# Patent

## Title
Image heating apparatus

## Abstract

An image heating apparatus configured to heat an image formed on a recording material includes a cylindrical rotatable member including a conductive layer, a magnetic core inserted through the rotatable member, a coil helically wound around an outer side of the magnetic core within the rotatable member, and an inverter configured to supply an alternating current to the coil. A frequency of the alternating current supplied from the inverter is within a range of 20.5 to 100 kHz. The conductive layer generates heat by electromagnetic induction due to an alternating magnetic field produced from the alternating current supplied to the coil. The coil is wound at an interval of 1 mm or longer.

## Background
BACKGROUND
Field of the Invention
The present disclosure relates to an image heating apparatus included in an image forming apparatus such as a copying machine and a printer, and, in particular, to an apparatus configured to heat an image by electromagnetic induction heating with use of a high frequency.
Description of the Related Art
Conventionally, there is provided an image forming apparatus such as a copying machine and a printer of the electrophotographic method or the like that includes an image heating apparatus configured to heat and fix an unfixed image (a toner image) formed on a recording material such as printing paper and an overhead projector (OHP) sheet by an appropriate image formation process, onto a surface of the recording material as a permanently fixed image. One of methods employed for the image heating apparatus is the electromagnetic induction heating method. This type of image heating apparatus includes a heated member configured to generate heat by an induced current and an exciting coil configured to produce a magnetic flux, and heats the unfixed image on the recording material with the aid of the heat of the heated member. As such a fixing apparatus, there is discussed a fixing apparatus in which a part of a core configured to form a closed magnetic path is inserted through a hollow portion of a roller-like heated member, and an alternating current of a low frequency (50 to 60 Hz) is supplied to an exciting coil helically wound around the core so that the roller-like heated member is heated (see Japanese Patent Application Laid-Open No. 10-319748).
Generally, a transformer can be downsized by an increase in a driving frequency with use of a switching power source or the like. The reason therefor is that the increase in the driving frequency can reduce a magnetic flux required to produce a same voltage, thereby allowing a magnetic core to be designed so as to have a small cross-sectional area.
However, in the fixing apparatus discussed in Japanese Patent Application Laid-Open No. 10-319748, the increase in the driving frequency raises the following problem. Relatively high power of several hundred watts or higher should be produced in the image heating apparatus included in the image forming apparatus. Therefore, the exciting coil has a large number of turns, and a parasitic capacitance (also referred to as a stray capacitance or a floating capacitance) tends to be formed between adjacent coil wires. This parasitic capacitance behaves as if a capacitor is connected in parallel with the exciting coil. As a result, if an alternating current of a high frequency (a frequency range from 20.5 kHz to 100 kHz) is supplied to the exciting coil with use of a switching power source using a resonance circuit, a switching loss and a switching noise may increase according to an undesired charge to and discharge from the parasitic capacitance, resulting in breakage of the power source.
SUMMARY
Disclosed is an image heating apparatus, which is configured to heat an image formed on a recording material, includes a cylindrical rotatable member including a conductive layer, a magnetic core inserted through the rotatable member, a coil helically wound around an outer side of the magnetic core within the rotatable member, and an inverter configured to supply an alternating current to the coil. A frequency of the alternating current supplied from the inverter is within a range of 20.5 to 100 kHz. The conductive layer generates heat by electromagnetic induction due to an alternating magnetic field produced from the alternating current supplied to the coil. The coil is wound at an interval of 1 mm or longer.
Also disclosed is an image heating apparatus, which is configured to heat an image formed on a recording material, includes a cylindrical rotatable member including a conductive layer, a magnetic core inserted through the rotatable member, a coil helically wound around an outer side of the magnetic core within the rotatable member, and an inverter configured to supply an alternating current to the coil. A frequency of the alternating current supplied from the inverter is within a range of 20.5 to 100 kHz. The conductive layer generates heat by electromagnetic induction due to an alternating magnetic field produced from the alternating current supplied to the coil. A resistance RSLV of the conductive layer in a circumferential direction thereof is expressed by an expression (1), assuming that LSLV [m] represents a length of the conductive layer in a generatrix direction of the rotatable member, dSLV [m] represents a diameter, tSLV [m] represents a thickness, and ρSLV [Ωm] represents a volume resistivity. An expression (2) is satisfied, assuming that tCOIL represents a width of a wire of the coil, LCOIL represents a length of a portion where the coil and the magnetic core overlaps each other in the generatrix direction, Ve represents an effective value voltage of a commercial power source, which is supplied to the inverter, and PSLV represents power generated on the conductive layer.




  
    
      
        
          R
          SLV
        
        =
        
          
            
              ρ
              SLV
            
            ×
            π
            ⁢
            
              
            
            ⁢
            
              d
              SLV
            
          
          
            
              t
              SLV
            
            ×
            
              L
              SLV
            
          
        
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        1
        ≤
        
          
            
              2
            
            ⁢
            
              V
              e
            
          
          
            π
            ⁢
            
              
                
                  P
                  SLV
                
                ⁢
                
                  R
                  SLV
                
              
            
          
        
