Chapter 3: Electromechanical-Energy-Conversion
Principles
This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy
- The electromechanical-energy-conversion process takes place through the medium of the electric or magnetic field of the conversion device of which the structures depend on their respective functions.
- Transducers: microphone, pickup, sensor, loudspeaker
- Force producing devices: solenoid, relay, electromagnet
- Continuous energy conversion equipment: motor, generator
This chapter is devoted to the principles of electromechanical energy conversion and the analysis of the devices accomplishing this function. Emphasis is placed on the analysis of systems that use magnetic fields as the conversion medium.
- The concepts and techniques can be applied to a wide range of engineering situations involving electromechanical energy conversion.
- Based on the energy method, we are to develop expressions for forces and torques in magnetic-field-based electromechanical systems.
§3.1 Forces and Torques in Magnetic Field Systems
- The Lorentz Force Law gives the force F on a particle of charge q in the presence of electric and magnetic fields.
F=q(E+v×B)F=q(E+v×B) size 12{F=q \( E+v times B \) } {} (3.1)
F : newtons, q : coulombs, E : volts/meter, B : telsas, v : meters/second
- In a pure electric-field system,
F qE (3.2)
- In pure magnetic-field systems,
F=q(v×B)F=q(v×B) size 12{F=q \( v times B \) } {}(3.3)
Figure 3.1 Right-hand rule for F=( q x v) B .
- For situations where large numbers of charged particles are in motion,
Fv=ρ(E+v×B)Fv=ρ(E+v×B) size 12{F rSub { size 8{v} } =ρ \( E+v times B \) } {}(3.4)
J=ρvJ=ρv size 12{J=ρv} {} (3.5)
Fv=J×BFv=J×B size 12{F rSub { size 8{v} } =J times B} {} (3.6)
(charge density): coulombs/
m3m3 size 12{m rSup { size 8{3} } } {}, F (force density): newtons/
m3m3 size 12{m rSup { size 8{3} } } {},
J=ρvJ=ρv size 12{J=ρv} {} (current density): amperes/
m2m2 size 12{m rSup { size 8{2} } } {}.
- Most electromechanical-energy-conversion devices contain magnetic material.
- Forces act directly on the magnetic material of these devices which are constructed of rigid, nondeforming structures.
- The performance of these devices is typically determined by the net force, or torque, acting on the moving component. It is rarely necessary to calculate the details of the internal force distribution.
- Just as a compass needle tries to align with the earth’s magnetic field, the two sets of fields associated with the rotor and the stator of rotating machinery attempt to align, and torque is associated with their displacement from alignment.
- In a motor, the stator magnetic field rotates ahead of that of the rotor, pulling on it and performing work.
- For a generator, the rotor does the work on the stator.
- Based on the principle of conservation of energy: energy is neither created nor destroyed; it is merely changed in form.
- Fig. 3.2(a): a magnetic-field-based electromechanical-energy-conversion device.
- A lossless magnetic-energy-storage system with two terminals
- The electric terminal has two terminal variables: e (voltage), i (current).
- The mechanical terminal has two terminal variables:
ffldffld size 12{f rSub { size 8{ ital "fld"} } } {}(force), x (position)
- The loss mechanism is separated from the energy-storage mechanism.
–Electrical losses: ohmic losses…
–Mechanical losses: friction, windage…
Fig. 3.2(b): a simple force-producing device with a single coil forming the electric terminal, and a movable plunger serving as the mechanical terminal.
- The interaction between the electric and mechanical terminals, i.e. the electromechanical energy conversion, occurs through the medium of the magnetic stored energy.
Figure 3.2(a) Schematic magnetic-field electromechanical-energy-conversion device;
(b) simple force-producing device.
- WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {}: the stored energy in the magnetic field
dWflddt=ei−fflddxdtdWflddt=ei−fflddxdt size 12{ { { ital "dW" rSub { size 8{"fld"} } } over { ital "dt"} } = ital "ei" - f rSub { size 8{"fld"} } { { ital "dx"} over { ital "dt"} } } {} (3.7)
e=dλdte=dλdt size 12{e= { {dλ} over { ital "dt"} } } {} (3.8)
dWfld=idλ−fflddxdWfld=idλ−fflddx size 12{ ital "dW" rSub { size 8{"fld"} } = ital "id"λ - f rSub { size 8{"fld"} } ital "dx"} {} (3.9)
- Equation (3.9) permits us to solve for the force simply as a function of the flux and the mechanical terminal position x .
- Equations (3.7) and (3.9) form the basis for the energy method.
§3.2 Energy Balance
- Consider the electromechanical systems whose predominant energy-storage mechanism is in magnetic fields. For motor action, we can account for the energy transfer as
Energy inputform electricsource=Mechanicalenergyouput+Increase in energystored in magneticfield+Energyconvertedinto heatEnergy inputform electricsource=Mechanicalenergyouput+Increase in energystored in magneticfield+Energyconvertedinto heat size 12{ left [ matrix {
"Energy input" {} ##
"form electric" {} ##
"source"
} right ]= left [ matrix {
"Mechanical" {} ##
"energy" {} ##
"ouput"
} right ]+ left [ matrix {
"Increase in energy" {} ##
"stored in magnetic" {} ##
"field"
} right ]+ left [ matrix {
"Energy" {} ##
"converted" {} ##
"into heat"
} right ]} {} (3.10)
{}
- Note the generator action.
- The ability to identify a lossless-energy-storage system is the essence of the energy method.
- This is done mathematically as part of the modeling process.
