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9. ALTERNATING CURRENT CIRCUITS
9.1. Harmonic time-varying quantities
1. A harmonic time–varying quantity is a quantity depending on time as a sine or
cosine function,
f  t   Fmax sin  t   or
f  t   Fmax cos t    .
The two alternatives are equivalent, since


sin  x    cos x ,
2

so that, for instance, the second can be readily turned into the first, as

 



f  t   Fmax cos t     Fmax sin   t       Fmax sin   t       
2
2 




 Fmax sin  t    , where    
.
2
Therefore the sine type of time–variation will be used in the following to describe a
harmonic time–varying quantity.
In the expression of a harmonic time–varying quantity,
f  t   Fmax sin  t  
,
Fig. 9.11.
the quantity Fmax is called the amplitude and the time–varying quantity and  t+  is
named the instantaneous phase. It is then obvious that  is the initial phase, and the
quantity  is called the angular frequency; the latter is linked to the frequency f and
the period T of the harmonic time–varying quantity according with the relations
2
  2 f 
.
T
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The time variation and the main characteristics of a harmonic time–varying function are
illustrated in fig. 9.1.
The most important feature of a harmonic time–varying function is the fact that it
is periodic, meaning that the values of the function repeat themselves after an integer
multiple of the period T ,
f t   f t  nT  , n  Z .
A definition can be formulated now: an alternating current circuit is a linear
dynamic circuit where all quantities have a harmonic time–variation of the same
frequency.
2. A slightly different expression of harmonic time–varying functions is however
used in the study of alternating current circuits – this is called the instantaneous
representative (or representation) of such a function and is formulated in terms of the so
called effective value of the harmonic time–varying function.
The effective value of a function is its root–mean–square (r.m.s.) value over its
definition domain; in the case of a periodic function, this means the r.m.s. value over a
period,
1  T
 f t 2 dt .
F


T
Invoking this definition, the effective value of a harmonic time–varying function is given
by
2
Fmax
1 T 2
1 T 2
2
F   f dt   Fmax sin  t  dt 
T 0
T 0
2T
2
0 1  cos 2 t  dt 
T
2
Fmax
F2
F2
F2
F2
T
T
t 0  max sin  2 t  2  0  max T  max sin  4  2   sin 2   max ,
2T
4 T
2T
4  2
2
where  = 0 was considered for the sake of simplicity. It follows that
F
F  max
, Fmax  F 2
,
2

and the instantaneous representative of the harmonic time–varying function f is
f t   F 2 sin  t  
.
It is now obvious that any harmonic time–varying quantity f(t) of a given
(angular) frequency is entirely characterised by two real parameters: the effective value
F  0, 
and the initial phase
  0,2  or   ,  .
Therefore, any convenable mathematical object characterised by such parameters only can
be a valid representative of a harmonic time–varying function,
F ,  .
f  t   F 2 sin  t   

given 
3.
There are, however, some conditions imposed to such additional
representatives of harmonic time–varying quantities: they have to satisfy in an acceptable
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manner the properties of these functions which are relevant in the operation of dynamic
circuits, properties related to linear and differential operations.
1. The sum of two harmonic time–varying quantities of the same frequency is
also a harmonic time–varying function of the same frequency. Indeed, it is simply of
matter of trigonometric calculus to prove that


f1  t   F1 2 sin  t   1 
 f  t   F 2 sin  t   


f 2  t   F2 2 sin  t   2    F  F12  F22  2 F1 F2 cos  1  2 .


F sin  1  F2 sin  2
f  t   f1  t   f 2  t 

 tan  1
F1 cos 1  F2 cos 2

2. The product by a real constant of a harmonic time–varying quantity of given
frequency is also a harmonic time–varying function of the same frequency. Indeed, it is
again simple to prove that


 f  t   F 2 sin  t   
f 0  t   F0 2 sin  t   0 
.
   F   F0
f  t    f 0  t  ,  R 

, if   0
 0
  

 0   , if   0
3. The derivative of a harmonic time–varying quantity of given frequency is also
a harmonic time–varying function of the same frequency,

 f  t   F 2 sin  t   
f 0  t   F0 2 sin  t   0 


,
d
   F   F0
f t  
f 0 t 



dt

    0  2
of effective value  times greater than and initial phase increased by  /2 with respect
to the original function. Indeed, it is again simple to see that
d
d
f t  
f 0 t  
F0 2 sin  t   0   F0 2 cos  t   0 
dt
dt















 



  F0 2 sin   t   0     F0  2 sin   t   0    ,
2
2 



where, by denoting

F   F0
,   0 
,
2
one obtains the given result,
d
f 0  t   f  t   F 2 sin  t    .
dt
4. A simple and illustrative representative of a harmonic time–varying function
f(t) is the geometric or phasor representative – this is a vector in the (real) plane,
denoted as
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
F  F 
of length equal to the effective value F of the function and rotated counterclockwise at
an angle equal to the initial phase  with respect to the horizontal axis (fig. 9.2). The
one–to–one correspondence between the instantaneous and phasor representatives is
obvious,

F ,   F  F  .
f  t   F 2 sin  t   

given 
Fig. 9.2.
Fig. 9.3.
Immediately related to this phasor representative there can be devised a more
convenient representative of harmonic time–varying functions. Let the imaginary unit be
denoted as
1  j
,
by the letter j instead of letter i , which is retained to designate the electric current
intensity. The complex representative of a harmonic time–varying function is a complex
number, denoted as
F  F e j   F cos  j sin   ,
of modulus equal to the effective value F and argument equal to the initial phase  of
the harmonic time–varying function, so that the real and imaginary parts of this complex
number are
Re F   F cos
, ImF   F sin  .
Thus, the one–to–one correspondence to the other representatives is simply illustrated by
the fact that the complex representative is represented in the complex plane by a point
indicated by a position vector identical to the phasor representative (fig. 9.3),

F ,   F  F   F  F e j .
f t   F 2 sin  t  

given 
That is why even the position vector of the complex number in the complex plane can be

directly designated by F instead of F .
The validity of using these additional representatives can be sustained by checking
if they satisfy the above mentioned properties of harmonic time–varying functions.
1. It is a simple matter to check that the sum of two harmonic time–varying
functions f 1 and f 2 of the same frequency, resulting in a time–varying function f of the
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same frequency, corresponds to the vector sum of their phasor representatives (fig. 9.4)
and to the sum of their complex representatives, respectively,
f  f1  f 2


 

f1  F1  F1 


F

F

F
 F  F1  F 2 .

1
2
f 2  F2  F 2 

f  F  F 
Fig. 9.4.
Fig. 9.5.
2. It is also obvious that the product by a real constant of a harmonic time–
varying function f 0 of a given frequency, resulting in a time–varying function f of the
same frequency, corresponds to the product by the same real constant of the phasor
representative (fig. 9.5) or the complex representative of the original function,
respectively,
f   f0




R


 F   F0 .
  F   F0
f 0  F0  F 0 

f  F  F 
3. Most important, there are simple geometric or complex operations performed
on associated representatives, respectively, corresponding to the derivation of a harmonic
time–varying function. According with the property of harmonic time–varying functions
relative to the their derivative, the latter is a harmonic time–varying function of the same
frequency, of effective value  times greater and initial phase increased with  /2 with
respect to the same parameters of the original function,

 f  t   F 2 sin  t   
f 0  t   F0 2 sin  t   0 


.
d
   F   F0
f t  
f 0 t 



dt

   0 

2
Thus, the phasor representative of the derivative of a harmonic time–varying function is a
phasor rotated counterclockwise – that is, in the positive trigonometric direction – by a
right angle  /2 and of length  times greater than that of the phasor representative of
the original function (fig. 9.6). Correspondingly, the complex representative of the


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derivative of a harmonic time–varying function is the product by the complex constant
j of the complex representative of the original function
d

f t  
f 0 t  
dt




f 0  F0  F 0   F   F0  0 
 F  j  F0 .

2
f  F  F 


The former correspondence is obvious; for the latter
correspondence one can readily compute
F  F e j   F0 e
j

j  0  2

   F e j 0 e j 2 
0
j

  F0 e 0 j  j  Y0 e 0  j  F 0 ,
where, according with Euler’s formula,
e j
 cos

 j sin

j .
2
2
The fact that the complex representation of
harmonic time–varying functions, while retaining
linear operations, reduces the derivation of time–
varying functions to a simple multiplication by j 
2
Fig. 9.6.
of their complex representatives is very important, and constitutes the main argument for
using the complex representation in the study of alternating current circuits.
5. A final important remark is concerned with the units used in the study of the
quantities characterising the alternating current operation of a circuit.
The instantaneous values of a harmonic time–varying quantity, as well as its
effective value, are obviously measured in corresponding I.S. units. For the complex
representatives of such harmonic time–varying quantities, however, this can no longer be
done: there is no ordering relationship possible on the set of complex numbers. On the
other hand, the modulus of such a complex representative – that is, the effective value of
the represented quantity – is still measured in the corresponding I.S. units.
The following rule can then be observed when operating with complex
representatives of harmonic time–varying quantities or with other complex parameters
used to characterise the alternating current operation of linear dynamic circuits: the
modulus (or, in some cases, the real and the imaginary parts) of a complex quantity is
(are) measured in the I.S. units corresponding to that quantity.
9.2. The electric dipole under alternating current conditions
1. The operation of an alternating current circuit can be easily understood if one
starts with the study of its simplest building block – the electric dipole corresponding to a
branch of the circuit. On the other hand, since an alternating current circuit is a set of
interconnected ideal circuit elements, one can distinguish between a purely passive dipole
and a purely active dipole, so that these different types of dipoles will be approached
separately.
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Let a purely passive dipole be considered first, carrying a current
i t   I 2 sin  t   