        ≤
        
          
            L
            COIL
          
          
            1.0
            +
            
              t
              COIL
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates an overview of a configuration of an image forming apparatus in which a heating apparatus is included.
FIG. 2 is a perspective view illustrating main portions of the heating apparatus.
FIG. 3 is a cross-sectional view illustrating the heating apparatus according to the exemplary embodiment that is taken along a line B-B.
FIG. 4 is a front view of the heating apparatus and a block diagram of a printer control unit.
FIG. 5 illustrates a magnetic field and an induced current at the moment at which a current increases in an exciting coil in a direction indicated by an arrow.
FIG. 6 illustrates a series resonance circuit, which is one specific example of a high-frequency converter.
FIG. 7 illustrates a model of a transformer corresponding to the exciting coil and a heat generation member.
FIG. 8 illustrates a shape of a conductive layer and a method for calculating a circumferential resistance.
FIGS. 9A and 9B are conceptual diagrams illustrating how metals of coil wires operate as a capacitor.
FIG. 10 is a conceptual diagram illustrating how metals of Litz wires operate as a capacitor.
FIGS. 11A and 11B illustrate graphs that indicate relationships between the number of turns of the coil and a coil interval, and a parasitic capacitance, respectively.
FIG. 12 illustrates a configuration of an opened magnetic path.
FIGS. 13A, 13B, and 13C illustrate shapes of lines of magnetic force.
FIGS. 14A and 14B are schematic cross-sectional views.
FIGS. 15A and 15B illustrate a method for calculating a magnetic permeance.
FIG. 16 illustrates a magnetic equivalent circuit.
FIG. 17 illustrates a configuration of a magnetic core in a longitudinal direction.
FIGS. 18A and 18b illustrate efficiency of the circuit.
FIGS. 19A, 19B, and 19C illustrate power conversion efficiency.
FIG. 20 illustrates a method for conducting an experiment for acquiring the power conversion efficiency.
FIG. 21 is a graph illustrating the conversion efficiency.
FIG. 22 illustrates a configuration of a fixing apparatus in the longitudinal direction.
FIGS. 23A and 23B are cross-sectional views of the fixing apparatus.