- For the lossless magnetic-energy-storage system of Fig. 3.3(a), rearranging (3.9) in form of (3.10) gives
dWelec=dWmech+dWflddWelec=dWmech+dWfld size 12{ ital "dW" rSub { size 8{ ital "elec"} } = ital "dW" rSub { size 8{ ital "mech"} } + ital "dW" rSub { size 8{ ital "fld"} } } {} (3.11)
where
dWelec=idλdWelec=idλ size 12{ ital "dW" rSub { size 8{ ital "elec"} } = ital "id"λ} {}differential electric energy input
dWmech=fflddxdWmech=fflddx size 12{ ital "dW" rSub { size 8{ ital "mech"} } =f rSub { size 8{ ital "fld"} } ital "dx"} {}differential mechanical energy output
dWflddWfld size 12{ ital "dW" rSub { size 8{ ital "fld"} } } {}differential change in magnetic stored energy
- Here e is the voltage induced in the electric terminals by the changing magnetic stored energy. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals.
dWelec=eidtdWelec=eidt size 12{ ital "dW" rSub { size 8{ ital "elec"} } = ital "eidt"} {} (3.12)
- The basic energy-conversion process is one involving the coupling field and its action and reaction on the electric and mechanical systems.
- Combining (3.11) and (3.12) results in
dWelec=eidt=dWmech+dWflddWelec=eidt=dWmech+dWfld size 12{ ital "dW" rSub { size 8{ ital "elec"} } = ital "eidt"= ital "dW" rSub { size 8{ ital "mech"} } + ital "dW" rSub { size 8{ ital "fld"} } } {} (3.13)
§3.3 Energy in Singly-Excited Magnetic Field Systems
- We are to deal energy-conversion systems: the magnetic circuits have air gaps between the stationary and moving members in which considerable energy is stored in the magnetic field.
- This field acts as the energy-conversion medium, and its energy is the reservoir between the electric and mechanical system.
- Fig. 3.3 shows an electromagnetic relay schematically. The predominant energy storage occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap.
Figure 3.3Schematic of an electromagnetic relay.
λ=L(x)Iλ=L(x)I size 12{λ=L \( x \) I} {}(3.14)
dWmech=fflddxdWmech=fflddx size 12{ ital "dW" rSub { size 8{ ital "mech"} } =f rSub { size 8{ ital "fld"} } ital "dx"} {}(3.15)
dWfld=idλ−fflddxdWfld=idλ−fflddx size 12{ ital "dW" rSub { size 8{ ital "fld"} } = ital "id"λ - f rSub { size 8{ ital "fld"} } ital "dx"} {}(3.16)
- WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} is uniquely specified by the values of
λλ size 12{λ} {} and x . Therefore,
λλ size 12{λ} {}and x are referred to as state variables.
- Since the magnetic energy storage system is lossless, it is a conservative system.
WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} is the same regardless of how
λλ size 12{λ} {} and x are brought to their final values. See Fig. 3.4 where tow separate paths are shown.
Figure 3.4Integration paths for Wfld
Wfld(λ0,x0)=∫path 2adWfld+∫path 2bdWfldWfld(λ0,x0)=∫path 2adWfld+∫path 2bdWfld size 12{W rSub { size 8{"fld"} } \( λ rSub { size 8{0} } ,x rSub { size 8{0} } \) = Int cSub { size 8{"path 2a"} } { ital "dW" rSub { size 8{"fld"} } } + Int cSub { size 8{"path 2b"} } { ital "dW" rSub { size 8{"fld"} } } } {} (3.17)
On path 2a,
dλ=0dλ=0 size 12{dλ=0} {} and
ffld=0ffld=0 size 12{f rSub { size 8{ ital "fld"} } =0} {}. Thus,
dWfld=0dWfld=0 size 12{ ital "dW" rSub { size 8{ ital "fld"} } =0} {} on path 2a.
On path 2b, dx 0 .
Therefore, (3.17) reduces to the integral of
idλidλ size 12{ ital "id"λ} {} over path 2b.
Wfld(λ0,x0)=∫0λ0i(λ,x0)dλWfld(λ0,x0)=∫0λ0i(λ,x0)dλ size 12{W rSub { size 8{"fld"} } \( λ rSub { size 8{0} } ,x rSub { size 8{0} } \) = Int rSub { size 8{0} } rSup { size 8{λ rSub { size 6{0} } } } {i \( λ,x rSub { size 8{0} } \) } dλ} {} (3.18)
For a linear system in which
λλ size 12{λ} {} is proportional to i , (3.18) gives
Wfld(λ,x)=∫0λi(λ',x)dλ'=∫0λλ'L(x)dλ'=12λ2L(x)Wfld(λ,x)=∫0λi(λ',x)dλ'=∫0λλ'L(x)dλ'=12λ2L(x) size 12{W rSub { size 8{"fld"} } \( λ,x \) = Int rSub { size 8{0} } rSup { size 8{λ} } {i \( { {λ}} sup { ' },x \) } d { {λ}} sup { ' }= Int rSub { size 8{0} } rSup { size 8{λ} } { { { { {λ}} sup { ' }} over {L \( x \) } } } d { {λ}} sup { ' }= { {1} over {2} } { {λ rSup { size 8{2} } } over {L \( x \) } } } {} (3.19)
- V : the volume of the magnetic field
Wfld=∫v(∫0BH.dB')dVWfld=∫v(∫0BH.dB')dV size 12{W rSub { size 8{ ital "fld"} } = Int rSub { size 8{v} } { \( Int rSub { size 8{0} } rSup { size 8{B} } {H "." d { {B}} sup { ' }} \) } ital "dV"} {} (3.20)
If
B=μHB=μH size 12{B=μH} {} ,
Wfld=∫v(B22μ)dVWfld=∫v(B22μ)dV size 12{W rSub { size 8{ ital "fld"} } = Int rSub { size 8{v} } { \( { {B rSup { size 8{2} } } over {2μ} } \) } ital "dV"} {} (3.21)
§3.4 Determination of Magnetic Force and Torque form Energy
- The magnetic stored energy
WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} is a state function, determined uniquely by the values of the independent state variables
λλ size 12{λ} {} and x.
