Continuous current case In the continuous domain, the current in the diagram of an electric circuit is represented by an arrow.But the actual direction of movement of the electrons is the opposite of the positive direction of the current:

Note: if after calculation in any branch, we find a negative sign of the value of I, this means that the real direction of the current is reversed compared to that chosen arbitrarily at the start.

Continuous current case
In the continuous domain, the current in the diagram of an electric circuit is
represented by an arrow. But the actual direction of movement of the electrons is the
opposite of the positive direction of the current:

Note:
if after calculation in any branch, we find a negative sign of the value of I, this means that
the real direction of the current is reversed compared to that chosen arbitrarily at the
start.

Similarly, a voltage is represented by an arrow near the element considered, as in the
following diagrams (we note in case (1) that the current-voltage arrows are in the opposite
direction: this is the receiver convention, unlike case (2) where we speak of the generator
convention):

(1) case of a receiver (2) case of a generator
convention (example : resistance) (example : storage battery)
In general, the "arrow" is oriented towards the highest potential (for example, for a battery the
arrow is oriented towards the +). However, if we ignore the values of the potentials, we give an
arbitrary direction to the voltage diffrence and if we find after calculation that VA−VB< 0, this
means that the direction is reversed.
Note: Elements with an input and an output (A and B in our examples) are called electric dipoles.

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 3
2) Kirchhoff laws
• Kirchhoff’s Current Law (KCL) (1st Law)
A node is a point where several conductors join. One of Kirchhoff's laws, called the law of
nodes, states that:

"The sum of the intensities of the currents flowing into a node (or a junction) is equal to
the sum of the intensities of the currents flowing out of the node."

In other words, we can also express the Kirchhoff Current law as follows:
" The algebraic sum of the currents flowing into a node is constantly zero.

∑Ik
n

k=1
= 0

Example :

In this case we have:
I1+ I2+ I3+ I4=0

(this proves that at least one of the currents is necessarily

negative).

• Kirchhoff Voltage Law (KVL) (2nd Law)
A Loop consists of a set of electrical conductors and components, starting from a point,
and arriving at this same point. This law states that the algebraic sum of the voltages along
the mesh is constantly zero:

∑Vk
n
k=1
= 0

Then , we choose an arbitrary direction in the loop and we express all the voltages
according to this choice.

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 4
Example:

In this case we have:

V1− V2− V3+ V4=0

3. Behavior of usual electric components in the frequency domain
   Important note

The arrows on the components are made in the same way as for continuous operation (current-
voltage arrows in the same direction for a generator, and in the opposite direction for a

receiver).

• The resistor (Resistance)
The resistor, which is usually referred to as : R has the property of
having a current response i(t) proportional to the applied voltage
v(t) at any instant. This is Ohm's law established in continuous
domain and which can be extended to frequency domain.
Therefore, we have:

v(t) = Ri(t)

• The inductor
According to Lenz's law, the inductance (denoted L) has the
property of producing at its terminals a voltage proportional to
the derivative of the current. Generally speaking, the relation
between current and voltage is given by:
v(t) =
dL(i(t))
dt = L
di(t)
dt + i(t)
dL
dt

i(t)
dL
dt
usually equals zero
In most cases (if L=const) , Lenz’s Law becomes : v(t) =
dL(i(t))
dt

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 5
Note :
From the obtained relation, we note that if the current is suddenly cut off for example
(discontinuity of the current in a very short time), we will have:

v(t) = L
di ⏞ ( t)
≠0
dt ⏟
→0
⟹ v(t) ⟶ ∞

In theory, this would correspond to an infinite voltage. But this means that in practice, a
sudden opening of an inductive circuit leads to an overvoltage which manifests itself by
more or less dangerous sparks. It is said that the inductance does not support sudden cuts
in the current flowing through it. Furthermore, in continuous operation, the inductor
behaves like a closed (short)circuit:

v(t) =
dL(i(t))
dt = 0 ⟺ closed switch(short circuit)

• The Capacitor
The relation between the instantaneous value of the current i(t)
and the value of the capacitance in such a circuit is given by:

i(t) =
dq
dt = C
dv(t)
dt

Thus, the capacitor is the opposite element of the inductor: it is
enough to exchange the roles of voltage and current. The
capacitor C corresponds to a proportionality between the current
and the derivative of the voltage.

Note :
From the obtained relation, we note that if the voltage varies abruptly, for example
(discontinuity of the voltage in a very short time), we will have:

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 6

i(t) = C
dv⏞ ( t)
≠0
dt⏟
→0
⟹ i(t) ⟶ ∞

In theory, this would correspond to an infinite current. In practice, it means that in a
sudden change in the voltage of a capacitive circuit leads to a current spike.
Furthermore, in continuous mode, the capacitor behaves like an open circuit.

i(t) = C
dv(t)
dt = 0 ⟺ open switch (open cicuit)

II- Representation of sinusoidal quantities : from sine to complex numbers
The sinusoidal mode is one of the most used modes in the electrical engineering. Also, and in
order to deal with easier exploitation of the associated electrical equations, same methods that
are used in continuous mode to establish those equations are also developed in the frequency
domain.
Note

Same rules are maintained, except that they are appliable at all times.