DESCRIPTION OF THE EMBODIMENTS
1-1. General Description of Image Forming Apparatus Including Image Heating Apparatus
FIG. 1 illustrates an overview of a configuration of an image forming apparatus 100 in which an image heating apparatus according to a first exemplary embodiment is included. The image forming apparatus 100 is an electrophotographic laser beam printer. A photosensitive drum 101 works as an image bearing member, and is rotationally driven at a predetermined process speed (a circumferential speed) in the clockwise direction indicated by an arrow. The photosensitive drum 101 is evenly charged so as to have a predetermined polarity and a predetermined electric potential by a charging roller 102 during this rotation process. A laser beam scanner 103 works as an image exposure unit. The scanner 103 outputs laser light L on-off keyed according to a digital image signal input from a not-illustrated external apparatus such as a computer and generated by an image processing unit, to scan and expose a charged surface of the photosensitive drum 101. Electric charges are removed from an exposed bright portion on the surface of the photosensitive drum 101 by this scanning and exposure, whereby an electrostatic latent image corresponding to the image signal is formed on the surface of the photosensitive drum 101. A development device 104 supplies a developer (toner) from a development roller 104a onto the surface of the photosensitive drum 101, as a result of which the electrostatic latent image formed on the surface of the photosensitive drum 101 is sequentially developed as a toner image, which is a transferable image. A sheet feeding cassette 105 contains recording materials P in a stacked state. A sheet feeding roller 106 is driven based on a sheet feeding start signal, and the recording materials P contained in the sheet feeding cassette 105 are each separated from the others and are fed one by one. Then, the recording material P is introduced at a predetermined timing to a transfer portion 108T, which is an abutment nip portion between the photosensitive drum 101 and a transfer roller 108 rotating by being driven by a contact with the photosensitive drum 101, via a registration roller pair 107. In other words, the conveyance of the recording material P is controlled by the registration rollers 107 in such a manner that a leading edge of the toner image on the photosensitive drum 101 and a leading edge of the recording material P reach the transfer portion 108T at the same time. After that, the recording material P is conveyed through the transfer portion 108T while being sandwiched by the transfer portion 108T, during which a transfer voltage (a transfer bias) controlled in a predetermined manner is applied from a transfer bias application power source (not illustrated) to the transfer roller 108. The transfer bias having a reverse polarity of the toner is applied to the transfer roller 108, and the toner image on the surface side of the photosensitive drum 101 is electrostatically transferred onto a surface of the recording material P at the transfer portion 108T. The recording material P after the transfer of the toner image is separated from the surface of the photosensitive drum 101, is conveyed through a conveyance guide 109, and is introduced into an image heating apparatus A as an image heating apparatus. The recording material P is subjected to processing for fixing the toner image with use of heat at the image heating apparatus A. On the other hand, the surface of the photosensitive drum 101 after the transfer of the toner image onto the recording material P is subjected to a removal of toner remaining after the transfer, paper powder, and the like by a cleaning device 110 to thereby return to a clean surface, and then is repeatedly provided to be used in image formation. The recording material P after passing through the image heating apparatus A is discharged onto a sheet discharge tray 112 via a discharge port 111.
1-2. General Description of Image Heating Apparatus
In the present exemplary embodiment, the image heating apparatus A is an apparatus that works according to the electromagnetic induction heating method. FIG. 2 is a perspective view illustrating main portions of the image heating apparatus A according to the present exemplary embodiment. FIG. 3 is a cross-sectional view taken along a line B-B illustrated in FIG. 2. Referring to FIG. 2, a fixing sleeve 1 is a rotatable member including a conductive layer (a heat generation layer). In the perspective view of FIG. 2, the fixing sleeve 1 is illustrated with use of a cutaway view of a central portion in a longitudinal direction for facilitating better understanding of the interior of the fixing sleeve 1. The fixing sleeve 1 is the rotatable member including a conductive layer 1a as a base layer, an elastic layer 1b formed around the conductive layer 1a, and a release layer 1c formed around the elastic layer 1b. A diameter of the fixing sleeve 1 is 10 to 50 mm. The conductive layer 1a is made of a metal having a film thickness of 10 to 50 μm. The elastic layer 1b is formed by shaping silicon rubber having a hardness of 20 degrees (Japanese Industrial Standards (JIS)-A, under a weight of one kg) into a layer having a thickness of 0.1 mm to 0.3 mm. Then, a fluorine-contained resin tube having a thickness of 10 μm to 50 μm is coated around the elastic layer 1b as the front layer 1c (i.e., the release layer) in such a manner that the elastic layer 1b is covered with this tube.
A pressure roller 2 works as a counter member, and includes a core metal 2a, an elastic layer 2b formed around the core metal 2a, and a release layer formed around the elastic layer 2b. The elastic layer 2b may desirably be made from a highly thermally-resistant material such as silicon rubber, fluorine-contained rubber, and fluorosilicone rubber. Both ends of the core metal 2a are rotatably held, and are rotationally driven by a driving source (not illustrated) in a direction indicated by an arrow M in FIG. 2 to apply a rotating force to the fixing sleeve 1 with the aid of a frictional force with an outer surface of the fixing sleeve 1, and also convey the recoding material P while sandwiching the recording material P. A U-shaped stay 3 receives a pressing force in a direction indicated by an arrow H in FIG. 2 to press a slidable member 4 illustrated in FIG. 3 toward the pressure roller 2, thereby forming a nip portion N. Flange members 5a and 5b illustrated in FIG. 2 are fitted around both ends of the fixing sleeve 1 on the left side and the right side, and regulate a lateral movement when the fixing sleeve 1 rotates. The flange members 5a and 5b may desirably be made from a highly thermally-resistant material such as liquid crystal polymer (LCP) resin.
An exciting coil 6 is disposed within the fixing sleeve 1. The exciting coil 6 is wound so as to form a helically shaped portion having an axis of a helix substantially in parallel with a generatrix direction of the fixing sleeve 1. The exciting coil 6 is used to produce an alternating magnetic field. The alternating magnetic field is a magnetic field having a magnitude and a direction repeatedly changing according to time. A magnetic core 7 is disposed within the helically shaped portion, and guides lines of magnetic force in the alternating magnetic field to form a magnetic path of the lines of magnetic force. The magnetic core 7 may desirably be made from a material having a small hysteresis loss and a high relative magnetic permeability, such as calcined ferrite, ferrite resin, an amorphous alloy, and a ferromagnetic material including an oxidized material or an alloy material having a high magnetic permeability such as a permalloy. In the present exemplary embodiment, calcined ferrite having a relative magnetic permeability of 1800 is used for the magnetic core 7.
A inverter circuit (not illustrated) is connected to both ends 6a and 6b of the exciting coil 6, and a high-frequency current (an alternating current) is supplied thereto. An alternating magnetic field produced by the high-frequency current induces an induced current in the conductive layer 1a, by which the fixing sleeve 1 (the conductive layer 1a) generates heat by electromagnetic induction. A commonly-used single conductive wire or the like can be used for the exciting coil 6. A high-frequency current within a range of 20.5 kHz to 100 kHz is supplied to this exciting coil 6 via the power supply contact portions 6a and 6b with use of a high-frequency converter or the like, by which a magnetic flux is produced. This magnetic flux causes an induced current to flow in the conductive layer 1a, leading to Joule heat generation. This heat is transmitted to the elastic layer 1b and the release layer 1c, thereby heating the entire fixing sleeve 1 to heat the recording material P conveyed through the fixing nip portion N to then fix the toner image.
1-3. Control of Printer
FIG. 4 is a front view of the image heating apparatus A and a block diagram of a printer control unit 10. A thermometry element 11 such as a non-contact type thermistor is disposed on an upstream side with respect to the conveyance of the recording material P into the image heating apparatus A, and at a central portion in the longitudinal direction. With the thermometry element 11, a temperature of the fixing sleeve 1 is maintained at a predetermined target temperature. A printer controller 10a performs communication with and receives image data from a host computer 12, and rasterizes the received image data into information that the printer can print. The printer controller 10a also exchanges signals and performs serial communication with an engine control unit 10b. The engine control unit 10b exchanges signals with the printer controller 10a, and further controls a power control unit 10c and a fixing temperature control unit 10d of a printer engine via serial communication. The fixing temperature control unit 10d controls the temperature of the image heating apparatus A based on a temperature detected by the thermometry element 11. The power control unit 10d serving as a power adjustment unit adjusts a voltage to be applied to the exciting coil 6 and controls power of a high-frequency converter 13.
In a printer system including the printer control unit 10 configured in this manner, the host computer 12 transfers the image data, and sets various printing conditions such as a size of the recording material P according to a request from a user.
1-4. Induced Current Produced in Body of Sleeve
FIG. 5 illustrates a magnetic field and an induced current at the moment at which a current increases in the exciting coil 6 in a direction indicated by an arrow I1. The magnetic core 7 functions as a member for guiding lines of magnetic force produced at the exciting coil 6 into the magnetic core 7 to form a magnetic path. Therefore, most of the lines of magnetic force are guided into the magnetic core 7 to pass through within the magnetic core 7 (the magnetic path), thereby forming a closed magnetic path. Then, the fixing sleeve 1 is set up so as to surround this magnetic path. An alternating magnetic field is produced within the magnetic core 7. Then, an induced electromotive force is produced in a circumferential direction of the conductive layer 1a according to Faraday's law. Faraday's law defines that “a magnitude of an induced electromotive force E produced on the conductive layer 1a is proportional to a change rate of a magnetic field Φ perpendicularly penetrating this conductive layer 1a”. Therefore, the induced electromotive force is expressed by the following expression, an expression (4-1).