dWfld(λ,x)=idλ−fflddxdWfld(λ,x)=idλ−fflddx size 12{ ital "dW" rSub { size 8{"fld"} } \( λ,x \) = ital "id"λ - f rSub { size 8{"fld"} } ital "dx"} {} (3.22)
dF(x1,x2)=∂F∂x1∣x2dx1+∂F∂x2∣x1dx2dF(x1,x2)=∂F∂x1∣x2dx1+∂F∂x2∣x1dx2 size 12{ ital "dF" \( x rSub { size 8{1} } ,x rSub { size 8{2} } \) = { { partial F} over { partial x rSub { size 8{1} } } } \rline rSub { size 8{x rSub { size 6{2} } } } ital "dx" rSub {1} size 12{+ { { partial F} over { partial x rSub {2} } } \rline rSub {x rSub { size 6{1} } } } size 12{ ital "dx" rSub {2} }} {} (3.23)
dWfld(λ,x)=∂Wfld∂λ∣xdλ+∂Wfld∂x∣λdxdWfld(λ,x)=∂Wfld∂λ∣xdλ+∂Wfld∂x∣λdx size 12{ ital "dW" rSub { size 8{"fld"} } \( λ,x \) = { { partial W rSub { size 8{"fld"} } } over { partial λ} } \rline rSub { size 8{x} } dλ+ { { partial W rSub { size 8{"fld"} } } over { partial x} } \rline rSub { size 8{λ} } ital "dx"} {} (3.24)
Comparing (3.22) with (3.24) gives (3.25) and (3.26):
i=∂Wfld(λ,x)∂λ∣xi=∂Wfld(λ,x)∂λ∣x size 12{i= { { partial W rSub { size 8{"fld"} } \( λ,x \) } over { partial λ} } \rline rSub { size 8{x} } } {} (3.25)
ffld=−∂Wfld(λ,x)∂x∣λffld=−∂Wfld(λ,x)∂x∣λ size 12{f rSub { size 8{ ital "fld"} } = - { { partial W rSub { size 8{ ital "fld"} } \( λ,x \) } over { partial x} } \rline rSub { size 8{λ} } } {}(3.26)
- Once we know
WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} as a function of
λλ size 12{λ} {} and as a function of
λλ size 12{λ} {} and i(
λλ size 12{λ} {}, x) .
- Equation (3.26) can be used to solve for the mechanical force
ffld(λ,x)ffld(λ,x) size 12{f rSub { size 8{ ital "fld"} } \( λ,x \) } {}.The partial derivative is taken while holding the flux linkages
λλ size 12{λ} {} constant.
- For linear magnetic systems for which
λ=L(x)iλ=L(x)i size 12{λ=L \( x \) i} {}, the force can be found as
ffld=−∂∂x12λ2L(x)∣λ=λ22L(x)2dL(x)dxffld=−∂∂x12λ2L(x)∣λ=λ22L(x)2dL(x)dx size 12{f rSub { size 8{"fld"} } = - { { partial } over { partial x} } left ( { {1} over {2} } { {λ rSup { size 8{2} } } over {L \( x \) } } right ) \rline rSub { size 8{λ} } = { {λ rSup { size 8{2} } } over {2L \( x \) rSup { size 8{2} } } } { { ital "dL" \( x \) } over { ital "dx"} } } {} (3.27)
{}ffld=i22dL(x)dxffld=i22dL(x)dx size 12{f rSub { size 8{ ital "fld"} } = { {i rSup { size 8{2} } } over {2} } { { ital "dL" \( x \) } over { ital "dx"} } } {} (3.28)
- For a system with a rotating mechanical terminal, the mechanical terminal variables become the angular displacement
θθ size 12{θ} {} and the torque
TfldTfld size 12{T rSub { size 8{ ital "fld"} } } {} .
dWfld(λ,θ)=idλ−TflddθdWfld(λ,θ)=idλ−Tflddθ size 12{ ital "dW" rSub { size 8{"fld"} } \( λ,θ \) = ital "id"λ - T rSub { size 8{"fld"} } dθ} {} (3.29)
Tfld=−∂Wfld(λ,θ)∂θ∣λTfld=−∂Wfld(λ,θ)∂θ∣λ size 12{T rSub { size 8{ ital "fld"} } = - { { partial W rSub { size 8{ ital "fld"} } \( λ,θ \) } over { partial θ} } \rline rSub { size 8{λ} } } {} (3.30)
- For linear magnetic systems for which
λ=L(θ)iλ=L(θ)i size 12{λ=L \( θ \) i} {} :
Wfld(λ,θ)=12λ2L(θ)Wfld(λ,θ)=12λ2L(θ) size 12{W rSub { size 8{ ital "fld"} } \( λ,θ \) = { {1} over {2} } { {λ rSup { size 8{2} } } over {L \( θ \) } } } {} (3.31)
Tfld=−∂∂θ12λ2L(θ)∣λ=12λ2L(θ)2dL(θ)dθTfld=−∂∂θ12λ2L(θ)∣λ=12λ2L(θ)2dL(θ)dθ size 12{T rSub { size 8{"fld"} } = - { { partial } over { partial θ} } left ( { {1} over {2} } { {λ rSup { size 8{2} } } over {L \( θ \) } } right ) \rline rSub { size 8{λ} } = { {1} over {2} } { {λ rSup { size 8{2} } } over {L \( θ \) rSup { size 8{2} } } } { { ital "dL" \( θ \) } over {dθ} } } {} (3.32)
Tfld=i22dL(θ)dθTfld=i22dL(θ)dθ size 12{T rSub { size 8{ ital "fld"} } = { {i rSup { size 8{2} } } over {2} } { { ital "dL" \( θ \) } over {dθ} } } {} (3.33)
§3.5 Determination of Magnetic Force and Torque from Coenergy
- Recall that in §3.4, the magnetic stored energy
WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} is a state function, determined uniquely by the values of the independent state variables
λλ size 12{λ} {} and x .