I  I e j
under a voltage at terminals
ut   U 2 sin  t   

U  U e j
,
associated with the current according to the load rule (fig. 9.7). The main problem related
to the study of such a (passive) dipole is to find the
dipole characteristic(s) that would allow one to
compute the current when the voltage is given, and
vice-versa, to compute the voltage when the current
is given,
?
Fig. 9.7.
u  t  
it  .
2. The solution to this problem is easily found if it is reformulated in terms of the
parameters characterising the harmonic time–varying quantities involved: how to
introduce dipole characteristic(s) that would allow to compute the effective value and
initial phase of the current when the effective value and initial phase of the voltage are
given, and vice-versa,
?
U ,  
I ,   .
The relationship between the effective values of the voltage and current is
obviously a multiplicative relation; one defines the dipole impedance as
U
Z

.
I
The relationship between the initial phase of the voltage and the current is obviously an
additive relation; one defines the dipole phase difference as
   

,
as the difference between the voltage and current (initial) phases, necessarily in this order.
The couple Z ,   is thus necessary and sufficient to describe the operation of a
passive dipole. Indeed, suppose that the voltage is given, as u  t   U 2 sin  t    ,
U
2 sin  t      ; similarly, let the
then the curent i  t   I 2 sin  t    is i  t  
Z
current be given, as i  t   I 2 sin  t    , then the voltage u  t   U 2 sin  t    is
readily obtained as u  t   Z I 2 sin  t      .
It is quite obvious that a similar manner to characterise a purely passive dipole is
given by the couple Y ,   , where
I
S
.
U
is the dipole admittance, and
    
Y
is again the dipole phase difference.
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In the subsequent section, it will be shown that the phase difference of a purely
passive dipole is restricted to the interval
  
    ,  .
 2 2
3. Another approach to the characterisation of a purely passive dipole is based on
the correspondence between the complex and instantaneous representatives of harmonic
time–varying functions: the link between the voltage and current at the terminals of the
dipole is translated in the link between their complex representatives,
?
u  t   U  I  i  t  .
The complex representatives of the voltage and current at the terminals of the
passive dipole can be directly related: one simply computes
U U e j U j    

 e
 Z e j .
j

I
I
Ie
The complex impedance of the purely passive dipole is then defined as
U
Z
;
I
it assembles the needed characteristics of the couple
Z  Z e j
Z ,  , according with
,
and is obviously necessary and sufficient to express I in terms of U or vice-versa, U
in terms of I , that is, to express the relationship between the voltage and the current at
the terminals of a passive dipole.
It is obvious that the reciprocal ratio can be considered as well; the complex
admittance of a purely passive dipole is defined as
I
Y
.
U
It is immediately related to the complex
impedance, as
1
Y
,
Z
and to the couple Y ,   , according
with
1
Y
 Y e  j .
j
Fig. 9.8.
Ze
These complex parameters allow the comprehensive description of the operation
under alternating current conditions of a passive dipole directly in terms of complex
representatives of harmonic time–varying quantities at terminals (fig. 9.8),
U  ZI
,
I  YU
.
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5. Alternative ways of characterising a passive dipole under alternating current
conditions are defined, starting from the complex characteristics introduced above.
The complex impedance can be specified not only in terms of its modulus and
argument, but also in terms of its real and imaginary parts,
Re  Re Z

X e  ImZ
,

,
which are called the (equivalent) dipole resistance and the (equivalent) dipole admittance,
respectively. The relationship between these new and the previously defined parameters is
obtained by a simple comparison,
Z  Re  j X e
it follows that
Re  Z cos 
Z  Z e j  Z cos  j sin    Z cos  j Z sin 

;
X e  Z sin  ,
,
Xe
.
Re
Quite similarly, the complex admittance can be expressed not only in terms of its
modulus and argument, but also in terms of its real and imaginary parts,
Z  Re2  X e2
,
Ge  Re Y 
S
  arctan
,
Be   ImY 
S
,
which are respectively called the (equivalent) dipole conductance and the (equivalent)
dipole susceptance. The relationship between these new and the previously defined
parameters is simply obtained by comparing
Y  Ge  j Be
it follows that
Ge  Y cos 

Y  Y e  j  Y cos     j sin      Y cos  jY sin  ;
Be  Y sin  ,
,
Be
.
Ge
Finaly, the relationship between these last equivalent sets of characteristic
parameters of a passive dipole, Re , X e  and Ge , Be  , is not as simple as it coud
seem. Indeed, from
Re  j X e
R  jXe
1
1
,
Y  Ge  j Be  

 e2
Z Re  j X e  Re  j X e  Re  j X e  Re  X e2
it follows that
R
X
Ge  2 e 2
, Be  2 e 2 ,
Re  X e
Re  X e
and, similarly, from
Ge  j Be
G  j Be
1
1
Z  Re  j X e  

 e2
,
Y Ge  j Be Ge  j Be Ge  j Be  Ge  Be2
Y  Ge2  Be2
,
  arctan
,
Xe 
it follows that
Re 
Ge
Ge2

Be2
Be
Ge2
 Be2
.
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6. The last sets of parameters characterising a passive dipole can be used in the
construction of equivalent circuits of a passive dipole.
Let some of the previous relations be combined as
U  Z I   Re  j X e  I  Re I  j X e I
U  UR  jU X

Fig. 9.9.
.
Fig. 9.10.
This decomposition of the complex voltage U into two components, one
proportional to the complex current I (said to be in phase with it), and the other
proportional to the product by j of the complex current I (said to be in quadrature with
it – meaning rotated by +  /2), is illustrated in fig. 9.9. On the other hand, the
decomposition of the total voltage into a sum of voltages,
u  uR  u X ,
each of them proportional to the current, corresponds to a series connection of two
complex circuit elements, as illustrated in fig. 9.10. It must be noted, however, that while
the resistive component uR is instantaneously proportional to the current i , the reactive
component uX is proportional to the current with the phase increased by  /2 .
Similarly, by combining some other previous relations results in
I  Y U  Ge  j Be  U  Ge U  j Be U
Fig. 9.11.

I  IG  jI B
.
Fig. 9.12.
This decomposition of the complex current I into two components, one
proportional to the complex voltage U (said to be in phase with it), and the other
proportional to the product by –j of the complex voltage U (said to be in quadrature
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with it – meaning rotated by –  /2), is illustrated in fig. 9.11. On the other hand, the
decomposition of the total current into a sum of currents,
i  iG  i B ,
each of them proportional to the voltage, corresponds to a parallel connection of two
complex circuit elements, as illustrated in fig. 9.12. It must be noted, however, that while
the conductive component iG is instantaneously proportional to the voltage u , the
susceptive component iB is proportional to the voltage with the phase decreased by  /2 .
7. Let a purely active dipole be considered now, carrying a current
i t   I 2 sin  t     I  I e j
under a voltage at terminals
ut   U 2 sin  t     U  U e j ,
associated with the current according to the source rule (fig. 9.13). The main problem
related to the study of such a (active) dipole is again
to find the dipole characteristic(s) that would allow
one to compute the current when the voltage is
given, and vice-versa, to compute the voltage when
the current is given,
?
Fig. 9.13.
u  t  
it  .
The first kind of ideal active dipole that needs consideration is the ideal voltage
generator (fig. 9.14), characterised by its harmonic time–varying e.m.f.,
et   E 2 sin  t     E  E e j .
The constitutive/operating equation of this element,
ue ,  i ,  t ,
can be immediately rewritten in terms of harmonic time–varying quantities as
U 2 sin  t     E 2 sin  t    ,  I 2 sin  t    ,  t .
This equation leads to
U  E ,  
,  I ,   ,  t ,
or simply
U ,   E,   ,  I ,   .
Finaly, this relation can be concisely expressed in terms of complex representatives as
UE
,
I
,
which gives the answer to the characterisation problem of this dipole.
Fig. 9.14.
Fig. 9.15.
The other kind of ideal active dipole to be considered is the ideal current generator
(fig. 9.15), characterised by its harmonic time–varying generated current,
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at   A 2 sin  t     A  A e j .
The constitutive/operating equation of this element,
ia ,  u ,  t ,
can be immediately rewritten in terms of harmonic time–varying quantities as
I 2 sin  t     A 2 sin  t    ,  U 2 sin  t    ,  t .
This equation corresponds to
I  A ,  
,  U ,  ,  t ,
or simply
I ,    A,   ,  U ,  .
Finaly, this relation can be concisely expressed in terms of complex representatives as
IA
,
U
,
which gives the answer to the characterisation problem of this dipole.
9.3. Electromagnetic power under alternating current conditions
1. The energy related operation of alternating current circuits is also easier
understood if the energy related quantities are first studied in the simple case of a dipole.
Let a dipole be considered, carrying a current
i t   I 2 sin  t   

I  I e j
under a voltage at terminals
ut   U 2 sin  t   

U  U e j
,
associated with the current according to, say, the
Fig. 9.16.
load rule (fig. 9.16).
The instantaneous electromagnetic power received by the dipole at terminals is
p t   u  t   i  t   U 2 sin  t     I 2 sin  t     2U I sin  t   sin  t    
 U I cos      cos  2 t       U I cos   U I cos  2 t      ,
where      is the phase difference of the dipole.
The instantaneous power is thus a periodic time–function, of period T/2 (i.e., of
frequency 2f), and obviously can not be used to characterise the overall power
performance of the dipole. Instead, taking into account this very periodicity, the average
electromagnetic power received over a period could be a reasonable measure of the
power actually used by the dipole,
1  T
P  pt   
p  t  dt
W
.
T 
This is called the active power received at the terminals by the dipole, and its value is
U I cos  T
UI
1 T
T
P   U I cos   cos  2 t     dt 
t 0 
sin  2 t      0 
0
T
T
2 T
U I cos 
UI
sin  4       sin      U I cos  .