  
    
      
        E
        =
        
          
            -
            N
          
          ⁢
          
            
              Δ
              ⁢
              
                
              
              ⁢
              Φ
            
            
              Δ
              ⁢
              
                
              
              ⁢
              t
            
          
        
      
    
    
      
        (
        
          4
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




    E: the induced electromotive force
    N: the number of turns of the coil 6
    ΔΦ/Δt: a change in the magnetic flux perpendicularly penetrating through the circuit during an extremely short time Δt


When the current I1 is supplied to the exciting coil 6, an alternating magnetic field is produced within the magnetic core 7, so that an induced electromotive force in the circumferential direction is produced over an entire region of the conductive layer 1a in the longitudinal direction thereof, whereby a circumferential current I2 flows. Because the conductive layer 1a has an electric resistance, the flow of this circumferential current I2 causes Joule heat generation. An operating principle for inducing this current I2 is equivalent to magnetic coupling of a coaxial transformer.
1-5. High-Frequency Converter
FIG. 6 illustrates a relationship among a series resonance circuit, which is one specific example of the high-frequency converter 13, the exciting coil 6, and the conductive layer 1a. This mechanism is divided into a commercial power source, a rectifier circuit, a high-frequency switching circuit, a resonance circuit, an ideal transformer, and the conductive layer 1a. A commercial alternating-current voltage (e.g., AC 100 V or 200 V, 50/60 Hz) acquired from the commercial power source is converted into an undulating current by the rectifier circuit, and is supplied into the high-frequency switching circuit. Then, a voltage Va converted into the undulating current is supplied into the resonance circuit as a high-frequency current (e.g., 20.5 kHz to 100 kHz) by a switching element such as an insulated gate bipolar transistor. Hereinafter, the insulated gate bipolar transistor will be referred to as the IGBT. Driving of this IGBT (i.e., switching between an ON state and an OFF state) is controlled by a driving circuit. In the resonance circuit, a resonance capacitor CR and the exciting coil LR constitute a series resonance circuit. In the series resonance circuit, an impedance is minimized when an output frequency matches a resonance frequency fR, so that a largest amount of a current flows therethrough. The resonance frequency fR of the series resonance circuit can be acquired by the following expression, an expression (5-1).




  
    
      
        
          f
          R
        
        =
        
          1
          
            2
            ⁢
            
              
            
            ⁢
            π
            ⁢
            
              
                
                  L
                  R
                
                ⁢
                
                  C
                  R
                
              
            
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




In the present exemplary embodiment, as a result of measuring an inductance of the exciting coil 6 with use of an inductance-capacitance-resistance (LCR) meter, LR=14 μH was acquired. Therefore, for example, when a capacitance of the resonance capacitor CR is set as CR=2 μF, the resonance frequency fR can be calculated as fR=30 kHz from the expression (5-1). Therefore, when a high-frequency current of 30 kHz is produced, the current flowing through the resonance circuit is maximized, so that a heat amount generated on the heat generation member is also maximized. The capacitance of this resonance capacitor CR can be selected according to the inductance LR of the exciting coil 6 and a frequency that the user wants to use.
A voltage Vsq(t) at a certain moment in the resonance circuit can be expressed by an expression (5-2) and an expression (5-3) with use of a Fourier series, assuming that fsw represents a switching frequency. In this high-frequency converter 13, a relationship between an effective value voltage Va supplied to the high-frequency switching circuit and an effective value voltage VFHA supplied to the resonance circuit can be expressed by an expression (5-4) with use of a primary high harmonic approximation.




  
    
      
        
          
            V
            sq
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              V
              a
            
            2
          
          +
          
            
              2
              π
            
            ⁢
            
              V
              a
            
            ⁢
            
              
                ∑
                
                  
                    n
                    =
                    1
                  
                  ,
                  3
                  ,
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    ...
                  
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  1
                  n
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        sw
                      
                      ⁢
                      t
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          2
        
        )
      
    
  
  
    
      
        
          
            V
            FHA
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            2
            π
          
          ⁢
          
            V
            a
          
          ⁢
          
            sin
            ⁡
            
              (
              
                2
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  f
                  SW
                
                ⁢
                t
              
              )
            
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          3
        
        )
      
    
  
  
    
      
        
          V
          FHA
        
        =
        
          
            
              2
            
            π
          
          ⁢
          
            V
            a
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          4
        
        )
      
    
  




In this case, assuming Va=Ve, VFHA can be expressed by the following expression, an expression (5-5).





  
    
      
        
          V
          FHA
        
        =
        
          
            
              2
            
            π
          
          ⁢
          
            V
            e
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          5
        
        )
      
    
  




Further, assuming that Vm represents a maximum value of the voltage of the commercial power source, VFHA is expressed by the following expression, an expression (5-6).





  
    
      
        
          V
          FHA
        
        =
        
          
            
              
                2
              
              π
            
            ×
            
              1
              
                2
              
            
            ×
            
              V
              m
            
          
          =
          
            
              V
              m
            
            π
          
        
      
    
    
      
        (
        
          5
          ⁢
          
            -
          
          ⁢
          6
        
        )
      
    
  




1-6. Method for Calculating Power According to Transformer Model

FIG. 7 illustrates a model of a transformer corresponding to the exciting coil 6 and the heat generation member. A relationship between the voltage VFHA applied to the exciting coil 6 and the heat amount generated on the cylindrical heat generation member (power PSLV used for the heat generation of the cylindrical heat generation member) can be approximated from an expression of a transformer ratio of a transformer. The high-frequency voltage VFHA is produced on a primary winging side (the exciting coil 6). As a result, an induced electromotive force VSLV is applied to a secondary winding side (the heat generation member) via the magnetic core F, and is consumed by a resistance RSLV as heat, leading to generation of the heat amount (=power) PSLV. In this case, the number of turns of the secondary-side coil can be regarded as one turn. Then, assuming that NCOIL represents the number of turns of the primary-side coil (the exciting coil 6), a relationship of the following expression, an expression (6-1) is established among VFHA, VSLV, and NCOIL from the expression of the transformer ratio.




  
    
      
        
          
            N
            COIL
          
          1
        
        =
        
          
            V
            FHA
          
          
            V
            SLV
          
        
      
    
    
      
        (
        
          6
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




The following expression, an expression (6-2) can be acquired by transforming the expression (6-1).





  
    
      
        
          V
          SLV
        
        =
        
          
            1
            
              N
              COIL
            
          
          ×
          
            V
            FHA
          
        
      
    
    
      
        (
        
          6
          ⁢
          
            -
          
          ⁢
          2
        
        )
      
    
  




Further, a relationship of the following expression, an expression (6-3) can be acquired with use of the expression (6-2), with PSLV representing the heat amount (=power) generated on the cylindrical heat generation member, and RSLV representing a circumferential resistance of the heat generation member.