dWfld(λ,x)=idλ−fflddxdWfld(λ,x)=idλ−fflddx size 12{ ital "dW" rSub { size 8{ ital "fld"} } \( λ,x \) = ital "id"λ - f rSub { size 8{ ital "fld"} } ital "dx"} {} (3.34)
dWfld(λ,x)=∂Wfld∂λ∣xdλ+∂Wfld∂x∣λdxdWfld(λ,x)=∂Wfld∂λ∣xdλ+∂Wfld∂x∣λdx size 12{ ital "dW" rSub { size 8{ ital "fld"} } \( λ,x \) = { { partial W rSub { size 8{ ital "fld"} } } over { partial λ} } \rline rSub { size 8{x} } dλ+ { { partial W rSub { size 8{ ital "fld"} } } over { partial x} } \rline rSub { size 8{λ} } ital "dx"} {} (3.35)
i=∂Wfld(λ,x)∂λ∣xi=∂Wfld(λ,x)∂λ∣x size 12{i= { { partial W rSub { size 8{ ital "fld"} } \( λ,x \) } over { partial λ} } \rline rSub { size 8{x} } } {} (3.36)
ffld=∂Wfld(λ,x)∂x∣λffld=∂Wfld(λ,x)∂x∣λ size 12{f rSub { size 8{ ital "fld"} } = { { partial W rSub { size 8{ ital "fld"} } \( λ,x \) } over { partial x} } \rline rSub { size 8{λ} } } {} (3.37)
- Coenergy: from which the force can be obtained directly as a function of the current.The selection of energy or coenergy as the state function is purely a matter of convenience.
- The coenergy
Wfld'(i,x)Wfld'(i,x) size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) } {} is defined as a function of i and x such that
Wfld'(i,x)=iλ−Wfld(λ,x)Wfld'(i,x)=iλ−Wfld(λ,x) size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) =iλ - W rSub { size 8{ ital "fld"} } \( λ,x \) } {} (3.38)
d(iλ)=idλ=λdid(iλ)=idλ=λdi size 12{d \( iλ \) = ital "id"λ=λ ital "di"} {} (3.39)
dWfld'(i,x)=d(iλ)−dWfld(λ,x)dWfld'(i,x)=d(iλ)−dWfld(λ,x) size 12{d { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) =d \( iλ \) - ital "dW" rSub { size 8{ ital "fld"} } \( λ,x \) } {} (3.40)
dWfld'(i,x)=λdi+fflddxdWfld'(i,x)=λdi+fflddx size 12{d { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) =λ ital "di"+f rSub { size 8{ ital "fld"} } ital "dx"} {} (3.41)
- From (3.37), the coenergy
Wfld'(i,x)Wfld'(i,x) size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) } {}can be seen to be a state function of the two independent variables i and x .
dWfld'(i,x)=∂Wfld'∂i∣xdi+∂Wfld'∂x∣idxdWfld'(i,x)=∂Wfld'∂i∣xdi+∂Wfld'∂x∣idx size 12{d { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } } over { partial i} } \rline rSub { size 8{x} } ital "di"+ { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } } over { partial x} } \rline rSub { size 8{i} } ital "dx"} {} (3.42)
λ=∂Wfld'(i,x)∂i∣xλ=∂Wfld'(i,x)∂i∣x size 12{λ= { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) } over { partial i} } \rline rSub { size 8{x} } } {} (3.43)
ffld=∂Wfld'(i,x)∂x∣iffld=∂Wfld'(i,x)∂x∣i size 12{f rSub { size 8{ ital "fld"} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) } over { partial x} } \rline rSub { size 8{i} } } {} (3.44)
- For any given system, (3.26) and (3.40) will give the same result.
- By analogy to (3.18) in §3.3, the coenergy can be found as (3.41)
Wfld(λ0,x0)=∫0λ0i(λ,x0)dλWfld(λ0,x0)=∫0λ0i(λ,x0)dλ size 12{W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{0} } ,x rSub { size 8{0} } \) = Int rSub { size 8{0} } rSup { size 8{λ rSub { size 6{0} } } } {i \( λ,x rSub { size 8{0} } \) } dλ} {} (3.42)
Wfld'(i,x)=∫λ(i',x)di'Wfld'(i,x)=∫λ(i',x)di' size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) = Int rSub {} rSup {} {λ \( { {i}} sup { ' },x \) } d { {i}} sup { ' }} {} (3.43)
For linear magnetic systems for which
λ=L(x)iλ=L(x)i size 12{λ=L \( x \) i} {} ,
Wfld'(i,x)=12L(x)i2Wfld'(i,x)=12L(x)i2 size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) = { {1} over {2} } L \( x \) i rSup { size 8{2} } } {} (3.44)
ffld=i22dL(x)dxffld=i22dL(x)dx size 12{f rSub { size 8{ ital "fld"} } = { {i rSup { size 8{2} } } over {2} } { { ital "dL" \( x \) } over { ital "dx"} } } {} (3.45)
- For a system with a rotating mechanical displacement,
Wfld'(i,θ)=∫0iλ(i',θ)di'Wfld'(i,θ)=∫0iλ(i',θ)di' size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,θ \) = Int rSub { size 8{0} } rSup { size 8{i} } {λ \( { {i}} sup { ' },θ \) } d { {i}} sup { ' }} {} (3.46)
Tfld=∂Wfld'(i,θ)∂θ∣iTfld=∂Wfld'(i,θ)∂θ∣i size 12{T rSub { size 8{ ital "fld"} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,θ \) } over { partial θ} } \rline rSub { size 8{i} } } {} (3.47)
If the system is magnetically linear,
Wfld'(i,θ)=12L(θ)i2Wfld'(i,θ)=12L(θ)i2 size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,θ \) = { {1} over {2} } L \( θ \) i rSup { size 8{2} } } {} (3.48)
Tfld=i22dL(θ)dθTfld=i22dL(θ)dθ size 12{T rSub { size 8{ ital "fld"} } = { {i rSup { size 8{2} } } over {2} } { { ital "dL" \( θ \) } over {dθ} } } {} (3.49)
(3.47) is identical to the expression given by (3.33).