T
T
4
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The active power is the electromagnetic power effectively used by the dipole and
transferred into some other form of energy (thermal, mechanical, etc.).
In the case when the dipole is a passive dipole, two equivalent formulae can be
used for computing the received active power,
P  U I cos   Z I  I  cos   Z cos   I 2  Re I 2 ,
P  U I cos   U  YU  cos   Y cos   U 2  Ge U 2 .
It can be concluded that the active power transferred at the terminals of a dipole is given
by
P  U I cos   Re I 2  Ge U 2
W
,
the last two relations being valid for a passive dipole only.
An important remark is now worth mentioning. The average power computed
above for the dipole under study corresponds to the reference directions of voltage and
current at terminals associated according with the load rule – it is a received power. The
considered dipole is a passive dipole if this received average power is positive, i.e.,
P  0  cos   0 ,
meaning that the phase difference of a passive dipole satisfy the condition


   
.
2
2
The preceding expressions of the active power found for a passive dipole can be
related to the series and parallel equivalent circuits of such a dipole. Moreover, they
suggest that the active power has to be assigned to the equivelent resistance or
conductance only. This suggestion can readily be checked.
Considering the series connected equivalent circuit of a passive dipole (fig. 9.17),
the instantaneous electromagnetic power received by its equivalent resistance is
p R t   u R i  Re i  i  Re  2 I 2 sin 2  t     Re I 2 1  cos 2 t  2  .
The corresponding average value over a period is computed as above and gives
1 T
PR  p R  t    Re I 2 1  cos 2 t  2  dt  Re I 2  P .
T 0
This result confirms that the active power received by a passive dipole is indeed asigned
exclusively to its equivalent resistance.
Fig. 9.17.
Fig. 9.18.
A similar argument is valid with reference to the parallel equvalent circuit of the
passive dipole (fig. 9.18): the instantaneous electromagnetic power received by its
equivalent conductance is
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pG t   u iG  u  Ge u  Ge  2U 2 sin 2  t     GeU 2 1  cos 2 t  2  .
The corresponding average value over a period is computed as above and gives
1 T
PG  pG  t    GeU 2 1  cos 2 t  2 dt  GeU 2  P .
T 0
This result also confirms that the active power received by a passive dipole is indeed
asigned exclusively to its equivalent conductance.
2. A natural question arises: if, under alternating current conditions, all the active
power – that is all the average effective power consumption of a passive dipole – is
asigned to its equivalent resistance or conductance, then what could be the part played by
the equivalent reactance or susceptance in the power transfer associated with a dipole ?
The answer can be found if the detailed analysis of the power transfer associated with the
equivalent reactance or susceptance is done.
If the series connected equivalent circuit of a passive dipole is considered (fig.
9.17), and the phase difference between the current and the reactive voltage is accounted
for, then the instantaneous electromagnetic power transferred to its equivalent reactance is


p X  t   u X i  X e I 2 sin   t      I 2 sin   t    
2

 X e  2 I 2 cos t   sin  t     X e I 2 sin  2 t  2   0 .
This result seems to contradict the fact that all the effective power consumption goes to
the equivalent resistance. However, this statement is valid with reference to average
values only; indeed, the corresponding average value over a period is
XeI 2
1 T
0
2
PX  p X  t    X e I sin  2 t  2   dt 
cos 2 t  2   T 
0
T
2 T
XeI 2
cos 4  2    cos 2    0 ,

4
and the conclusion of the preceding subsection is thus confirmed.
A similar analysis can be done starting from the parallel connected equivalent
circuit of a passive dipole (fig. 9.18): if the phase difference between the voltage and the
susceptive current is accounted for, the instantaneous electromagnetic power transferred
to its equivalent susceptance is


p B  t   u i B  U 2 sin   t     Be U 2 sin   t     
2

2
2
 Be  2U sin  t   cos t     BeU sin  2 t  2   0 .
This result seems to contradict the fact that all the effective power consumption goes to
the equivalent susceptance. However, this statement is valid with reference to average
values only; indeed, the corresponding average value over a period is
B U2
1 T
0
PB  p B  t    BeU 2 sin  2 t  2  dt  e
cos 2 t  2  T 
T 0
2 T
BeU 2
cos 4  2   cos 2   0 ,
4
and the conclusion of the preceding subsection is again confirmed.

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The preceding analysis shows that the presence of the equivalent reactance/
succeptance is associated with a non-zero power circulation between the reactive part of
the dipole and the rest of the circuit: during a part of a period some power is received by
the reactive part of the dipole and during the rest of the period the same power is
delivered back to the rest of the circuit, so that the average value of the corresponding
power transfer over a period is zero.
Such a behaviour can not be characterised by its average value. Instead, the
amplitude (the multiplicative factor) of the harmonic time–varying factor of this power
circulation can be used as a measure of this power transfer process – by definition, this is
the reactive power received by the passive dipole under consideration,
Q  multiplica tive factor of the harmonic time  varying, zero average value, power
.
circulatio n associated with the reactive/s usceptive part of the dipole
var
For this definition to be consistent, the equivalence of the two approaches
considered above has to be checked. Indeed,
Q  X e I 2  Z sin   I 2  Z I  I  sin   U I sin  ,
Q  Be U 2  Y sin   U 2  U  YU  sin   U I sin  ,
and an additional formula for the reactive power, dependent on quantities at terminals
only is thus obtained, which can be interpreted as the reactive power received at the
terminals of the dipole.
It can be concluded that the reactive power transferred at the terminals of a dipole
is given by
Q  U I sin   X e I 2  Be U 2
var
,
the last two relations being valid for a passive dipole only.
3. The two aspects of the power transfer at the terminals of a dipole – the actual
average power consumption and the power circulation of zero average – can be assembled
into a single complex quantity, the complex power, defined as
S  P  jQ
.
This relation can be detailed, accorcing with the formulae obtained before for its
real and imaginary parts:
S  U I cos  jU I sin   U I cos  jsin    U I e j  U I e j     
 U e j I e  j  U  I ,
S  Re I 2  j X e I 2   Re  j X e  I 2  Z  I 2 ,
S  Ge U 2  j Be U 2  Ge  j Be U 2  Y   U 2 .
Finaly, the complex power is given by three equivalent formulae,
S  U  I   Z  I 2  Y  U 2
,
the last two, however, being valid for passive dipoles only. It must be noted that the first
expression of the complex power involves the complex voltage and (conjugate) current,
while the last two formulae involve the effective voltage or current. There is no unit of
the complex power; however, its numerical value has to be such that its real and
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_________
imaginary parts,
P  Re S W
Q  ImS
,
var
,
are expressed in their corresponding I.S. units.
4. Another power related quantity can be defined: the apparent power is the
modulus of the complex power,
S  S  P2  Q2
VA
.
Correspondingly, the formulae for the calculus of the apparent power transfer associated
with a dipole is
S  U I  Z I 2  YU 2
VA
.
A comparison of the first formula above and the corresponding formula of the
active power,
S U I
and
P  U I cos  ,
shows that, since cos   1 , the maximum value of the active power is just the apparent
power,
P  S cos   S .
A rigorous analysis shows that, indeed, the apparent power is the maximum value that the
active power could reach, under constant effective values of the dipole voltage and
current,
S
max
U  const. , I  const.
P
.
Finaly, an associated power performance parameter can be defined – the power
factor is the ratio of the active power to its maximum admissible value, the apparent
power,
P
k
;
S
in the case of a dipole operating under alternating current conditions, its value is just
k  cos 
.
This parameter characterises the efficiency of using the most of the theoreticaly
available active power: the greater the power factor k , the greater part of the apparent
power is effectively used. That is why the problem of increasng the power factor
k  cos  is extremely important in the case when large amounts of power are involved.
9.4. Ideal circuit elements under alternating current conditions
It is now useful to summarise the characterisation of the operation of an
alternating current circuit, with application to the building blocks of such circuits – the
ideal circuit elements. These circuit elements are here analysed especially in what
concerns the characterisation of their operation in terms of the complex representation of
the quantities at terminals and the power transfer at their terminals.
1. Let a purely passive dipole be considered first, carrying a current
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_________
i t   I 2 sin  t     I  I e j
under a voltage at terminals
ut   U 2 sin  t     U  U e j ,
associated with the current according with the load rule (fig. 9.19). The operation of such
a dipole is characterised, in terms of complex representatives of the voltage and current,
by the equations
Fig. 9.19.
U  Z I