  
    
      
        
          P
          SLV
        
        =
        
          
            
              V
              SLV
              2
            
            
              R
              SLV
            
          
          =
          
            
              
                (
                
                  
                    V
                    FHA
                  
                  
                    N
                    COIL
                  
                
                )
              
              2
            
            
              R
              SLV
            
          
        
      
    
    
      
        (
        
          6
          ⁢
          
            -
          
          ⁢
          3
        
        )
      
    
  




The circumferential resistance RSLV of the heat generation member is an electric resistance when a current flows in the circumferential direction of the conductive layer 1a. 

FIG. 8 illustrates parameters of the conductive layer 1a required to calculate the circumferential resistance RSLV of the conductive layer 1a. These parameters are a length LSLV [m] of the conductive layer 1a in the longitudinal direction thereof, a diameter (an outer diameter) dSLV [m], a thickness tSLV [m], and a volume resistivity ρSLV [Ωm]. In this case, the electric resistance (the circumferential resistance) RSLV in the circumferential direction can be expressed by the following expression, an expression (6-4).




  
    
      
        
          R
          SLV
        
        =
        
          
            
              ρ
              SLV
            
            ×
            π
            ⁢
            
              
            
            ⁢
            
              d
              SLV
            
          
          
            
              t
              SLV
            
            ×
            
              L
              SLV
            
          
        
      
    
    
      
        (
        
          6
          ⁢
          
            -
          
          ⁢
          4
        
        )
      
    
  




In this case, a value indicated in a table 1 is acquired by calculating the circumferential resistance RSLV of the conductive layer 1a according to the first exemplary embodiment using the expression (6-4). Stainless steel is used as the material of the conductive layer 1a.











TABLE 1








NUMERICAL




SYMBOL
VALUE
UNIT



















VOLUME RESISTIVITY
ρ
7.2E−07
Ωm


DIAMETER
d
3.0E−02
m


THICKNESS
t
3.5E−05
m


LONGITUDINAL LENGTH
L
2.3E−01
m


CIRCUMFERENTIAL
R
8.4E−03
Ω


RESISTANCE









A value indicated in a table 2 is acquired by calculating the power generated from the heat generation member when the effective value voltage of the commercial power source is 100 V according to the expression (6-3) with use of the expressions (5-6) and (6-4). Therefore, 939 [W] can be acquired as the generated heat amount.











TABLE 2








NUMERICAL




SYMBOL
VALUE
UNIT



















EFFECTIVE VALUE VOLTAGE
Ve
100
V


FHA VOLTAGE
VFHA
45.0
V


NUMBER OF TURNS OF COIL
NCOIL
16
NONE


CIRCUMFERENTIAL
RSLV
8.4E−03
Ω


RESISTANCE


GENERATED HEAT AMOUNT
PSLV
939
W









1-7. Number of Turns of Exciting Coil and Parasitic Capacitance

An electrostatic capacitance is inevitably formed between adjacent metals. Among such capacitances, an electrostatic capacitance formed at a portion unintended by a designer is referred to as a parasitic capacitance (a stray capacitance or a floating capacitance). Also in the image heating apparatus A according to the present exemplary embodiment, if the exciting coil 6 is wound by a large number of turns, metals of adjacent coil wires behave like electrode plates of capacitors, and store electric charges, as indicated by dotted lines in FIG. 9A. As illustrated in FIG. 9B, these parasitic capacitances between the wound wires of the coil 6 behave as if a capacitor having a capacitance of ΣCs (a sum of parasitic capacitances Cs between the respective wires) is connected in parallel with the coil 6, resulting in a flow of an undesired current to charge and discharge these capacitances. If the supplied current is a low-frequency current (e.g., 50 to 60 Hz), this undesired current can be ignored, provided that the voltage is changed at a relatively low speed. However, if the voltage is changed at a high speed (e.g., 20.5 kHz to 100 kHz), this charging amount also increases, leading to occurrence of oscillation and then generation of a noise. A parameter that most largely contributes to a magnitude of this parasitic capacitance is a coil interval.
In the following description, a method for approximately calculating the parasitic capacitance CSTR from the number of turns of the coil 6, and how much the coil interval contributes thereto will be described, assuming that a naked wire having a square shape in cross-section (for simplification of the description) is used as the coil 6. An expression (7-1) can be acquired as an expression for calculating the electrostatic capacitance from an electric permittivity ε0 of a vacuum, a relative electric permittivity ε of air, an area SCOIL of facing surfaces between the coil wires, and the coil interval dCOIL, when air exists between the wound wires of the coil 6.




  
    
      
        
          C
          STR
        
        =
        
          
            ɛ
            0
          
          ⁢
          ɛ
          ⁢
          
            
              S
              COIL
            
            
              d
              COIL
            
          
        
      
    
    
      
        (
        
          7
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




The coil interval dCOIL can be acquired according to an expression (7-2) from a length LCOIL of a portion of the core 7 around which the coil 6 is wound in the longitudinal direction, the number of turns NCOIL, and a wire width tCOIL. The length LCOIL can be also defined as a length where the helically shaped portion of the coil 6 and the core 7 overlap each other in the generatrix direction of the fixing sleeve 1.




  
    
      
        
          d
          COIL
        
        =
        
          
            
              L
              COIL
            
            
              N
              COIL
            
          
          -
          
            t
            COIL
          
        
      
    
    
      
        (
        
          7
          ⁢
          
            -
          
          ⁢
          2
        
        )
      
    
  




The area SCOIL of the facing surfaces between the coil wires can be calculated according to an expression (7-3) from a length πdCORE of one turn of the coil 6 (dCORE is a diameter of the core 7), the wire width tCOIL, and the number of turns NCOIL. The wound wire of the coil 6 has a square shape in cross-section.