- In field-theory terms, for soft magnetic materials
Wfld'=∫V∫0H0BdHdVWfld'=∫V∫0H0BdHdV size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = Int rSub {V} { left ( Int rSub { size 8{0} } rSup { size 8{H rSub { size 6{0} } } } { ital "BdH"} right )} size 12{ ital "dV"}} {} (3.50)
Wfld'=∫vμH22dVWfld'=∫vμH22dV size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = Int rSub { size 8{v} } { { {μH rSup { size 8{2} } } over {2} } } ital "dV"} {} (3.51)
For permanent-magnet (hard) materials
Wfld'=∫v∫HcH0BdHdVWfld'=∫v∫HcH0BdHdV size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = Int rSub {v} { left ( Int rSub { size 8{H rSub { size 6{c} } } } rSup {H rSub { size 6{0} } } { ital "BdH"} right )} size 12{ ital "dV"}} {} (3.52)
- For a magnetically-linear system, the energy and coenergy (densities) are numerically equal:
λ2/2L=12Li2,B2/2μ=12μH2λ2/2L=12Li2,B2/2μ=12μH2 size 12{λ rSup { size 8{2} } /2L= { {1} over {2} } ital "Li" rSup { size 8{2} } ," "B rSup { size 8{2} } /2μ= { {1} over {2} } μH rSup { size 8{2} } } {}. For a nonlinear system in which
λλ size 12{λ} {} and i or B and H are not linearly proportional, the two functions are not even numerically equal.
Wfld+Wfld'=λiWfld+Wfld'=λi size 12{W rSub { size 8{ ital "fld"} } + { {W}} sup { ' } rSub { size 8{ ital "fld"} } =λi} {} (3.53)
Figure 3.5Graphical interpretation of energy and coenergy in a singly-excited system.
- Consider the relay in Fig. 3.3. Assume the relay armature is at position x so that the device operating at point a in Fig. 3.6. Note that
f
fld
=
−
∂
W
fld
(
λ
,
x
)
∂
x
∣
λ
≃
lim
Δx
→
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−
ΔW
fld
Δx
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λ
and
f
fld
=
∂
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fld
'
(
i
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x
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∂
x
∣
i
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lim
Δx
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Δ
W
fld
'
Δx
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i
f
fld
=
−
∂
W
fld
(
λ
,
x
)
∂
x
∣
λ
≃
lim
Δx
→
0
−
ΔW
fld
Δx
∣
λ
and
f
fld
=
∂
W
fld
'
(
i
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∂
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Δx
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i
size 12{f rSub { size 8{ ital "fld"} } = - { { partial W rSub { size 8{ ital "fld"} } \( λ,x \) } over { partial x} } \rline rSub { size 8{λ} } simeq {"lim"} cSub { size 8{Δx rightarrow 0} } { { - ΔW rSub { size 8{ ital "fld"} } } over {Δx} } \rline rSub { size 8{λ} } " and "f rSub { size 8{ ital "fld"} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i,x \) } over { partial x} } \rline rSub { size 8{i} } simeq {"lim"} cSub { size 8{Δx rightarrow 0} } { {Δ { {W}} sup { ' } rSub { size 8{ ital "fld"} } } over {Δx} } \rline rSub { size 8{i} } } {}
Figure 3.6Effect of
ΔΔ size 12{Δ} {}x on the energy and coenergy of a singly-excited device:
(a) change of energy with
λλ size 12{λ} {} held constant; (b) change of coenergy with i held constant.
- The force acts in a direction to decrease the magnetic field stored energy at constant flux or to increase the coenergy at constant current.
- In a singly-excited device, the force acts to increase the inductance by pulling on members so as to reduce the reluctance of the magnetic path linking the winding.
§3.6 Multiply-Excited Magnetic Field Systems
- Many electromechanical devices have multiple electrical terminals.
- Measurement systems: torque proportional to two electric signals; power as the product of voltage and current.
- Energy conversion devices: multiply-excited magnetic field system.
- A simple system with two electrical terminals and one mechanical terminal: Fig.3.7.
- Three independent variables:
θ,λ1,λ2θ,λ1,λ2 size 12{ left lbrace θ,λ rSub { size 8{1} } ,λ rSub { size 8{2} } right rbrace } {} ,
θ,i1,i2θ,i1,i2 size 12{ left lbrace θ,i rSub { size 8{1} } ,i rSub { size 8{2} } right rbrace } {} ,
θ,λ1,i2θ,λ1,i2 size 12{ left lbrace θ,λ rSub { size 8{1} } ,i rSub { size 8{2} } right rbrace } {}, or
θ,i1,λ2θ,i1,λ2 size 12{ left lbrace θ,i rSub { size 8{1} } ,λ rSub { size 8{2} } right rbrace } {}
dWfld(λ1,λ2,θ)=i1dλ1+i2dλ2−TflddθdWfld(λ1,λ2,θ)=i1dλ1+i2dλ2−Tflddθ size 12{ ital "dW" rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1} } ,λ rSub { size 8{2} } ,θ \) =i rSub { size 8{1} } dλ rSub { size 8{1} } +i rSub { size 8{2} } dλ rSub { size 8{2} } - T rSub { size 8{ ital "fld"} } dθ} {} (3.54)
Figure 3.7Multiply-excited magnetic energy storage system.