I  Y U ,
and its power transfer performance is concisely characterised in terms of the received
complex power,
S  U  I .
The complex impedance is obtained as
Z  Z e j ,
where
U
Z
,     ;
I
accordingly,
Re  Re Z  ,
X e  Im Z  ,
1
Y
and Ge  Re Y  , Be   Im Y  .
Z
As well,
P  Re S  , Q  ImS  .
In the case of an ideal resistor (fig. 9.20), characterised by the operation equation
Fig. 9.20.
u  Ri ,
simple substitutions result in
U 2 sin  t     R I 2 sin  t    ,
whence
U  RI
,  .
It follows that
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U
R
I
and, successively,
Z
Z  Re
j0
     0 ,
,
R
,
1 1
,

Z R
X e  Im Z   0 ,
Y
Re  Re Z   R ,
1
Ge  Re Y  
, Be   Im Y   0 .
R
In terms of complex representatives of the voltage and current, the ideal resistor is then
concisely characterised by the complex equation
U  RI
,
where
ZR
is the complex impedance of the ideal resistor.
The power transfer performance is characterised by a received complex power
S  U I  R I I  R I 2 ,
to which it corresponds
P  Re S  R I 2
,
Q  ImS  0
.
Moreover, according with the considerations in section 9.3 above,
P  R I 2  R i 2  pR :
the active power received by an ideal resistor is the average over a period of the power
dissipated in it.
In the case of an
equation
du
iC
,
dt
ideal capacitor (fig. 9.21), characterised by the operation
Fig. 9.21.
simple substitutions and the rule of derivation of harmonic time–varying functions result
in


I 2 sin  t     C U 2 sin   t     ,
2

whence

I  CU
,   
.
2
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It follows that
U
1
Z 
I C
and, successively,
1  j
Z
e
C
     
,

2
,
1
j
1

, Y   j C ,
j C  C
Z
C e
1
Re  Re Z   0 ,
X e  Im Z   
,
C
Ge  Re Y   0 , Be   Im Y     C .
In terms of complex representatives of the voltage and current, the ideal capacitor is then
concisely characterised by the complex equation
1
U
I ,
j C
2
1

j 2

where
Z
1
j C
is the complex impedance of the ideal capacitor.
The power transfer performance of the ideal capacitor is characterised by a
received complex power
S  U I  U  j C U    j C U U   j C U 2 ,
to which it corresponds
P  Re S  0
,
Q  ImS   CU 2
.
Moreover, according with the considerations in section 9.3 above, and the expession of
the electric energy,
C u2
Q    C U   2
  2 wel. :
2
the reactive power received by an ideal capacitor is negative and is the average over a
period of the electric energy stored in it.
2
In the case of an ideal coil (fig. 9.22), characterised by the operation equation
di
uL
,
dt
Fig. 9.22.
simple substitutions and the rule of derivation of harmonic time–varying functions result
in
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_________


U 2 sin  t     L  I 2 sin   t     ,
2

whence
U LI

,
It follows that
U
Z  L
I
and, successively,

2
.
     
,

2
,
j
1
1


,
Z j L  L
Re  Re Z   0 ,
X e  Im Z    L ,
1
Ge  Re Y   0 , Be   Im Y  
.
L
In terms of complex representatives of the voltage and current, the ideal coil is then
concisely characterised by the complex equation
Z  Le
j 2
U  j L I
 j L
,
Y
,
where
Z  j L
is the complex impedance of the ideal coil.
The power transfer performance of an ideal coil is characterised by a received
complex power
S  U I  j L I I   j L I 2 ,
to which it corresponds
P  Re S  0
,
Q  ImS    L I 2
.
Moreover, according with the considerations in section 9.3 above, and the expession of
the magnetic energy,
Li2
  2 wmg. :
2
the reactive power received by an ideal coil is positive and is the average over a period of
the magnetic energy stored in it.
Q    L I 2   2
An immediate extension of the preceding case is that of a system of ideal coupled
coils (fig. 9.23), characterised by the system of equations
n
di
u k   Lks s
, k  1, 2 , ... , n .
dt
s 1
By analogy with similar terms associated to the single coil approached above, it folows
that such a system of coupled coils is characterised, in terms of complex representatives
of currents and voltages, by the system of complex equations
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_________
n
U k   j Lks I s
k  1, 2 , ... , n
,
,
s 1
where
Zkk  j Lkk  j Lk
,
k  1, 2 ,... , n
are the complex (self–) impedances of each coil k and
Zks  j Lkk  j Lsk  Z sk
k , s  1, 2 ,... , n
,
are the mutual (coupling) complex impedances between coils k and s .
Fig. 9.23.
As well, the complex power received at terminals by the system of coupled coils
can be shown to be
n
n

n

S   U k I k    j Lks I k I s cos  k   s 
k 1
k 1s 1
n
n

n
   j Lk I k2   j Lks I k I s cos  k   s
k 1

,
k 1 s 1
sk
to which it corresponds
P  Re S   0
n
,

n
Q  Im S      Lks I k I s cos  k   s

,
k 1s 1
where the terms are to be assigned the plus sign when the currents flow in the same
direction through the associated marked teminal and the minus sign when the currents
flow in the opposite direction through the associated marked teminal. Moreover,
according with the considerations in section 9.3 above, and the expession of the magnetic
energy,
n

n

n
n
Lks ik i s
  2 wmg .
2
k 1s 1
Q  Im S      Lks I k I s cos  k   s   2 
k 1s 1
:
the reactive power received by the system of ideal coupled coils is positive and is the
average over a period of the magnetic energy stored in it.
2. Let a purely active dipole be considered now, carrying a current
i t   I 2 sin  t   
under a voltage at terminals

I  I e j
9. ALTERNATING CURRENT CRCUITS
170
_____________________________________________________________________________________________________________________________________________________________
_________
ut   U 2 sin  t   

U  U e j ,
Fig. 9.24.
associated with the current according with the source rule (fig. 9.24). The two types of
such active dipolar elements are recalled bellow.
In the case of an ideal voltage generator (fig. 9.25), characterised by its harmonic
time–varying e.m.f.,
et   E 2 sin  t     E  E e j ,
the operating equation,
ue ,
can be immediately rewritten in terms of harmonic time–varying quantities as
U 2 sin  t     E 2 sin  t    .
This equation corresponds to
U  E ,   ,
whence the characterisation of the ideal voltage generator in terms of complex quantities,
UE
,
I
.
Fig. 9.25.
According with the supposed reference directions of the current and voltage at
terminals and the operating equation in terms of complex quantities, the power transfer
performance of the ideal voltage generator is concisely characterised in terms of the
complex power delivered at terminals,
S  U  I   E  I   E I cos      j E I sin      PG  jQG
.
In the case of an ideal current generator (fig. 9.26), characterised by its harmonic
time–varying generated current,
at   A 2 sin  t     A  A e j .
The operating equation of this element,
ia ,
can be immediately rewritten in terms of harmonic time–varying quantities as
I 2 sin  t     A 2 sin  t    .
This equation corresponds to
I  A ,   ,
9. ALTERNATING CURRENT CRCUITS
171
_____________________________________________________________________________________________________________________________________________________________
_________
whence the characterisation of the ideal current generator in terms of complex quantities,
IA
,
U
.
Fig. 9.26.
According with the supposed reference directions of the current and voltage at
terminals and the operating equation in terms of complex quantities, the power transfer
performance of the ideal current generator is concisely characterised in terms of the
complex power delivered at terminals,
S  U  I   U  A  U A cos      j U Asin      PG  jQG
.
9.5. Complex calculus of alternating current circuits
1. The preceding section illustrates quite clearly that the alternating current
operation of ideal circuit elements is treated simply in terms of complex representatives of
harmonic time–varying quantities. This remark is valid as well for an alternating current
circuit of any complexity, and this can be proved by re–stating the fundamental relations
regarding dynamic circuits in terms of complex representatives of harmonic time–varying
quantities.
The starting point is the observation that Kirchhoff’s equations,
 ik  0 ,  u t k  0 ,
k  a 
k  q 
along with Joubert’s equation,
u t k  u R k  u C k  u L k  u k  ek ,
are essentially linear relations in terms of currents and voltages at the terminals of
branches or circuit elements. On the other hand, the correspondence between harmonic
time–varying quantities and their complex representatives preserves linear operations,
f 1  f 2  F1  F2 ,  f   F ,
and, moreover, reduces the differentiation with respect to time of harmonic time–varying
quantities to a simple algebraic operation performed on their complex representatives,
df
 j F .
dt
As a consequence, when all quantities in a circuit are harmonic time–varying
quantities of the same frequency, then to each such quantity it corresponds a complex
representative and, according to the above mentioned properties, to Kirchhoff’s and
Joubert’s equations it directly corresponds complex Kirchhoff’s and Joubert’s equations,
 ik  0   I k  0 ,
k  a 
k  a 
9. ALTERNATING CURRENT CRCUITS
172
_____________________________________________________________________________________________________________________________________________________________
_________
 ut k  0
k  q 

Utk  0
k  q 
u t k  uR k  uC k  uL k  uk  ek
,

U t k  U R k  UC k  U L k  U k  E k .
If now the complex Joubert’s equation is accounted for in the complex Kirchhoff’s
voltage equation,
 U t k  0   U R k  UC k  U L k  Uk  Ek  0 ,
k  q 
k  q 
the complex e.m.f.’s are transferred to the right hand side and the expressions for the
complex voltages at the terminals of circuit elements are expressed as in the preceding
section, the explicite complex Kirchhoff’s voltage equation takes the form