SCOIL=πdCORE×tCORE×(NCOIL−1)  (7-3)

If the expressions (7-2) and (7-3) are substituted into the expression (7-1), the parasitic capacitance CSTR is expressed by an expression (7-4).




  
    
      
        
          C
          STR
        
        =
        
          
            ɛ
            0
          
          ⁢
          ɛ
          ⁢
          
            
              (
              
                π
                ⁢
                
                  
                
                ⁢
                
                  d
                  CORE
                
                ×
                
                  t
                  COIL
                
                ×
                
                  (
                  
                    
                      N
                      COIL
                    
                    -
                    1
                  
                  )
                
              
              )
            
            
              (
              
                
                  
                    L
                    COIL
                  
                  
                    N
                    COIL
                  
                
                -
                
                  t
                  COIL
                
              
              )
            
          
        
      
    
    
      
        (
        
          7
          ⁢
          
            -
          
          ⁢
          4
        
        )
      
    
  




The following table 3 indicates a result of the calculation of the parasitic capacitance CSTR according to the present exemplary embodiment, which is performed with use of the expression (7-4).











TABLE 3








NUMERICAL




SYMBOL
VALUE
UNIT



















DIAMETER OF CORE
dCORE
14.0
mm


NUMBER OF TURNS OF COIL
NCOIL
16
NONE


LONGITUDINAL LENGTH
LCOIL
230
mm


WIDTH OF COIL WIRE
tCOIL
2
mm


ELECTRIC PERMITTIVITY OF
ε0
8.85E−12
F/m


VACUUM


RELATIVE ELECTRIC
ε
1.00059
NONE


PERMITTIVITY


AREA OF FACING SURFACES
SCOIL
1319
mm2


INTERVAL
dCOIL
12.4
mm


PARASITIC CAPACITANCE
CSTR
0.94
pF









The image heating apparatus A according to the first exemplary embodiment is designed in such a manner that the parasitic capacitance is sufficiently reduced.

FIG. 11A illustrates a graph that indicates a relationship between the number of turns of the coil 6 and the parasitic capacitance. The calculation is made with the width of the coil wire categorized into three types, 2 mm, 1 mm, and 0.5 mm. Further, the calculation is made assuming that the diameter of the core 7 is 14 mm, and the length of the core 7 in the longitudinal direction is 230 mm. FIG. 11B illustrates a graph that indicates a relationship between the coil interval and the parasitic capacitance. The parasitic capacitance increases as the coil interval decreases, and increase rates of the respective widths are generally similar to one another almost regardless of the width of the coil wire. In other words, it can be understood from the graph of FIG. 11B that the parasitic capacitance depends little on the width of the coil wire, and is largely affected by the relationship between the coil interval and the parasitic capacitance. This result is only an approximate calculation, but provides knowledge about the coil interval that can sufficiently reduce the influence of the parasitic capacitance.
For reference, it is desirable to reduce the parasitic capacitance to approximately 100 pF or smaller. This is because a voltage resonance capacitor may be provided in the resonance circuit to eliminate or reduce a switching loss and a switching noise, and a capacitance thereof is approximately 500 pF to 2000 pF. An increase in the parasitic capacitance to a non-negligible degree with respect to this capacitance makes it difficult to work out a design for reducing a switching loss and a switching noise. It can be concluded from this requirement together with the above-described approximate calculation that “it is possible to sufficiently reduce the influence of the parasitic capacitance by setting the coil interval to 1 mm or longer”.
This design is difficult to be achieved in a normal transformer design. This is because the length LCOIL illustrated in FIG. 9A is short in this case. The present exemplary embodiment is a design that makes best use of the fact that this apparatus is an image heating apparatus and therefore requires the dimension of the length LCOIL substantially equal to a length of an image heating region.
If a Litz wire formed by bundling thin wires together is used for the exciting coil 6, one bundle of the Litz wire can be handled in a similar manner to the single conductive wire described in the present exemplary embodiment. This is because electric potentials are completely the same within one bundle of the Litz wire, whereby no parasitic capacitance is formed between portions away from the contact point by equal distances. Therefore, as illustrated in FIG. 10, parasitic capacitances are formed at similar portions to the configuration illustrated in FIG. 9A.
1-8. Condition Required for Circumferential Resistance of Sleeve
A condition for achieving the coil interval of 1 mm or longer will be described in detail. First, an input voltage of the commercial power source and maximum power of the image heating apparatus are determined according to specifications of a product. It is necessary to control the circumferential resistance of the sleeve to realize an image heating apparatus that can reduce the parasitic capacitance and prevent generation of a noise under these constraint conditions.
In the following description, a relationship between the circumferential resistance of the sleeve and the parasitic capacitance will be described.
Regarding the number of turns, a relationship of an expression (8-1) can be acquired by transforming the expression (6-3).




  
    
      
        
          N
          COIL
        
        =
        
          
            V
            FHA
          
          
            
              
                P
                SLV
              
              ⁢
              
                R
                SLV
              
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




Then, a relationship of an expression (8-2) can be acquired by substituting the expression (5-5) into the expression (8-1) to eliminate VFHA.




  
    
      
        
          N
          COIL
        
        =
        
          
            
              
                
                  2
                
                π
              
              ⁢
              
                V
                e
              
            
            
              
                
                  P
                  SLV
                
                ⁢
                
                  R
                  SLV
                
              
            
          
          =
          
            
              
                2
              
              ⁢
              
                V
                e
              
            
            
              π
              ⁢
              
                
                  
                    P
                    SLV
                  
                  ⁢
                  
                    R
                    SLV
                  
                
              
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          2
        
        )
      
    
  




As a condition that the number of turns NCOIL should satisfy, first, the number of turns NCOIL should be one or larger as a minimum value. This is because the coil cannot fulfill the function as the exciting coil unless the coil is wound at least once or more. Therefore, the number of turns NCOIL should satisfy a relationship of the following expression (8-3).