i1=∂Wfld(λ1,λ2,θ)∂λ1∣λ2,θi1=∂Wfld(λ1,λ2,θ)∂λ1∣λ2,θ size 12{i rSub { size 8{1} } = { { partial W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1} } ,λ rSub { size 8{2} } ,θ \) } over { partial λ rSub { size 8{1} } } } \rline rSub { size 8{λ rSub { size 6{2} } ,θ} } } {} (3.55)
i2=∂Wfld(λ1,λ2,θ)∂λ2∣λ1,θi2=∂Wfld(λ1,λ2,θ)∂λ2∣λ1,θ size 12{i rSub { size 8{2} } = { { partial W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1} } ,λ rSub { size 8{2} } ,θ \) } over { partial λ rSub { size 8{2} } } } \rline rSub { size 8{λ rSub { size 6{1} } ,θ} } } {}(3.56)
Tfld=−∂Wfld(λ1,λ2,θ)∂θ∣λ1,λ2Tfld=−∂Wfld(λ1,λ2,θ)∂θ∣λ1,λ2 size 12{T rSub { size 8{ ital "fld"} } = - { { partial W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1} } ,λ rSub { size 8{2} } ,θ \) } over { partial θ} } \rline rSub { size 8{λ rSub { size 6{1} } ,λ rSub { size 6{2} } } } } {} (3.57)
To find
WfldWfld size 12{W rSub { size 8{ ital "fld"} } } {} , use the path of integration in Fig. 3.14.
Wfld(λ10,λ20,θ0)=∫0λ20i2(λ1=0,λ2,θ=θ0)dλ2+∫0λ10i1(λ1,λ2=λ20,θ=θ0)dλ1Wfld(λ10,λ20,θ0)=∫0λ20i2(λ1=0,λ2,θ=θ0)dλ2+∫0λ10i1(λ1,λ2=λ20,θ=θ0)dλ1 size 12{W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1 rSub { size 6{0} } } } ,λ rSub {2 rSub { size 6{0} } } size 12{,θ rSub {0} } size 12{ \) = Int rSub {0} rSup {λ rSub { size 6{2 rSub {0} } } } {i rSub {2} } } size 12{ \( λ rSub {1} } size 12{ {}=0,λ rSub {2} } size 12{,θ=θ rSub {0} } size 12{ \) dλ rSub {2} } size 12{+ Int rSub {0} rSup {λ rSub { size 6{1 rSub {0} } } } {i rSub {1} } } size 12{ \( λ rSub {1} } size 12{,λ rSub {2} } size 12{ {}=λ rSub {2 rSub { size 6{0} } } } size 12{,θ=θ rSub {0} } size 12{ \) dλ rSub {1} }} {} (3.58)
Figure 3.8Integration path to obtain
Wfld(λ10,λ20,θ0)Wfld(λ10,λ20,θ0) size 12{W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1 rSub { size 6{0} } } } ,λ rSub {2 rSub { size 6{0} } } size 12{,θ rSub {0} } size 12{ \) }} {}.
- In a magnetically-linear system,
λ1=L11i1+L12i2λ1=L11i1+L12i2 size 12{λ rSub { size 8{1} } =L rSub { size 8{"11"} } i rSub { size 8{1} } +L rSub { size 8{"12"} } i rSub { size 8{2} } } {} (3.59)
λ2=L21i1+L22i2λ2=L21i1+L22i2 size 12{λ rSub { size 8{2} } =L rSub { size 8{"21"} } i rSub { size 8{1} } +L rSub { size 8{"22"} } i rSub { size 8{2} } } {} (3.60)
L12=L21L12=L21 size 12{L rSub { size 8{"12"} } =L rSub { size 8{"21"} } } {} (3.61)
Note that
Lij=Lij(θ)Lij=Lij(θ) size 12{L rSub { size 8{ ital "ij"} } =L rSub { size 8{ ital "ij"} } \( θ \) } {}
i1=L22λ1−L12λ2Di1=L22λ1−L12λ2D size 12{i rSub { size 8{1} } = { {L rSub { size 8{"22"} } λ rSub { size 8{1} } - L rSub { size 8{"12"} } λ rSub { size 8{2} } } over {D} } } {} (3.62)
i2=−L21λ1+L11λ2Di2=−L21λ1+L11λ2D size 12{i rSub { size 8{2} } = { { - L rSub { size 8{"21"} } λ rSub { size 8{1} } +L rSub { size 8{"11"} } λ rSub { size 8{2} } } over {D} } } {} (3.63)
D=L11L22−L12L21D=L11L22−L12L21 size 12{D=L rSub { size 8{"11"} } L rSub { size 8{"22"} } - L rSub { size 8{"12"} } L rSub { size 8{"21"} } } {} (3.64)
The energy for this linear system is
Wfld(λ10,λ20,θ0)=∫0λ20L11(θ0)λ2D(θ0)dλ2+∫(L22(θ0)λ1−L12(θ0)λ20)D(θ0)=12D(θ0)L11(θ0)λ202+12D(θ0)L22(θ0)λ102−L12(θ0)D(θ0)λ10λ20Wfld(λ10,λ20,θ0)=∫0λ20L11(θ0)λ2D(θ0)dλ2+∫(L22(θ0)λ1−L12(θ0)λ20)D(θ0)=12D(θ0)L11(θ0)λ202+12D(θ0)L22(θ0)λ102−L12(θ0)D(θ0)λ10λ20alignl { stack {
size 12{W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1 rSub { size 6{0} } } } ,λ rSub {2 rSub { size 6{0} } } size 12{,θ rSub {0} } size 12{ \) = Int rSub {0} rSup {λ rSub { size 6{2 rSub {0} } } } { { {L rSub {"11"} size 12{ \( θ rSub {0} } size 12{ \) λ rSub {2} }} over { size 12{D \( θ rSub {0} size 12{ \) }} } } } } size 12{dλ rSub {2} } size 12{+ Int rSub {} rSup {} { { { \( L rSub {"22"} size 12{ \( θ rSub {0} } size 12{ \) λ rSub {1} } size 12{ - L rSub {"12"} } size 12{ \( θ rSub {0} } size 12{ \) λ rSub {2 rSub { size 6{0} } } } size 12{ \) }} over {D \( θ rSub {0} size 12{ \) }} } } }} {} #