1
 Rk I k 
I k  j Lk I k  U k
j Ck
k  q  



   E k
 k  q 
.
In particular, if the coil in branch k is part of a system of n coupled coils, the inductive
term within the parantheses has to be changed accordingly, and the last equation becomes
n


1

R
I

I

  k k j C k  j Ls I s  U k    E k .
k
k  q  
s 1
 k  q 
Along with the complex Kirchhoff’s current equation,
 Ik  0
k  a 
,
these complex equations completely describe the operation of any alternating current
circuit in terms of the complex representatives of harmonic time–varying quantities.
2. The power performance of an alternating current circuit can also be evaluated
in terms of complex quantities, by using a consequence of the complex Kirchhoff’s
equations which represents the theorem of complex power conservation. In the following,
however, a simpler argument will be used, based on a consequence of Poynting’s theorem
on electromagnetic energy, namely the theorem of power conservation in a dynamic
circuit. Using lower-case letters to indicate time–dependent quantities, the theorem of
power conservation in a dynamic circuit is written as
d wel . d wmg .
pt  pG  p J 

.
dt
dt
In the case of an alternating current circuit, the average value of power related
quantities is of a primary interest – it represents the active power transfer associated with
the circuit,
d wel . d wmg .
.
pt  pG  p J 

dt
dt
Such time–average value over a period of the power transfer at terminals pt , generated
power pG , and dissipated power pJ is just the corresponding active power: the active
power received at the teminals of the circuit,
1 T
Pt  pt   pt  t dt ,
T 0
9. ALTERNATING CURRENT CRCUITS
173
_____________________________________________________________________________________________________________________________________________________________
_________
the active power delivered by the generators in the circuit,
1 T
PG  pG   pG  t  dt ,
T 0
and the active power dissipated in (the resistors of) the circuit,
1 T
PJ  p J   p J  t dt .
T 0
The time–average value over a period of the electromagnetic energy stored in (the reactive
elements of) the circuit is, however, zero, due to the periodicity of all quantities in the
circuit,
d wel . d wmg . d w 1 T d w
1 T
1


 
dt   dw   w t   w 0   0 .
dt
dt
dt T 0 dt
T 0
T
A first consequence of these results is the equation of active power conservation in
an alternating current circuit,
Pt  PG  PJ .
A second consequence is the confirmation of the fact that the power transfer
associated with the reactive elements reduces to a simple power circulation between these
elements and the rest, with a zero time–average. This is quite normal, since the reactive
elements in a circuit exhibit precisely such a behaviour, their associated instantaneous
power transfer presenting a harmonic time–variation (of twice the alternating current
frequency). Therefore, the preceding results also imply that the multiplicative coefficients
of such harmonic time–variations of the zero–average power circulation associated with
the alternating current circuit must be the same for the two sides of the power
conservation equation. It thus follows that an equation of reactive power conservation in
an alternating current circuit is also valid,
Qt  QG  Q .
As for a single dipole, the two conservation equations can be assembled in a single
complex equation: let the second one be multiplied by the complex unit j and added to
the first. After reordering the terms one obtains
 Pt  j Qt    PG  j QG   P  j Q .
The relation can be written concisely as
St  SG  S
,
and represents the theorem of complex power conservation in an alternatimg current
circuit: the sum between the complex power received at the terminals and the complex
power delivered by (all) the generators equals the complex received by (all) the passive
elements in an alternaticg current circuit.
Taking into account the preceding analysis of the power transfer under alternating
current conditions, the detailed form of the theorem is
N
B
a 1
k 1


 V a Ia   E k Ik  U k Ak 
B

1
  Rk I k2  j C
k 1 
B B
k

I k2  j Lk I k2  


  j Lks I k I s cos  k   s
k 1s 1

9. ALTERNATING CURRENT CRCUITS
174
_____________________________________________________________________________________________________________________________________________________________
_________
The first sum on the left refers to the accessible termials (nodes) of the circuit, where V a
is the potential of the access node a and the current I a connecting the node to outside
circuits is assigned a positive sign when entering the node and a negative sign when
leaving it. The second sum on the left refers to all the generators in the circuit: the terms
of the first kind are assigned a positive sign when the reference direction of the complex
e.m.f. E k and current I k are the same and a negative sign when these reference
directions are opposite; the terms of the second kind are assigned a positive sign when the
reference direction of the complex voltage U k and generated current A k are associated
according with the source rule and a negative sign when these reference directions are
associated according with the load rule. The first sum on the right refers to the (self–)
impedances in the circuit and its terms, involving complex impedances and effective
currents, are all taken with a positive sign. Finaly, the last sum on the right refers to
mutual (coupling) impedances (inductances); its terms are assigned a positive sign when
the implied currents Ik and Is have the same reference direction with respect to the
marked terminal of the corresponding coil and a negative sign when the implied currents
have opposite reference direction with respect to the marked terminal of the
corresponding coil.
3. In principle, an alternating current circuit can be solved by first formulating its
dynamic circuit equations, then translating these equations into their complex form and
finaly solving the resulting system of complex algebraic equations. However, the complex
Kirchhoff’s equations, along with the theorem of complex power conservation, can be
used directly in the study of alternating current circuits. The following algorithm can be
applied to find the solution to a typical alternating current circuit problem:
1. The given data are: the frequency, the topology (configuration) of the circuit,
the characteristic values of passive elements (Rk , Ck , Lk , Lks ) , the instantaneous
characteristics of the generators ek t  , a k  t  , and, eventualy, in the case of unisolated
circuits, the instantaneous quantities at the terminals – either va t  or ia  t  .
The unknown quantities to be determined are the instantaneous currents ik  t  in
branches without a current generator and instantaneous voltages u k  t  across the current
generators, and, eventualy, in the case of unisolated circuits, unknown instantaneous
quantities at the terminals – either ia  t  or va t  . Reference directions are to be fixed
for these quantities and indicated on the circuit representation.
2. The complex circuit associated with the problem is first constructed: each
passive element is represented by its complex impedance and each active element is
represented by the corresponding complex generator characterised by the complex
representative of its instantaneous representative, as discussed in phe preceding section.
Complex representatives I k of the instantaneous currents ik  t  are carried by the
branches and complex representatives U k of the instantaneous voltages u k  t  appear
in the complex circuit.
3. The complex Kirchhoff’s equations are then directly formulated for the
complex circuit and the resulting system of complex algebraic equations is solved for the
complex representatives of the unknown quantities.
9. ALTERNATING CURRENT CRCUITS
175
_____________________________________________________________________________________________________________________________________________________________
_________
4. Finaly, the instantaneous representatives of the unknown quantities are
determined from their complex representatives by applying the simple correspondence
rules.
5. The theorem of complex power conservation can eventualy be used for
checking the result: the correct solution must satisfy the power conservation.
An example is here considered for the application of the above algorithm.
Let the circuit presented in fig. 9.27 be considered, where


e1  t   40 2 sin  t , a 2  t   2 sin   t   , f  50 Hz ,
4

500
100
C1 
 F , R3  10  , L3 
mH .


Fig. 9.27.
Fig. 9.28.
1. The unknown quantities obviously are i1 t  , u 2 t  , i3 t  , with the
reference directions indicated in fig. 9.28.
2. The complex circuit is constructed as in fig. 9.29, where the complex
characteristics of the active elements are
e1  t   40 2 sin  t  E1  40 e j 0  40 ,

2  j

a 2  t   2 sin   t    A 2 
e
4
2

4

 
 2  cos  
 4


 
  j sin  

 4

 

 2
2
  1 j ,
 2 
j
2 
 2
the angular frequency is
  2f  2  50  100 rad s ,
and the complex impedances of passive elements are
1
j
j 10 6
 2


 20 j ,
j C1 j  100  500   10  6
5  10 4
100
j L3  j 100 
 10  3  10 j
,
R3 = 10 .

