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Electrical Systems 2
Basilio Bona
DAUIN – Politecnico di Torino
Semester 1, 2016-17
B. Bona (DAUIN)
Electrical Systems 2
Semester 1, 2016-17
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Generalized Coordinates in Electrical Systems
The Lagrange approach to electrical circuits on the other hand requires
the definition of the kinetic co-energies and potential energies, and
consequently it is necessary to define a different set of generalized
coordinates and velocities.
Two formulations are possible:
one is called charge formulation and is based on generalized charge
coordinates,
the other is called flux formulation and is based on generalized flux
coordinates.
In the first case the generalized coordinates and velocities are charges and
currents, in the second case fluxes and voltages.
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According to the choice made, the kinetic co-energies and the potential
energies will be different
Given the power P(t) = i(t)e(t), the energy or work is given by
Z t
W (t) =
Z t
P(τ)dτ =
0
e(τ)i(τ)dτ
0
This relation is used to define the Lagrange state function L (q, q̇), as
specified in the following.
Unless otherwise stated, we will consider only ideal linear circuits, i.e.,
circuits where all the involved the components are linear and ideal.
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Generalized charge coordinates
The generalized coordinates are the charges q(t) stored inside the
capacitive components of the circuit.
The generalized velocities are the time derivatives of the charges, i.e., the
currents flowing into the capacitors
i(t) =
dq(t)
= q̇(t).
dt
The voltage across the capacitive component is proportional to the charge
e(t) =
B. Bona (DAUIN)
1
q(t)
C
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The electrostatic energy stored in the capacitive element, also called the
capacitive energy Wc (q) is defined as
Z t
Wc (q(t)) =
Z q
e(q) i dt =
e(q)dq
0
0
The capacitive co-energy Wc∗ (q) represents the electrostatic energy
expressed as a function of the voltage e(t).
This energy has no clear physical significance, as in the mechanical case,
but is useful for the definition of the Lagrange function.
The co-energy Wc∗ (e) is
Wc∗ (e) = qe − Wc (q) =
Z e
q(e)de
0
Both the energy and the co-energy do not depend on time. A time
inversion, i.e., time flowing anti-causally in the negative direction, does not
affect the results; from a physical point of view this means that capacitive
energy/co-energy can be stored or released at will to and from an ideal
capacitive element in the circuit.
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The energy differential is
dWc (q) = e(q) dq
and the co-energy differential is
dWc∗ (e) = q(e) de
from which the following relations are established
∂ Wc (q(e))
= e(t)
∂q
B. Bona (DAUIN)
and
Electrical Systems 2
∂ Wc∗ (e(q))
= q(t)
∂e
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If the element is linear with respect to the capacity i.e., q(e) = Ce and C
is a constant, the well known relations follow
Wc∗ (e)
Z e
=
and
Z e
q(e)de =
0
0
Z q
Wc (q) =
e(q)dq =
0
Ce de =
Z q
q
0
C
dq =
1 2
Ce
2
1 q2
2C
In this case the capacitive energy and co-energy are equal, and the
characteristic function is a straight line.
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Generalized flux coordinates
The generalized coordinates are the fluxes λ (t) generated inside the
inductive components of the circuit.
The generalized velocities are the time derivatives of the fluxes, i.e., the
voltages
dλ (t) ˙
e(t) =
= λ (t).
dt
The current flowing into the inductive component is proportional to the
flux
1
i(t) = λ (t)
L
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The electromagnetic energy stored in the inductive element, also called the
inductive energy Wi (λ ), is defined as
Z t
Wi (λ ) =
Z λ
i(λ ) e dt =
0
i(λ )dλ
0
It is also possible to express the inductive co-energy Wi∗ (i), i.e., the
electromagnetic energy expressed as a function of the current i(t).
This energy has no clear physical significance, as in the mechanical case,
but nevertheless is useful for the definition of the Lagrange function.
The co-energy Wi∗ (i) is defined as
Wi∗ (i) = iλ − Wi (λ ) =
Z i
λ (i)di
0
Both the energy and the co-energy do not depend on time. A time
inversion does not affect the results; from a physical point of view this
means that capacitive energy/co-energy can be stored or released at will
to and from an ideal inductive element in the circuit.
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The energy differential is
dWi (λ ) = i dλ
while the co-energy differential is
dWi∗ (i) = λ di
from which the following relations are established
∂ Wi (λ )
= i(t)
∂λ
B. Bona (DAUIN)
e
∂ Wi∗ (i)
= λ (t)
∂i
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If the element is linear with respect to the inductance i.e., λ (i) = Li and L
is a constant, the well known relations follow
Wi∗ (i)
Z i
=
Z i
λ (i)di =
0
Li di =
0
1 2
Li
2
and
Z λ
Wi (λ ) =
i(λ )dλ =
0
Z λ
λ
0
L
dλ =
1 λ2
2 L
In this case the inductive energy and co-energy are equal, and the
characteristic function is a straight line.