size 12{" "= { {1} over {2D \( θ rSub { size 8{0} } \) } } L rSub { size 8{"11"} } \( θ rSub { size 8{0} } \) λ rSub { size 8{2 rSub { size 6{0} } } } rSup {2} size 12{+ { {1} over {2D \( θ rSub {0} size 12{ \) }} } L rSub {"22"} } size 12{ \( θ rSub {0} } size 12{ \) λ rSub {1 rSub { size 6{0} } } rSup {2} } size 12{ - { {L rSub {"12"} size 12{ \( θ rSub {0} } size 12{ \) }} over {D \( θ rSub {0} size 12{ \) }} } λ rSub {1 rSub { size 6{0} } } } size 12{λ rSub {2 rSub { size 6{0} } } }} {}
} } {} (3.65)
- Coenergy function for a system with two windings can be defined as:
Wfld'(i1,i2,θ)=λ1i1+λ2i2−WfldWfld'(i1,i2,θ)=λ1i1+λ2i2−Wfld size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ \) =λ rSub { size 8{1} } i rSub { size 8{1} } +λ rSub { size 8{2} } i rSub { size 8{2} } - W rSub { size 8{ ital "fld"} } } {} (3.66)
{}dWfld'(i1,i2,θ)=λ1di1+λ2di2+TflddθdWfld'(i1,i2,θ)=λ1di1+λ2di2+Tflddθ size 12{d { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ \) =λ rSub { size 8{1} } ital "di" rSub { size 8{1} } +λ rSub { size 8{2} } ital "di" rSub { size 8{2} } +T rSub { size 8{ ital "fld"} } dθ} {} (3.67)
λ1=∂Wfld(i1,i2,θ)∂i1∣i2,θλ1=∂Wfld(i1,i2,θ)∂i1∣i2,θ size 12{λ rSub { size 8{1} } = { { partial W rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ \) } over { partial i rSub { size 8{1} } } } \rline rSub { size 8{i rSub { size 6{2} } ,θ} } } {} (3.68)
λ2=∂Wfld'(i1,i2,θ)∂i2∣i1,i2λ2=∂Wfld'(i1,i2,θ)∂i2∣i1,i2 size 12{λ rSub { size 8{2} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ \) } over { partial i rSub { size 8{2} } } } \rline rSub { size 8{i rSub { size 6{1} } ,i rSub { size 6{2} } } } } {} (3.69)
{}Tfld=∂Wfld'(i1,i2,θ)∂θ∣i1,i2Tfld=∂Wfld'(i1,i2,θ)∂θ∣i1,i2 size 12{T rSub { size 8{ ital "fld"} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ \) } over { partial θ} } \rline rSub { size 8{i rSub { size 6{1} } ,i rSub { size 6{2} } } } } {} (3.70)
Wfld'(i1,i2,θ0)=∫0i20λ2(i1=0,i2,θ=θ0)di2+∫0λ10λ1(i1,i2=i20,θ=θ0)di1Wfld'(i1,i2,θ0)=∫0i20λ2(i1=0,i2,θ=θ0)di2+∫0λ10λ1(i1,i2=i20,θ=θ0)di1 size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ rSub { size 8{0} } \) = Int rSub { size 8{0} } rSup { size 8{i rSub { size 6{2 rSub {0} } } } } {λ rSub { size 8{2} } } \( i rSub {1} size 12{ {}=0,i rSub {2} } size 12{,θ=θ rSub {0} } size 12{ \) ital "di" rSub {2} } size 12{+ Int rSub {0} rSup {λ rSub { size 6{1 rSub {0} } } } {λ rSub {1} } } size 12{ \( i rSub {1} } size 12{,i rSub {2} } size 12{ {}=i rSub {2 rSub { size 6{0} } } } size 12{,θ=θ rSub {0} } size 12{ \) ital "di" rSub {1} }} {} (3.71)
W'(i1,i2,θ0)=12L11(θ)i12+12L22(θ)i22+L12(θ)i1i2W'(i1,i2,θ0)=12L11(θ)i12+12L22(θ)i22+L12(θ)i1i2 size 12{ { {W}} sup { ' } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ rSub { size 8{0} } \) = { {1} over {2} } L rSub { size 8{"11"} } \( θ \) i rSub { size 8{1} } rSup { size 8{2} } + { {1} over {2} } L rSub { size 8{"22"} } \( θ \) i rSub { size 8{2} } rSup { size 8{2} } +L rSub { size 8{"12"} } \( θ \) i rSub { size 8{1} } i rSub { size 8{2} } } {} (3.72)
Tfld=∂Wfld'(i1,i2,θ0)∂θ∣i1,i2=i122dL11(θ)dθ+i222dL22(θ)dθ+i1i2dL12(θ)dθTfld=∂Wfld'(i1,i2,θ0)∂θ∣i1,i2=i122dL11(θ)dθ+i222dL22(θ)dθ+i1i2dL12(θ)dθ size 12{T rSub { size 8{ ital "fld"} } = { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,θ rSub { size 8{0} } \) } over { partial θ} } \rline rSub { size 8{i rSub { size 6{1} } ,i rSub { size 6{2} } } } = { {i rSub {1} rSup {2} } over { size 12{2} } } { { size 12{ ital "dL" rSub {"11"} size 12{ \( θ \) }} } over { size 12{dθ} } } size 12{+ { {i rSub {2} rSup {2} } over { size 12{2} } } { { size 12{ ital "dL" rSub {"22"} size 12{ \( θ \) }} } over { size 12{dθ} } } } size 12{+i rSub {1} } size 12{i rSub {2} { { size 12{ ital "dL" rSub {"12"} size 12{ \( θ \) }} } over { size 12{dθ} } } }} {} (3.73)
- Note that (3.70) is simpler than (3.57). That is, the coenergy function is a relatively simple function of displacement.