The complex unknown quantities are I1 , U 2 , I 3 , with the same reference directions as
their instantaneous correspondents.
9. ALTERNATING CURRENT CRCUITS
176
_____________________________________________________________________________________________________________________________________________________________
_________
3. The complex Kirchhoff’s equations written for the upper node and the left and
right loops with respect to the clockwise direction are
  I1  A 2  I 3  0
 1
I1  R3 I 3  j L3 I 3  E1 .

j

C
1

 U 2  R3 I 3  j L3 I 3  0
Fig. 9.29.
Fig. 9.30.
Using the numeric values, these equations are successively rewritten as
 I1  I 3  1  j 
 I1  I 3  1  j 


 20 j  I 3  1  j   10  10 j   I 3  40  10  10 j   I 3  40  20 j  20 ,
 U  10  10 j   I
 U  10  10 j   I
3
3
 2
 2
whence
20  20 j

 I3  2
 I 3  10  10 j

.
 I1  1  j
 I1  2  1  j 
U  10  10 j   2  U  20  20 j
2
 2

4. The instantaneous representatives corresponding to the determined complex
quantities are directly obtained if the complex quantities are expressed in their
exponential form. In turn, this is obtained easier if the phasors associated to the complex
quantities are represented in the complex plane (fig. 9.30). It immediately follows that
 I1  1  j
 I1  2 e j  4

j 4
,
U 2  20  20 j  U 2  20 2 e
j0
 I 2
 I3  2 e
 3
and, according to the instantaneous – complex representatives correspondence rules,
9. ALTERNATING CURRENT CRCUITS
177
_____________________________________________________________________________________________________________________________________________________________
_________





j 4
 i1  t   2  2 sin   t    2 sin   t  
 I1  2 e
4
4









j 4
 u2  t   20 2  2 sin   t    40 sin   t   .
U 2  20 2 e
4
4



j0
 I3  2 e
 i3  t   2  2 sin  t  0   2  2 sin  t

5. The computation of the complex power balance constitutes a good check of the
solution. Since the circuit is isolated, the equation of complex power conservation reduces
to
SG  S .
The two sides are separately computed as
SG  E1 I1  U 2 A2  40  1  j    20  20 j   1  j  
 40  40 j 20  20 j 20 j  20  40  0 j ,
1
S
I12  R3 I 32  j L3 I 32  20 j 2  10  4  10 j 4  40  0 j .
j C1
The resulting power balance,
SG  40  0 j  S ,
detailed as
PG  P  40 W , QG  Q  0 var ,
validates the correctness of the solution.
9.6. Theorems on alternating current circuits
1. The study of alternating current circuits is facilitated by the use of additional
theorems, formulated in the terms of the complex representation approach.
The starting point is the important remark on the similarity between Kirchhoff’s
theorems for direct current circuits and the complex Kirchhoff’s theorems for alternatimg
current circuits. Let the complex proper impedance of a branch be introduced as
1
Z k  Rk 
 j Lk ,
j C k
along with the complex mutual (coupling) impedances,
Z ks  j Lks  j Lsk  Ζ sk .
Then, in the case of alternating current circuits without coupled coils, there is a
perfect similarity between the corresponding Kirchhoff’s equations,
  Ik  0

 Ik  0
 k  a 
k  a 
.


 Zk I k  U k    Ek
   Rk I k  U k    Ek
k  a 
k  a 
k  a 
 k  a 
The important immediate consequence is that all theorems proved for direct current
circuits translate into valid theorems for complex alternatimg current circuits without
coupled coils.
9. ALTERNATING CURRENT CRCUITS
178
_____________________________________________________________________________________________________________________________________________________________
_________
In the case of alternating current circuits with coupled coils, Kirchhoff’s equations
for direct current circuits and complex alternating current circuits bear some similarity in
the sense of their linearity,
  Ik  0

 Ik  0
 k  a 
k  a 

.
B






   Rk I k  U k    E k
  Z k I k   Z ks I s  U k    E k
 k  a 
k  a 
k  a  
s 1
 k  a 
The important immediate consequence is that all theorems proved for direct current
circuits, based exclusively on the linearity of equations, translate into valid theorems for
complex alternatimg current circuits with coupled coils.
Based on these statements, the theorems regarding the alternating current circuits
presented in the following are given without proof.
2. The important equivalence theorem for alternating current circuits, which are
linear by hypothesis, represented by the superposition theorem, remains valid: the state
I k , U k  of a complex alternating current circuit is the superposition (algebraic sum) of
particular states I   , U   corresponding to maintaining, in turn, a single complex

k
k

generator in the circuit while all other are passivated. A passivated generator is a
generator whose active part (complex electromotive force E or generated current A )
is equated to zero, so that its internal complex impedance (or admittance) only remains to
be considered (fig. 9.31). This is equivalent to saying that passivating an ideal complex
source means short–circuiting each ideal complex voltage source and open–circuiting
each ideal complex current source. In a more compact way,
E   or A    I k  , U k 






all E   and A    I k   I k  , U k   U k   .




Fig. 9.31.
3. The important group of equivalence theorems are immediately translated for
complex alternating circuits without coupled coils. It is recalled that, for circuits with the
same number of accessible terminals, the equivalence criterion is: two complex
alternating current circuits are said to be equivalent if, under the same set of complex
voltages at terminals, one obtains the same set of complex currents through terminals, or
vice–versa, for any such given set.
9. ALTERNATING CURRENT CRCUITS
179
_____________________________________________________________________________________________________________________________________________________________
_________
The equivalence theorems for passive complex circuits are first considered – these
refer to circuits including complex self impedances only.
Fig. 9.32.
The equivalent complex impedance of a group of n series connected complex
self impedances (fig. 9.32) is given by
Z   Zk
,
k
and the associated theorem of the complex voltage divider is
Uk  U
Zk
Z
.
Fig. 9.33.
The equivalent complex impedance(admittance) of a group of
connected complex self impedances (admittances) as in fig. 9.33 is given by
1
1

 Y   Yk
,
Z k Zk
k
and the associated theorem of the complex current divider is
Ik  I
Y
Z
I k
Zk
Y
.
n
parallel
9. ALTERNATING CURRENT CRCUITS
180
_____________________________________________________________________________________________________________________________________________________________
_________
Fig. 9.34.
As well, the star–delta and delta–star equivalence theorems for star– and delta–
connected complex self impedances (fig. 9.34) are given by a system of symmetric
equations, for which it is worth noticing that in each case a single relation is sufficient to
be retained, since the other two can be derived by taking a cyclic permutation of
subscripts, 1  2  3  1 ,


Z1 Z 2  Z 2 Z 3  Z 3 Z1
Z12 Z 31
Z1 
 Z12 

Z3
Z12  Z 23  Z 31


Z
Z

Z
Z

Z
Z
Z 23 Z12


1 2
2 3
3 1
and
.
 Z 23 
 Z2 
Z1
Z12  Z 23  Z 31


Z1 Z 2  Z 2 Z 3  Z 3 Z1
Z 31 Z 23


Z

Z

3
31


Z12  Z 23  Z 31
Z2


4. The equivalence theorems for active complex circuits are stated within the
limits of the same equivalence criterion as above.
Fig. 9.35.
The equivalence theorem of non–ideal complex sources (with internal self
impedances, fig. 9.35) states the conditions which ensure that a non–ideal complex
voltage source E, z and a non–ideal complex current source A , y are equivalent:
 
E
A
y
,
z
1
y
and, conversely,
A
E
z
,
y
1
z
.
The very important theorem of equvalent complex generators is as well valid: with
respect to a pair A , B of its terminals, a complex active circuit (with internal coupled
9. ALTERNATING CURRENT CRCUITS
181
_____________________________________________________________________________________________________________________________________________________________
_________
coils, eventualy) is equivalent to a non–ideal complex voltage generator or a non–ideal
complex current generator (fig. 9.36). The complex electromotive force of the equivalent
complex voltage source is equal to the open–circuit complex voltage U AB 0 of the given
complex circuit with respect to the terminals A and B , the complex generated current of
the equivalent complex current source is equal to the short–circuit complex current I ABS
of the given complex circuit with respect to the terminals A and B , and the internal
complex impedance (or admittance) of either one of the equivalent complex generators,
1
Z AB 0 
, is the equivalent complex impedance (or admittance) of the passivated
Y AB 0
complex circuit, taken with respect to the terminals A and B (fig. 9.37). It must be
recalled that passivating a complex circuit means substituting each ideal complex source
by its internal complex impedance (admittance), that is short–circuiting each ideal
complex voltage source and open–circuiting each ideal complex current source. It is to be
underlined that the theorem is only valid for alternating current circuits ahich interact with
external circuits at the sole terminals A and B .
Fig. 9.36.
Fig. 9.37.
5. The simplest connection of a complex source and a complex load (without
copled coils) used in practice – a given non–ideal complex source E, z delivering
power (or signals) to an arbitrary complex load Z (fig. 9.38) – is studied quite similarly
to the corresponding direct current case.
The complex current and voltage at the source/load terminals are, respectively,
I Z  
E
zZ
,
U Z  
EZ
,
zZ
9. ALTERNATING CURRENT CRCUITS
182
_____________________________________________________________________________________________________________________________________________________________
_________
and, generaly, the effective value of the current decreases when increasing the load
impedance, while the effective value of the voltage increases when increasing the load
impedance.
The complex generated power, the active power loss and transferred active power
are, respectively
E  E
E2
SG  Z   E  I 

,
zZ zZ

E
Pi  Z   r I  r
zZ
2

2
E
P Z   R I  R
zZ
2
2

r E2
zZ
2
R E2
zZ
2
,
,
where the complex impedances have been considered given respectively as
z  r  jx , Z  R  jX .
The active power transfered to the load is then
R E2
R E2
R E2
R E2
,
P Z  



2
2
2
2
2




r

R

x

X








zZ
r  jx  R  j X
rR  j x X
and reaches its maximum value when the denominator attains its minumum. Since the
resistances are non-negative, minimising the denominator supposes first cancelling its
second term,
x  X  0  X  x .
The maximum value of the resulting
expression,
R E2
,
P Z  X   x 
r  R  2
is reached under the conditions analised
previously, for the corresponding direct
current problem,
Fig. 9.38.
Rr .
These arguments lead to the theorem of the maximum active power transfer: a given non–
ideal complex source delivers the maximum active power to a complex load impedance
equal to the complex conjugate of the complex internal source impedance,
E2
.
4r
The load is said to be matched to the source when operating under maximum power
transfer conditions, which are therefore also named (load) matching conditions.
Z  R  j X  r  j x  z