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Lagrange Function in Electromagnetic Systems
To avoid confusion between the symbol for generalized coordinates and
the symbol for charges, we will use the symbol ξ and ξ˙ for generalized
coordinates and generalized velocities, respectively.
In the electromagnetic systems the Lagrange function Le , is the difference
between the “kinetic” co-energy Ke∗ (ξ , ξ˙ ) and the “potential” energy
Pe (ξ ):
Le (ξ , ξ˙ ) = Ke (ξ , ξ˙ ) − Pe (ξ )
The energy/co-energy functions are different if we use the flux or the
charge coordinates.
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Charge coordinates and Lagrange function
Using the charge coordinates ξ = q and velocities ξ˙ = q̇ = i, we write
Le,charge (ξ , ξ˙ ) = Wi∗ (q̇) − Wc (q)
where the “kinetic” co-energy coincides with the inductive co-energy
stored into the inductive element:
Ke∗ (ξ , ξ˙ ) ≡ Wi∗ (q̇) = Wi∗ (i)
and the “potential” energy coincides with the capacitive energy stored into
the capacitive elements
Pe (ξ ) ≡ Wc (q)
We notice that the kinetic co-energy does not depend on the generalized
coordinates, but only on the generalized velocities q̇ = i.
The vectors q and i are the collection of all the charges on the capacitive
elements and all the currents flowing into the inductive elements.
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Since the energies are additive, assuming a linear circuit with Ni inductors
and Nc capacitors, we can write
(
)
Ni
1
∗
∗
2
Ke (ξ , ξ˙ ) = Wi (i) =
∑ Lk ik
2 k=1
where ik is the current flowing into the k-th inductive component with
inductance Lk . Similarly
(
)
1 Nc qk2
Pe (ξ ) = Wc (q) =
∑ Ck
2 k=1
where qk is the charge stored into the k-th capacitive component with
capacity Ck .
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Flux coordinates and Lagrange function
Using the flux coordinates ξ = λ and velocities ξ˙ = λ˙ = e, we can write
Le,flux (ξ , ξ˙ ) = Wc∗ (λ˙ ) − Wi (λ )
where the “kinetic” co-energy coincides with the capacitive co-energy
stored into the capacitive element:
Ke (λ˙ ) ≡ Wc∗ (λ˙ ) = Wc∗ (e)
and the “potential” energy coincides with the inductive energy stored into
the inductive elements:
Pe (λ ) ≡ Wi (λ )
We notice that the kinetic co-energy does not depend on the generalized
coordinates, but only on the generalized velocities λ˙ = e.
The vectors λ and e are, respectively, the collection of all the fluxes on the
inductive elements and all the voltages across the capacitive elements.
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Since the energies are additive, assuming again a linear circuit with Ni
inductors and Nc capacitors, we can write
(
)
Nc
1
2
∗
∗
Ke (ξ , ξ˙ ) = Wc (e) =
∑ Ck e k
2 k=1
where ek is the voltage across the k-th capacitive component with
capacity Ck . Similarly
(
)
1 Ni λk2
Pe (ξ ) = Wi (λ ) =
∑ Lk
2 k=1
where λk is the flux across the k-th inductive component with inductance
Lk .
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In conclusion
1
Le,charge =
2
and
1
Le,flux =
2
(
Ni
Nc
q2
∑ Lk ik2 − ∑ Ckk
k=1
k=1
(
Nc
Ni
λ2
∑ Ck ek2 − ∑ Lkk
k=1
k=1
)
)
Notice that, for ideal linear components,
Le,flux = −Le,charge
or
Le,flux + Le,charge = 0
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Dissipative Elements
The dissipative elements are represented by the resistors. The dissipated
power is
e 2 (t)
Pr = R i 2 (t) =
R
In order to include the dissipative effects into the Lagrange equations, it is
customary, as it has been in mechanical systems with the definition of the
dissipative function, to define an electric dissipative function De as
follows
1 Nr
1 Nr
De,charge (q̇) = ∑ Rk q̇k2 = ∑ Rk ik2
2 k=1
2 k=1
or
1 Nr 1 ˙ 2 1 Nr 1 2
De,flux (λ˙ ) = ∑
λ = ∑
e
2 k=1 Rk k
2 k=1 Rk k
where Nr is the total number of dissipative elements in the circuit, i.e., the
resistors, and Rk their resistance.
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Generalized forces - charge approach
Using charge coordinates, the k-th generalized electric force is a
generalized voltage, indicates as Fe,k = Ek and can be computed
considering the set of k = 1, . . . , NE ideal voltage generators Ek (t) present
in the circuit, starting from the general expression of the virtual work
δ W nc =
NE
∑ Fknc δ qk
k=1
and the virtual work related to the electrical system
NE
dW =
NE
we can write
δ W nc =
k=1
k=1
k=1
NE
NE
∑
Ek (t)δ qk =
k=1
B. Bona (DAUIN)
NE
∑ Pk dt = ∑ ek ik dt = ∑ ek dqk
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∑ Ek δ qk
k=1
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In order to compute each Ek it is necessary to consider the currents
flowing into the ideal voltage generators, compute the product (voltage ×
current) and assign to each virtual charge variation δ qk the contribution
of the related voltages.