- The use of a coenergy function of the terminal currents simplifies the determination of torque or force.
- Systems with more than two electrical terminals are handled in analogous fashion.
- System with linear displacement:
Wfld(λ10,λ20,x0)=∫0λ20i2(λ1=0,λ2,x=x0)dλ2+∫0λ10i1(λ1,λ2=λ20,x=x0)dλ1Wfld(λ10,λ20,x0)=∫0λ20i2(λ1=0,λ2,x=x0)dλ2+∫0λ10i1(λ1,λ2=λ20,x=x0)dλ1 size 12{W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1 rSub { size 6{0} } } } ,λ rSub {2 rSub { size 6{0} } } size 12{,x rSub {0} } size 12{ \) = Int rSub {0} rSup {λ rSub { size 6{2 rSub {0} } } } {i rSub {2} } } size 12{ \( λ rSub {1} } size 12{ {}=0,λ rSub {2} } size 12{,x=x rSub {0} } size 12{ \) dλ rSub {2} } size 12{+ Int rSub {0} rSup {λ rSub { size 6{1 rSub {0} } } } {i rSub {1} } } size 12{ \( λ rSub {1} } size 12{,λ rSub {2} } size 12{ {}=λ rSub {2 rSub { size 6{0} } } } size 12{,x=x rSub {0} } size 12{ \) dλ rSub {1} }} {} (3.74)
Wfld'(i10,i20,x0)=∫0λ20λ2(i1,i2,x=x0)di2+∫0λ10λ1(i1,i2=i20,x=x0)di1Wfld'(i10,i20,x0)=∫0λ20λ2(i1,i2,x=x0)di2+∫0λ10λ1(i1,i2=i20,x=x0)di1 size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1 rSub { size 6{0} } } } ,i rSub {2 rSub { size 6{0} } } size 12{,x rSub {0} } size 12{ \) = Int rSub {0} rSup {λ rSub { size 6{2 rSub {0} } } } {λ rSub {2} } } size 12{ \( i rSub {1} } size 12{,i rSub {2} } size 12{,x=x rSub {0} } size 12{ \) ital "di" rSub {2} } size 12{+ Int rSub {0} rSup {λ rSub { size 6{1 rSub {0} } } } {λ rSub {1} } } size 12{ \( i rSub {1} } size 12{,i rSub {2} } size 12{ {}=i rSub {2 rSub { size 6{0} } } } size 12{,x=x rSub {0} } size 12{ \) ital "di" rSub {1} }} {}
ffld=−∂Wfld(λ1,λ2,x)∂x∣λ1,λ2ffld=−∂Wfld(λ1,λ2,x)∂x∣λ1,λ2 size 12{f rSub { size 8{ ital "fld"} } = - { { partial W rSub { size 8{ ital "fld"} } \( λ rSub { size 8{1} } ,λ rSub { size 8{2} } ,x \) } over { partial x} } \rline rSub { size 8{λ rSub { size 6{1} } ,λ rSub { size 6{2} } } } } {} (3.76)
ffld=−∂Wfld'(i1,i2,x)∂x∣i1,i2ffld=−∂Wfld'(i1,i2,x)∂x∣i1,i2 size 12{f rSub { size 8{ ital "fld"} } = - { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,x \) } over { partial x} } \rline rSub { size 8{i rSub { size 6{1} } ,i rSub { size 6{2} } } } } {} (3.77)
For a magnetically-linear system,
Wfld'(i1,i2,x)=12L11(x)i12+L22(x)i22+L12(x)i1i2Wfld'(i1,i2,x)=12L11(x)i12+L22(x)i22+L12(x)i1i2 size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } \( i rSub { size 8{1} } ,i rSub { size 8{2} } ,x \) = { {1} over {2} } L rSub { size 8{"11"} } \( x \) i rSub { size 8{1} } rSup { size 8{2} } +L rSub { size 8{"22"} } \( x \) i rSub { size 8{2} } rSup { size 8{2} } +L"" lSub { size 8{"12"} } \( x \) i rSub { size 8{1} } i rSub { size 8{2} } } {} (3.78)
ffld=i122dL11(x)dx+i222dL22(x)dx+i1i2dL12(x)dxffld=i122dL11(x)dx+i222dL22(x)dx+i1i2dL12(x)dx size 12{f rSub { size 8{ ital "fld"} } = { {i rSub { size 8{1} } rSup { size 8{2} } } over {2} } { { ital "dL" rSub { size 8{"11"} } \( x \) } over { ital "dx"} } + { {i rSub { size 8{2} } rSup { size 8{2} } } over {2} } { { ital "dL" rSub { size 8{"22"} } \( x \) } over { ital "dx"} } +i rSub { size 8{1} } i rSub { size 8{2} } { { ital "dL" rSub { size 8{"12"} } \( x \) } over { ital "dx"} } } {} (3.79)