P  Pmax 
9.7. Applications of alternating current circuits
1. A first application to be considered is directly linked to the last subject
approached in the preceding section: it concerns the analysis of a simple source–and–load
alternating circuit and the improvement of the power factor of the latter.
9. ALTERNATING CURRENT CRCUITS
183
_____________________________________________________________________________________________________________________________________________________________
_________
Fig. 9.39.
Fig. 9.40.
Let a very simple electric energy distribution network be considered, consisting in
a nonideal complex voltage generator E, z g , a connection line z l  , and a load


characterised, as usual, by its nominal (effective) voltage U2 , nominal active power P2 ,
and nominal (supposedly inductive) power factor cos 2 (fig. 9.39). A simpler equivalent
circuit is immediately constructed, by assembling the source and line complex
impedances into an equivalent internal complex impedance,
z  z g  z l  rg  j x g   rl  j xl   rg  rl  j x g  xl  r  j x ,



 

so that an equivalent generator E, z remains to be considered (fig. 9.40).
Starting from the given data, simple calculations give other values of interest,
P2
P2  U 2 I 2 cos  2
 I2 
I ,
U 2 cos  2
2
U 2 U 2 cos  2
U2  Z2I2
 Z2 

.
I2
P2
The equivalent circuit is described by the complex equation
z I 2  U 2  E   r  j x I 2  U 2  E ,
corresponding to the phasor diagram in fig. 9.41, whence the approximate value of the
effective voltage U2 is obtained as
U 2  OD  AD  E  AB  E   AC  CD   E  r  cos 2  x  sin  2 I 2 

 E  r  cos  2  x  sin  2
U
P2
2 cos  2

P
 E  r  x  tg  
U
2
2

.
2
It follows that there exists a voltage drop between the source and the load, of relative
value
U 2
P
U 2 E  U 2
 r  x  tg  2 22


.
U2
U2
U2
U2


9. ALTERNATING CURRENT CRCUITS
184
_____________________________________________________________________________________________________________________________________________________________
_________
Fig. 9.41.
Fig. 9.42.
The analysis of the last result shows that, generaly, the voltage drop between the
source and the load is reduced quadraticaly if the voltage U2 is increased; that is why the
electric energy is transmitted over long distances on high voltage lines. On the other hand,
the formula obtained above allows one to evaluate the needed source effective e.m.f. E if
the load nominal effective voltage U2 is imposed or the available load effective voltage
U2 if the source effective e.m.f. E is imposed. One may note that an inductive load
determines a larger voltage drop than a resistive load (since tg  2 is positive for the first
and is zero for the second), and that a capacitive load (of negative value for tg  2 )
determines a smaller voltage drop than a resistive load. Moreover, if the capacitive load
presents a capacitive phase difference satisfying the condition
r
tg  2  
,
x
then the load effective voltage U2 may even be greater than the source effective e.m.f. E .
There exists as well an active power loss in the equivalent internal resistance r
(or, for that matter, in the source resistance rg and the line resistance rl ) ,
2
2


 S2 
P2


 ,
P  r I  r
 r 
 U 2 cos  2 
U
2




with a relative value (with respect to the load nominal active power)
2
2
1  tg 2  2
P
 S2 
P22  Q22
 r P2

  r
.

P2
U 22
P2 U 22
U2 
The analysis of this result shows again that the active power loss is reduced when
the voltage is increased, with the same conclusion of using high voltage for the
transmission of electric energy over long distances. On the other hand, at a local level, a
reduction of the power loss can be obtained by increasing the equivelent load inductive
power factor (under the same values of load nominal voltage U2 and active pover P2 )
toward the optimum unitary value, for which  2 = 0 and tg  2 = 0 .
The improvement of the load power factor under the above specified conditions
can be achieved simply by connecting a suitable (ideal) capacitor parallel to the load (fig.
9.42). The load voltage remains the same for such a parallel connection, and the load
active power is not changed since an ideal capacitor requires no active power.
The total complex current in the circuit changes from the value I 2 to the value
P
r

P2
P2
9. ALTERNATING CURRENT CRCUITS
185
_____________________________________________________________________________________________________________________________________________________________
_________
I1  I 2  I C ,
and the power factor of the equivalent load changes from the value cos  2 to the new
value cos 2' , corresponding to the new phase difference  2' between the complex
current I1 and the complex voltage U 2 . Suppose that an improved (increased) value
is required, corresponding to a reduction in the phase difference; the load
cos 2'
reactive power is correspondingy reduced from the initial value Q2  P2 tg  2 to the new
value Q2'  P2 tg  2' . The reduction of the load reactive power is covered by the
reactive power associated with the connection of the capacitor,
QC  Q2' Q2  P2 tg 2' P2 tg  2 .
Since the reactive power received by a capacitor is
 C U  2   C U 2 ,
I2

C
C
the value of the correction capacitance results then as
P2 tg  2  tg  2'
 QC
C
.
C

 U 22
 U 22
QC  


2. Another interesting application concerns a phenomenon somehow related to the
improvement of the power factor discussed above; it is the resonance in an
alternating current circuit.
There are two accepted definition of the resonance. An alternating current circuit
is said to operate at resonance if its exchanged reactive power equals zero. This first
definition is quite general, but it concerns a quantity – the reactive power – which is
difficult to be evaluated or measured. A definition more adequate for measurement is the
following: an alternating current circuit is said to operate at resonance if, under defined
source conditions, the effective value of a specified quantity reaches an extremum value
(either a maximum or a minimum).
As typical examples, let two simple circuits be considered, consisting in a series or
a parallel connection of a nonideal coil L, r  and a nonideal capacitor C , R  . The
associated complex circuits are directly studied. If the general definition of resonance is
considered, then simple resonance cconditions result as
 U I sin   0
  0


2
Q0
 Xe I  0
 X e  0 .
B U 2  0
B  0
 e
 e
If the experimental definition of resonance is considered, then the typical source condition
is that of a given effective value of the voltage or current at the circuit terminals.
U = const
or
I = const.
Finaly, the resulting resonance condition is usualy expressed as the resonance (angular)
frequency, depending on the characteristics of the elements in the circuit.
The complex impedance of the series connected complex circuit (fig. 9.43) is
9. ALTERNATING CURRENT CRCUITS
186
_____________________________________________________________________________________________________________________________________________________________
_________
Z  r  j L  Z p
that is
Z  r  j L 
, where
1
1 1  j CR
,
 Y p  j C  
Zp
R
R
R 1  j CR 
R
,
 r  j L 
1  j CR
1   CR  2
whence

 
 CR 2 
R
Z  Re  j X e  r 

j

L


 .
2
1   CR  2 
 1   CR   
The general resonance condition is
 CR 2
Xe  0  L 
 L  L  CR  2  CR 2 ,
2
1   CR 
where the fact that   0 was accounted for. The resonant (angular) frequency results as

CR 2  L
LCR 
2

1
LC
1
L
CR
2

   0 1
L
CR 2
,
where
1
LC
is the resonant (angular) frequency of the reference series r–L–C circuit (fig. 9.44),
where an ideal capacitor  R    is considered.
0 
Fig. 9.43.
Fig. 9.44.
The experimental resonance conditions for the series connected circuit in fig. 9.43
are rather difficult to obtain. That is why the simpler series r–L–C complex circuit (fig.
9.44) will be analysed in the following.
Proceeding as above, the complex impedance of the series r–L–C circuit is

1
1 
 ,
Z  Re  j X e  r  j L 
 r  j   L 
j C
 C 

and the general resonance condition results simply in
1
1
0 
Xe  0  L 

.
C
LC
Invoking the experimental definition of resonance the (modulus of the complex)
impedance is computed,
9. ALTERNATING CURRENT CRCUITS
187
_____________________________________________________________________________________________________________________________________________________________
_________
2

1 

,
Z  r    L 
 C 

for which the extremum condition is
2
1
.
LC
Now more than one experimental resonance can be defined if, for instance, constant
effective voltage U at terminals is assumed. The current resonance is defined as
U
U
U
1
max I  max


at  I   0 
.
U  const .
U  const . Z  
Z min
r
LC
On the other hand, for instance, the resonance of the capacitive voltage, given by
I
U
U
max U C  max
 max

,
U  const .
U  const .  C U  const .  C Z   C  min  Z  
obviously happens at an angular frequency  C   0 , and the resonance of the inductive
Z min  r
 0 
for
voltage, given by
 LU

 L U  max
,
U const .
U const .
U const . Z  
Z  
happens again at still another angular frequency  L   0 .
max U L  max  L I  max
The complex admittance of the parallel connected complex circuit (fig. 9.45) is
1
1
1
Y
 
, where Z s  r  j L ,
Z s R j C
that is
Y
r  j L
1
1
1
  j C  2

 j C
r  j L R
r   L  2 R
whence
1
 

L
r
Y  Ge  j Be    2

j


C


 .
2
2
2
 R r   L    r   L 

The general resonance condition is
L
Be  0 
  C  L  C r 2  C  L  2 ,
2
2
r   L 
where the fact that   0 was accounted for. The resonant (angular) frequency results as