If ideal current generators are present, they do not contribute to the
generalized voltages, but act as constraints on the node current
summation.
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Generalized forces - flux approach
Using flux coordinates, the k-th generalized electric force is a generalized
current, indicates as Fe,k = Ik and can be computed considering the set
of k = 1, . . . , NI ideal current generators Ik (t) present in the circuit,
starting from the general expression of the virtual work
δ W nc =
NI
∑ Fknc δ qk
k=1
and the virtual work related to the electrical system
NI
dW =
NI
we can write
δ W nc =
k=1
k=1
k=1
NI
NI
∑
Ik (t)δ λk =
k=1
B. Bona (DAUIN)
NI
∑ Pk dt = ∑ ik ek dt = ∑ ik dλk
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∑ Ik δ λk
k=1
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In order to compute each Ik it is necessary to consider the voltage across
the ideal current generators, compute the product (voltage × current) and
assign to each virtual flux variation δ λk the contribution of the related
currents.
If ideal voltage generators are present, they do not contribute to the
generalized currents, but act as constraints on the mesh voltage
summation.
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Lagrange Equations in Electromagnetic Systems
After having chosen the n generalized coordinates ξ and velocities ξ˙ ,
either charge or flux coordinates, we write the n Lagrange equations
!
d ∂ Le
∂ Le ∂ De
−
+
= Fk
k = 1, . . . , n
˙
dt ∂ ξk
∂ ξk
∂ ξ˙k
If the k-th coordinate involves both capacitive and inductive elements, the
corresponding equation will in general be a second-order differential
¨k .
equation expressed in terms of q̈k or λ
In all the other cases, e.g., when only resistive elements are present in the
circuit, a first-order differential equation or an algebraic equation will
result.
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If we collect all equations and organize them in matrix form, we can write
the following second-order linear matrix equations
Zq̈(t) + Rq̇(t) + Qq(t) = E (t)
or
Yλ¨ (t) + Gλ˙ (t) + Λλ (t) = I (t)
where Z, R, Q, Y, G, Λ are suitable n × n matrices
Z
inductance matrix
Y = Z−1 reluctance matrix
R
G=R
resistance matrix
−1
Q
capacitance matrix
−1
Λ=Q
B. Bona (DAUIN)
conductance matrix
elastance matrix
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It is often convenient to rewrite the equations as functions of the
generalized velocities
ξ˙ = {q̇k = ik , λ˙ k = ek }
In this case we obtain mixed integro-differential equations. If we define the
voltage across a capacitor and the current in an inductor respectively as
Z
vCk =
1
qk dt
Ck
Z
iLk =
1
λk dt
Lk
the Lagrange equations become
d
i(t) + Ri(t) + vC (t) = E (t)
dt
(1)
d
e(t) + Ce(t) + iL (t) = I (t)
dt
(2)
Z
or
Y
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Comments
The Lagrange approach, that makes use of the energy formulation to
obtain the differential equations, does not represent the tool commonly
adopted for linear circuits analysis: the Kirchhoff equations applied to
meshes and nodes is by far the preferred tool, in conjunction with the
Laplace transforms.
Nevertheless there are some advantages related to the use of the
Lagrangian approach
has a general validity, independent of the specific field of application,
mechanical or electrical;
can be easily applied when nonlinear components are present;
is very beneficial when modelling systems where electrical and
mechanical parts interact.
is a good starting point for state-variable equation.
Furthermore, the Lagrange approach allows to introduce some analogies
between electrical and mechanical quantities.
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Electrical and mechanical analogies
Recalling the definition of the k-th generalized momentum
µk =
∂L
∂ q̇k
and the generalized velocities
q̇k (t) = ik (t)
λ˙ k (t) = ek (t)
we see that the power P(t) can be expressed as the product of two
abstract quantities, the first one called effort s(t), the other called flow
φ (t), not to be confused with the flux of the magnetic field Φ(t),
P(t) = s(t)φ (t)
Using these two quantities it is possible to define a number of analogies
between the mechanical and the electrical variables, making possible to
interpret a quantity in one class with the analogous quantity in the other
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Type
ξ
ξ˙
s
φ
µ
Transl
x
ẋ
f = mẍ
ẋ
mẋ
Rot
θ
θ̇
τ = Γ θ̈
θ̇
Charge
q
i
e = Lq̈
Flux
λ
e
¨
i = Cλ
K
P
D
F
1 2
mẋ
2
1
kl ∆x 2
2
1
βl ẋ 2
2
f
Γ θ̇
1 2
Γ θ̇
2
1
ka ∆θ 2
2
1
βa θ̇ 2
2
τ
q̇ = i
Li = λ
1 2
Li
2
1 2
q
2C
1 2
R q̇
2
E
λ˙ = e
Ce = q
1 2
Ce
2
1 2
λ
2L
1 ˙2
λ
2R
I
∗
Table: Electro-mechanical analogies for one-dimensional linear ideal elements.
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