined from the condition of charge
continuity,
C uC 0, A   C uC 0
 uC 0, A   uC 0 .
Substituting the known value of uC 0 and that of
uC 0, A  
uC t, A   A  E2 ,
lim
t 0,t  0
it results the equation
A  E2  E1 ,
whence
A  E1  E2
and

t
RC
uC  t    E1  E 2  e
 E2
, t0 .
The evoluution of the voltage across the capacitor is obtained by assembling the
solutions to the circuit equations before and after the initial moment (fig. 8.13),
 E1
t
uC  t   


 E 2   E1  E 2 e
,
,
t0
t0
,
  RC
.
It can be easily verified that indeed the voltage is continuous at the moment t = 0 ,
lim uC t   lim uC t   uC 0   E1 ,
t 0,t  0
t 0,t  0
while for t > 0 the voltage across the capacitor tends assymptoticaly toward the limit
8. DYNAMIC CRCUITS
147
_____________________________________________________________________________________________________________________________________________________________
_________
lim uC  t   E2 .
t 
This circuit illustrates the switching of a capacitor from a voltage E1 to a voltage E2 : if
E1 < E2 , this corresponds to charging the capacitor; if E1 > E2 , this corresponds with
discharging the capacitor.
Fig. 8.13.
The current carried by the circuit is readily computed, for t > 0 , as
t 
t
t



 1 
d uC
E 2  E1  R C
d 
RC 
RC
iC
 C  E 2   E1  E 2  e

e
,
  C   RC   E1  E 2  e
dt
dt 
R





so that the evolution of the current in the circuit is (fig. 8.14)
0

it    E2  E2  t
e 
 R
,
t0
,
t0
,
  RC
.
Fig. 8.14.
It is obvious that if E1 < E2 (charging process), then i > 0 for t > 0 and the opposite, i
< 0 for t > 0 , is valid when E1 > E2 (discharging process). The current is discontinuous
at the switching moment t = 0 ,
8. DYNAMIC CRCUITS
148
_____________________________________________________________________________________________________________________________________________________________
_________
E2  E1
 lim i  t  ,
t 0,t  0
t 0,t  0
R
and it tends assymptoticaly to zero when the charging/discharging process is completed.,
lim i  t   0 .
lim
it   0 
t 
The time constant  = RC of the circuit is a measure of the slowness of transient
(time–varying) processes in the circuit. Indeed, it can be shown that the time constant is
the duration, measured on the assymptote, between the initial moment and the intersection
of the assymptote and the tangent to the representative curve at t = 0 (fig. 8.13 or 8.14).
Fig. 8.15.
Fig. 8.16.
This means that if the time constant is very small, then, for instance, the current decays
quite rapidly to zero (as in fig. 8.15, where a four times shorter time constant  is
considered, as compared with the reference one 0 ), while when the time constant is very
large the current decays significantly slower as in fig. 8.16, where a three times longer
time constant  is considered, as compared with the reference one 0 ). Moreover, it can
be seen that after a time interval of three time constants from the initiation of the transient
processes the time–varying quantity is within 5% from its assymptotic value.