L  Cr2
CL2

1
LC
Cr2
1
L

 0
Cr2
1
L
,
where
1
LC
is the resonant (angular) frequency of the reference parallel R–L–C circuit (fig. 9.46),
obtained when an ideal coil  r  0  is considered.
0 
9. ALTERNATING CURRENT CRCUITS
188
_____________________________________________________________________________________________________________________________________________________________
_________
Fig. 9.45.
Fig. 9.46.
The experimental resonance conditions for the parallel connected circuit in fig.
9.45 are significantly more difficult to obtain. That is why the simpler parallel R–L–C
complex circuit (fig. 9.46) will be analysed in the following.
Proceeding as above, the complex admittance of the parallel R–L–C circuit is

1
1 1  1
Y  Ge  j Be 
 j C    j 
  C  ,
j L
R R L

and the general resonance condition results simply in
1
1
0 
Be  0 
 C

.
L
LC
For the experimental definition of resonance the (modulus of the complex)
admittance is computed,
2
 1

,
Y

  C 
2 L
R


for which the extremum conditions are
1
1
.
Ymin 
for    0 
R
LC
Now more than one experimental resonance can be defined if, for instance, constant
effective current I at terminals is assumed. The voltage resonance is defined as
I
I
1
max U  max

 RI
at U   0 
.
I  const .
I  const . Y   Ymin
LC
On the other hand, for instance, the resonance of the capacitive current, given by
C I

max I C  max  C U  max
 C I  max
,
I const .
I const .
I const . Y  
Y  
obviously happens at an angular frequency  C   0 , and the resonance of the inductive
1
voltage, given by
U
I
I
 max

,
I const .
U const .  L U const .  L Y  
L  min  Y  
happens again at still another angular frequency  L   0 .
max I L  max
The most important use of resonant circuits is linked to their frequency selective
property. Indeed, let a simple parallel connected R–L–C circuit be considered as shown in
9. ALTERNATING CURRENT CRCUITS
189
_____________________________________________________________________________________________________________________________________________________________
_________
fig. 9.46; when operating, for instance, under constant effective current conditions, the
dependence of the effective voltage at terminals on frequency is that presented in fig.
9.47.
Fig. 9.47.
Let the current carried by the circuit contain three components of the same
effective voltage but different frequencies,
i  t   i1  t   i2  t   i3  t   I 2 sin  1t  I 2 sin  2 t  I 2 sin  3t .
Since the circuit is linear, each component current determines an associated component
voltage of the same frequency. The effective values of the component voltages are related
to the frequency according with the dependence indicated in fig. 9.47,
u t   u1  t   u2  t   u3  t  





 U1 2 sin  1t   1  U 2 2 sin  2 t   2  U 3 2 sin  3t   3

.
Let the component frequencies be ordered, for example, as
 1   0 ,  2   0 ,  3   0 ;
then, according to the voltage dependence on frequency, the effective values of the
component voltages are then
U1  R I
, U2  RI
, U 3  R I .
These values show that the resonant circuit performs a selection of component signals
according with their frequency: the voltage u at the terminals of the resonant circuit
under consideration consists almost entirely in the component u2 , of a frequency near the
resonant frequency of the circuit, while the components of frequencies different from the
resonant frequency are rejected.
3. A very important application of alternating current circuits is represented by a
particular assembly of active and passive elements into so called three–phase circuits.
A three–phase circuit is an alternating current circuit consisting in three parts of
approximately the same configuration, where the corresponding electric quantities have
approximately the same effective value and a phase difference of  2 3 (120)
between them. The three–phase circuits are studied directly in terms of the complex
representation of the electric quantities.
9. ALTERNATING CURRENT CRCUITS
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In a three–phase circuit one can distinguish the three–phase generator (or
generators), the three–phase load (or loads) and the lines connecting them. Let the
simplest three–phase circuit be considered, where the three–phase generator consists into
three alternating current voltage sources – the phases of the generator, the three–phase
load consists into three loads operating under alternating current conditions – the phases
of the load, and three (or, sometimes, four) lines connecting the corresponding phases
(components) of the generator and load.
The reason for using three–phase circuits is twofold. First, supposing that different
corresponding generator and load phases do not influence one another, there is an
important reduction in the material needed for the lines: a three–phase circuit needs three
(or four) connecting lines, while three separate generator–load couples would have needed
six connecting lines. Second, there are special applications in electric three–phase
machinery which are significantly more efficient than other constructive approaches.
The generator and the load phases can be either star– or triangle–connected.
The star (wye) connection of the three–phase generator and load is presented in
fig. 9.48. The three generator phases have a common point called the null (or neutral)
generator terminal O . The three load phases have as well a common point called the null
(or neutral) load terminal N . The corresponding phase terminals of the three–phase
generator and load are connected by associated lines; eventualy, the null terminals of the
three–phase generator and load can also be connected by a null line. The complex
quantities associated with the (ideal) three–phase generator are the phase e.m.f.’s
Fig. 9.48.
E1 , E 2 , E 3 , the phase voltages U10 , U 20 , U 30 , and the phase currents I1 , I 2 , I 3 ,
along with the line voltages U'12 , U'23 , U'31 at the generator end. The complex
quantities associated with the three–phase load are the phase impedances Z1 , Z 2 , Z 3 ,
the phase voltages U1N , U 2 N , U 3N , and the phase currents I1 , I 2 , I 3 , along with the
line voltages U12 , U 23 , U 31 at the load end. The complex quantities associated with the
lines are their impedances Z l 1 , Z l 2 , Z l 3 (and, eventualy, Z0 ), the line currents
I1 , I 2 , I 3 and the line voltages U12 , U 23 , U 31 , generaly position–dependent along the
lines. The series connection of phases and lines in the star connection determines the
equation
9. ALTERNATING CURRENT CRCUITS
191
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I ph  I l :
the phase and line currents of the same subscript are equal.
Fig. 9.49.
The triangle (delta) connection of the three–phase generator and load is presented
in fig. 9.49. The three generator phases are connected in a loop between the accessible
terminals and, similarly, the three load phases are connected in a loop between the
accessible terminals. The corresponding terminals of the three–phase generator and load
are connected by associated lines; obviously, there is not a null line. The complex
quantities associated with the (ideal) three–phase generator are the phase e.m.f.’s
E12 , E 23 , E 31 , the phase voltages U'12 , U'23 , U'31 , and currents I' 12 , I'23 , I' 31 .
Similarly, the complex quantities associated with the three–phase load are the phase
impedances Z12 , Z 23 , Z31 , the phase voltages U12 , U 23 , U 31 , and the phase curents
I12 , I 23 , I 31 . The complex quantities associated with the lines are their impedances
Z l 1 , Z l 2 , Z l 3 , the line currents I1 , I 2 , I 3 and the line voltages U12 , U 23 , U 31 ,
generaly position–dependent along the lines. The connection of phases and lines in the
triangle connection determines the equation
U ph  U l :
the phase and line voltages of the same subscript are equal, separately at the generator and
the load ends.
A three–phase circuit operates under symmetric conditions when the phase or
line quantities of the same type have precisely the same effective values and present
between them phase differencies of precisely  2 3 (120) .
For instance, a symmetric system of phase voltages at the generators end in a star
connection is (fig. 9.50)
U1  U ph
,
U 2  U ph e  j 2
3
,
U 3  U ph e  j 2
3
.
It can be easily shown that, in such a case, the associated line voltages obtained by
considering Kirchhoff’s voltage equation in fig. 9.51,
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Fig. 9.50.
Fig. 9.51
U12  U1  U 2
, U 23  U 2  U 3
are also symmetric,
U12  U l
, U 23  U l e  j 2 3 ,
where
U l  3 U ph e j 6 .
,
U 31  U 3  U1 ,
U 31  U l e  j 2
3
,
Similarly, a system of symmetric phase currents of a triangle–connected load could be
(fig. 9.52)
I12  I ph
,
I 23  I ph e  j 2
3
,
I 31  I ph e  j 2
3
.
In this case it is easy to obtain the corresponding line curents by considering Kirchhoff’s
current equation at the terminals (fig. 9.53),
Fig. 9.52.
I1  I12  I 31 , I 2  I 23  I12
which are also symmetric,
I1  I l
, I 2  I l e  j 2 3 ,
where
I l  3 I ph e  j 6 .
Fig. 9.53
,
I 3  I 31  I 23
I 3  I l e  j 2
3
,
A three–phase load is said to be balanced if its phase complex impedances are
the same,
Z1  Z 2  Z3  Z Y
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193
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for the star–connection, or
Z12  Z 23  Z31  Z 
for the triangle–connection. In such cases simple application of the star–triangle
equivalence theorems show that a balanced star–connected load of impedances Z Y is
equivalent to a balanced triangle–connected load of impedances
Z  3ZY ;
conversely, a balanced triangle–connected load of impedances Z  is equivalent to a
balanced star–connected load of impedances
Z
ZY   .
3
The three–phase circuits essentialy are and can be studied as alternating current
circuits. However, due to specific symmetries in their constituency and operation, special
procedures for the study of three–phase circuits have been also developed.
In particular, it can be easily shown that the study of a three–phase balanced load
operating under a three–phase symmetric system of voltages can be reduced to the study
of a single phase of it, the quantities in the other two phases resulting afterwards by
simply changing by  2 3 the phase of the corresponding quantities in the reference
phase under study.
In the same particular case of a three–phase balanced load of phase impedance
Z  Z e j ,
operating under a three–phase symmetric system of line voltages of effective value Ul , it
results also a three–phase symmetric system of line currents of effective value Il .
Moreover, the active and reactive power received by the circuit can be computed to be
P  3 U l I l cos  , Q  3 U l I l sin  .