Important !:
Conventional symbols
Instantaneous values are represented by a lowercase letter, effective values by an uppercase
letter, and maximum values by an uppercase letter with the letter 'm' as a subscript.

1. Instantaneous representation of sinusoidal quantities
   This involves expressing the sinusoidal quantity in its analytical. Let g(t) be a sinusoidal quantity
   with pulsation ω ( ω = 2πf , f being the frequency of g(t)) , which is expressed as follows:

g(t) = Gm sin(ωt + φ) = √2G sin(ωt + φ)

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 7
g being the instantaneous value, Gm the maximum value, φ the phase shift relative to a chosen
origin and G the effective value of the quantity considered. If we want to represent the
characteristic g as a function of time, then we should consider the figure given above.
Note that the maximum value Gm and the effective value G are related by the following:

G = √
1
T
∫ Gm
2
T
0
sin2(ωt + φ)

In the electrical field, the effective value of an alternating voltage is equal to the value of a
direct voltage which would produce the same heating in the same resistor. In the particular
case of a sinusoidal function, we naturally find:

Gm = √2G

2. Vectorial representation
   In alternating current based circuits, voltages and currents evolve periodically over time.
   Depending on the nature of the considered circuit (resistive, capacitive or inductive), the
   currents and voltages have distinct phase shifts relative to a given origin. Also, the
   analytical resolution of electrical equations often gives rise to more or less tedious
   calculations.
   However, if we only take into account the steady state, it is possible to represent the sinusoid
   by a rotating vector, then arrive to a complex representation of magnitude that does not
   include the time factor and where only the effective value and the phase at the origin matter.

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 8
However, in order to fully understand how we can associate a sinusoidal quantity with a
complex number, we must first return to the Fresnel representation.
Indeed, let’s suppose that, on an orthonormal XOY frame (figure given above), a vector g⃗⃗⃗⃗m⃗ (t)
has:
• A magnitude that is equal to Gm
• A phase φ relative to the origin at t=0

Note :
If g(t) is expressed by a cosine, it is enough to project gm (t) on the OX axis to find the value
of g(t). In reality, expressing g(t) by a cos or a sin is not important in electricity: only the
phase shift between two electrical quantities matters: it is then enough to choose an origin
of the phases (arbitrary choice) and to express all the other quantities (currents, voltages)
with respect to this origin.

3)Complex and temporal representation of sinusoidal quantities
We know that a vector can be represented by a
complex number whose real part expresses the
projection of this vector on the X axis and the
imaginary part on the Y axis (figure opposite). Thus
we have :

g⃗ (t) ⟺ ǧ(t) = Gm[cos(ωt + φ) + jsin(ωt + φ)]
In order to find the instantaneous value, we take the imaginary part of ǧ(t), as follows :

g(t) = Im[ǧ(t)]

4)Complex and Non temporal representation of sinusoidal quantities
The easiest and most efficient way to represent g(t) is to associate it with a complex
quantity which no longer depends on time, unlike the temporal representation g~(t). Of
course, this is only possible in the case of the continuous mode : all currents and voltages are
therefore assumed perfectly sinusoidal. Indeed, we can express ǧ(t) as Gme
−j(ωt+φ)
, which is

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 9
the case of all currents and voltages of any circuit. Therefore we can say that all quantities are
expressed by:

ǧ(t) = √2e
jωtGe
jφ = √2e
jωtG̅

With

G̅ = Ge
jφ

Finally, we can consider that the new magnitude G̅ (figure opposite)
represents g(t). In order to return to the instantaneous value, we can simply write :

g(t) = Im [√2e
jωtG̅]

Important !
The use of non-temporal complex notation should not make us lose sight of the fact that
the time factor is implicit in complex expressions. During a derivation with respect to
time, it will be necessary to multiply the expression by jω and during an integration, it
will be necessary to divide by jω. Indeed, all sinusoidal quantities are expressed as :
g(t) = Gme
j(ωt+φ)
. Consequently, we obtain :

∂g̃(t)
∂t = jωg̃(t) and ∫ g̃(t)dt =
1
jω
g̃(t)

III-Representation of currents and voltages in elementary circuits
In the following examples, we consider applying complex numbers to electrical circuits. It will involve
determining the current absorbed by an electrical circuit supplied by a source assumed to be perfectly
sinusoidal.

v(t) = V√2sin(ωt + φv

) ⟺ V̅ = Ve
jφV
Then we obtain a sinusoidal current referred to as j(t) given by :

j(t) = J√2sin (ωt + φj

) ⟺ J̅= Je
jφJ

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 10
The phase shift φ between the applied voltage and the absorbed current is defined as:

φ = φJ − φV

Note :
If we choose the opposite then all the phase shifts will have an inversed sign, which refers to
which parameter is ahead and which is not.