1≦NCOIL  (8-3)

Next, a relationship of an expression (8-4) can be acquired from the relationship among LCOIL, dCOIL, and tCOIL, which is acquired by transforming the expression (7-2).




  
    
      
        
          N
          COIL
        
        =
        
          
            L
            COIL
          
          
            
              d
              COIL
            
            +
            
              t
              COIL
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          4
        
        )
      
    
  




    LCOIL: the length of the portion of the core 7 around which the coil 6 is wound in the longitudinal direction
    NCOIL: the number of turns of the coil 6
    tCOIL: the width of the coil wire


Then, a maximum value N(MAX) of the number N, which is expressed by an expression (8-5), can be acquired by substituting d=1 mm into the expression (8-4).




  
    
      
        
          
            N
            COIL
          
          ⁡
          
            (
            MAX
            )
          
        
        =
        
          
            L
            COIL
          
          
            
              
                d
                COIL
              
              ⁡
              
                (
                
                  =
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    mm
                  
                
                )
              
            
            +
            
              t
              COIL
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          5
        
        )
      
    
  




Therefore, the condition that NCOIL should satisfy is as indicated by an expression (8-6).




  
    
      
        1
        ≤
        
          N
          COIL
        
        ≤
        
          
            L
            COIL
          
          
            
              
                d
                COIL
              
              ⁡
              
                (
                
                  =
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    mm
                  
                
                )
              
            
            +
            
              t
              COIL
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          6
        
        )
      
    
  




A relationship of an expression (8-7) can be acquired from the expressions (8-6) and (8-2).




  
    
      
        1
        ≤
        
          
            
              2
            
            ⁢
            
              V
              e
            
          
          
            π
            ⁢
            
              
                
                  P
                  SLV
                
                ⁢
                
                  R
                  SLV
                
              
            
          
        
        ≤
        
          
            L
            COIL
          
          
            
              
                d
                COIL
              
              ⁡
              
                (
                
                  =
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    mm
                  
                
                )
              
            
            +
            
              t
              COIL
            
          
        
      
    
    
      
        (
        
          8
          ⁢
          
            -
          
          ⁢
          7
        
        )
      
    
  




The following table 4 indicates calculated values of a central term of the expression (8-7).












TABLE 4








NUM-






ERICAL
NUMERICAL




SYMBOL
VALUE
VALUE
UNIT




















EFFECTIVE VALUE
Ve
100
100
V


VOLTAGE


GENERATED HEAT
PSLV
1000
1000
W


AMOUNT


NUMBER OF
NCOIL
1.0
115.0
NONE


TURNS OF COIL


CIRCUMFERENTIAL
RSLV
10
8E−04
Ω


RESISTANCE









The following table 5 indicates calculated values of a term on the right side of the expression (8-7).











TABLE 5








NUMERICAL




SYMBOL
VALUE
UNIT



















INTERVAL
dCOIL
1.0
mm


LONGITUDINAL LENGTH
LCOIL
230
mm


WIDTH OF COIL WIRE
tCOIL
1
mm


NUMBER OF TURNS OF COIL
NCOIL
115
NONE


WHEN INTERVAL IS 1 mm









Because NCOIL=15.5, this configuration satisfies the condition 1≦X≦115 according to the expression (8-7), and therefore satisfies the “condition required for the circumferential resistance of the sleeve”. Accordingly, the configuration according to the first exemplary embodiment can provide a fixing apparatus that does not generate a radiated noise and the like and stably operates even when a part of the core 7 is inserted through the hollow portion of the fixing sleeve 1 (the conductive layer 1a), and a high-frequency alternating current is supplied to the exciting coil 6 helically wound around the core 7.
Specific examples of numerical values that satisfy the “condition required for the circumferential resistance of the sleeve” will be described. These values are only one example and one rough standard for realizing an output of 1000 W with use of the exciting coil having a width of 230 mm. A range of the circumferential resistance that can satisfy 1≦X≦115 is 0.8 mΩ≦RSLV≦10Ω. A table 6 indicates a result of calculating how large a design value of the thickness is for each of the minimum value and the maximum value of the circumferential resistance when the image heating apparatus is designed with use of metals having different volume resistivities under this condition.












TABLE 6









STAINLESS






STEEL
IRON
NICKEL
ALUMINUM




















MINIMUM
MAXIMUM
MINIMUM
MAXIMUM
MINIMUM
MAXIMUM
MINIMUM
MAXIMUM



SYMBOL
UNIT
VALUE
VALUE
VALUE
VALUE
VALUE
VALUE
VALUE
VALUE





















CIRCUM-
R
Ω
8.0E−04
10   
8.0E−04
10   
8.0E−04
10   
8.0E−04
10   


FERENTIAL


RESISTANCE


VOLUME
ρ
Ωm
7.2E−07
7.2E−07
8.9E−08
8.9E−08
6.8E−08
6.8E−08
2.7E−08
2.7E−08


RESISTIVITY


DIAMETER
d
m
3.0E−02
3.0E−02
3.0E−02
3.0E−02
3.0E−02
3.0E−02
3.0E−02
3.0E−02


LONGI-
L
m
2.3E−01
2.3E−01
2.3E−01
2.3E−01
2.3E−01
2.3E−01
2.3E−01
2.3E−01


TUDINAL


LENGTH


THICKNESS
t
μm
369
0.030
45.6
0.004
35.0
0.003
13.6
0.001









1-9. Result of Comparison Experiment

In the following description, a result of an experiment for comparing the image heating apparatus A according to the present exemplary embodiment and a conventional image heating apparatus will be described.
<Comparative Example 1 >
A comparative example 1 was configured in such a manner that a cylindrical heat generation member had a low volume resistivity, compared to the first exemplary embodiment.
The heat generation member of the comparative example 1 was made from iron, and had a diameter of 6 cm, a thickness of 5 mm, and a length of 230 mm in the longitudinal direction. The heat generation member in this case had a circumferential resistance as indicated in the following table 7.