1. Purely resistive circuit
   Consider a circuit with resistor R, connected under the
   sinusoidal voltage v(t). According to Ohm's law, the
   voltage applied to a resistor is determined at each instant
   by the expression:

v(t) = Rj(t)
which in the complex domain corresponds to:

V̅ = RJ
̅

Consequently, we have :

J
̅=
V̅
R̅

V
R
e
jφv = Je
φJ

Where

J =
V
R
and φv = φJ

Therefore, we have: φ = φJ − φV. The two representations, on the complex and then graphic level a

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 11
Current and voltage are therefore in phase (cancel each other out at the same time and reach
their extreme values at the same time).
It is also important to emphasize that this predictable result does not depend on the
chosen value of φv

2)Purely inductive circuit
There are no purely inductive circuits in nature.
The study of a pure inductor based circuit is a scientific
hypothesis that allows us to get an idea of the properties
of an inductor with an internal resistance.
The variation of the current flowing through a circuit
containing an inductor L causes the appearance
of a self-induction emp e(t), according to Lenz's law.
This is opposite to the variation of the current and is expressed as:

e(t) = −L
∂j
∂t

With resepect to the supply voltage , we can write :

v(t) = −e(t) = L
∂j
∂t

Which in the complex domain coresponds to :

V̅ = jLωJ
̅

Consequently :

J
̅=
V̅
jLω

V
Lω e
jφVe
−j
π
2 = Je
jφJ

Where :

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 12

J =
V
Lω
and φJ = φV −
π
2
⟹ φ = φJ − φV = −
π
2

The two representations, on the complex and then graphic level, are:

From the Figure above, we can see that the current presents a delay compared to the voltage.
Otherwise stated, the voltage reaches its maximum value before the current does.
Furthermore, the phase shift observed and expressed as an angle exactly equals π
2

Note :

It should also be noted that angles must be taken 'with their signs'. In the complex
representation (figure above), we see that φV is positive while φJ

is negative. The

phase-shift φ in this case becomes negative and equals −
π
2

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 13
3)Purely capacitive circuit
In a circuit with a capacitor C connected under a
variable voltage, there is a permanent movement of
electric charges. When the voltage increases, the
capacitor 'charges' and when the voltage decreases, it
'discharges', etc.
At every moment we have:

j =
dq
dt = C
dV
dt

Consequently , in the complex domain, we obtain :
J̅= V̅jCω = VCωe
jφVe
j
π
2 = Je
jφJ

Where

J = VCω

φJ = φV +
π
2
⟹ φ = φJ − φV = +
π
2

The two representations, on the complex and then graphic level, are:

From the above, we see that, unlike pure inductance, the current is 'ahead' of the voltage
(the voltage reaches its extremum after the current). The phase shift observed is exactly

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 14
π
2
. In the complex representation (figure above), we see that both φJand φV are positive.
The phase shift becomes positive and equals +
π
2

Important ! :
In electronics, for very high frequencies, the capacitors, even of very low value, offers only a
very low impedance to the alternating current: it is equivalent to a short circuit, unlike the
inductance which behaves like an open circuit at very high frequencies. We can clearly see
that the respective characteristics of the capacitor and the inductor go in opposite directions
since:
If f ↗⟹Lω↗ and 1
Cω
↘
4)RLC circuit
Consider an alternating current circuit
comprising a resistor, an inductor and a
capacitor connected in series. These
elements are represented separately,
located on distinct sections.
each element of the circuit can have not one
, but two or even three properties.
The methods used previously can be used to solve any electrical circuit.
We apply Kirchoff's law instantaneously:

v = Rj + L
dj
dt +
1
C
∫ jdt

Consequently we have :
V̅ = RJ
̅+ jLωJ
̅+
J
̅
jCω
= [R + j (Lω −
1
Cω)]J̅

Let

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 15

X = [(Lω −
1
Cω)]

X represents the circuit Reactance
We can write Z̅= [R + jX] = Ze
jφZ

Z̅represents the complex impedance of the electrical circuit, and we have:

Z = √R2 + (Lω −
1
Cω)
2
and Arg(Z̅) = arctg
Lω−
1
Cω
R

The current is written as :
J̅=
V̅
Z̅

Ve
jφV
Ze
jφZ

V
Z
e
φV−φZ = Je
jφJ

With J =
V
Z

V
√R2+X2
and φJ = φV − Artcg X
R

φ = φJ − φV = −Arctg
X
R

We also have

Z̅= Ze
jφZ =
V̅
J̅

V
J
e
(φV−φJ)

Consequently ,we always have

φZ = −φ
The representation in the complex domain is the following :

Ecole Nationale Polytechnique Alger Electricity/ Teacher: N.Nouar/

Chapter 1 Kirchhoff Laws in Frequency domain 16
• The figure on the left represents the case where the current lags behind the voltage
(inductive circuit), which leads to a negative value of φ.
• The figure on the right represents the case where the current is ahead of the
voltage (capacitive circuit), which leads to a positive value of φ.

This is due to the convention initially chosen when we defined φ as being :

φ = φJ − φV

Therefore it is important to note that :
• If φ0⟹Lω>
1
Cω
⟹ The circuit is inductive

• If φ>0⟹X