TABLE 7








NUMERICAL




SYMBOL
VALUE
UNIT



















VOLUME RESISTIVITY
ρ
9.0E−08
Ωm


DIAMETER
d
6.0E−02
m


THICKNESS
t
5.0E−03
m


LONGITUDINAL LENGTH
L
2.3E−01
m


CIRCUMFERENTIAL
R
1.5E−05
Ω


RESISTANCE









Under this circumferential resistance, the number of turns of the coil should be 371 turns to realize the output of 1000 W.











TABLE 8








NUMERICAL




SYMBOL
VALUE
UNIT



















EFFECTIVE VALUE VOLTAGE
Va
100
V


CIRCUMFERENTIAL
RSLV
1.5E−05
Ω


RESISTANCE


GENERATED HEAT AMOUNT
PSLV
1000
W


NUMBER OF TURNS OF COIL
NCOIL
371.0
NONE









Because X=371, this configuration does not satisfy the condition 1≦X≦115, and therefore does not satisfy the “condition required for the circumferential resistance of the sleeve”.
The following table 9 indicates an approximate calculation of the parasitic capacitance, and a result of evaluation of a switching noise when the first exemplary embodiment and the comparative example 1 were actually used as the image heating apparatus.










TABLE 9







STRAY CAPACITANCE
NOISE



[pF]
LEVEL


















FIRST EXEMPLARY
0.65
∘


EMBODIMENT


COMPARATIVE EXAMPLE
591
Δx









The comparative example 1 generated a large switching noise, while the first exemplary embodiment generated no noise and was in an excellent state.
As described above, the configuration according to the first exemplary embodiment has an effect of being able to provide an image heating apparatus that can prevent the high-frequency current from oscillating and therefore can reduce generation of a switching loss and a switching noise from this oscillation.
A second exemplary embodiment will be described as a configuration in which the magnetic core inserted in the hollow portion of the cylindrical rotatable member forms an opened magnetic path. In this case, a substantially even strong magnetic path should be formed in an entire region in the longitudinal direction of the cylindrical rotatable member. FIG. 12 illustrates the apparatus configuration. The magnetic core 7 is inserted through the hollow portion of the fixing sleeve 1 as the cylindrical rotatable member, and forms a continuous magnetic path over the entire fixing sleeve 1 in the longitudinal direction of the fixing sleeve 1. Calcined ferrite having a relative magnetic permeability of 1800 is used as the material of the magnetic core 7. The magnetic core 7 has a diameter of 14 mm in cross-section, and has an equal longitudinal length to the fixing sleeve 1.
The second exemplary embodiment is similar to the first exemplary embodiment except for use of an opened magnetic path. The conductive layer, the elastic layer, and the front layer of the fixing sleeve 1 are similar to those of the first exemplary embodiment, and the exciting coil, the thermometry element, and the temperature control method are similar to those of the first exemplary embodiment. However, a condition that will be described below should be satisfied to achieve the operating principle (described in detail in the section 1-4) equivalent to magnetic coupling of a coaxial transformer with use of an opened magnetic path.
2-1. Condition for Achieving Operating Principle Equivalent to Magnetic Coupling of Coaxial Transformer
In the section 1-4 described above, an induced magnetomotive force is produced in the circumferential direction of the conductive layer 1a according to Faraday's law. Faraday's law defines that “the magnitude of the induced electromotive force E produced on the conductive layer 1a is proportional to the change rate of the magnetic field Φ perpendicularly penetrating this conductive layer 1a”. Therefore, a design guideline is to design “a state in which more perpendicular components of lines of magnetic force pass through inside the conductive layer 1a of the fixing sleeve 1”, so as to efficiently produce the induced electromotive force E on the conductive layer 1a of the fixing sleeve 1. Therefore, an example illustrated in FIG. 13A is a desirable state, while an example illustrated in FIG. 13B is an undesirable state. The reason therefor is as follows. In the state illustrated in FIG. 13B, lines of magnetic force pass through within the material of the cylindrical rotatable member, and this case corresponds to a method for generating heat with use of an eddy current produced in the body of the heat generation rotatable member, as the conventional technique. Such shapes of lines of magnetic force are established, for example, when the cylindrical rotatable member has a high relative magnetic permeability, when the cylindrical rotatable member has a large cross-sectional area, when the magnetic core 7 has a small cross-sectional area, when the magnetic core 7 has a low relative magnetic permeability, and when the magnetic core 7 is divided in the longitudinal direction with a gap formed between divided core pieces.
Therefore, when lines of magnetic force are produced in the configuration illustrated in FIG. 13B, the roller base layer 1a as the cylindrical member serves as a main magnetic path, and no magnetic path is formed outside the body of the cylindrical member. The shapes of lines of magnetic force in this case are such shapes that a magnetic flux produced from the magnetic core 7 is immediately introduced into the body of the conductive layer 1a of the fixing roller 1, and returns through the body of the conductive layer 1a of the fixing roller 1. FIG. 14A is a cross-sectional view of a central position. This is a schematic view illustrating lines of magnetic force at the moment at which a current in the coil 6 increases in a direction indicated by an arrow I. Lines of magnetic force Bin passing through the magnetic path are indicated by arrows pointing in the forward direction in FIG. 14A (eight white circles with black circles contained therein). Then, arrows pointing in the backward direction in FIG. 14A (eight white circles with cross marks contained therein) represent lines of magnetic force Bni returning through the body of the conductive layer 1a of the fixing roller 1. As illustrated in FIG. 14B, a large number of eddy currents E// are produced in the body of the conductive layer 1a of the fixing roller 1 so as to form a magnetic field that disturbs a change in the magnetic field indicated by the white circles with the cross marks contained therein. FIG. 14B is an enlarged view illustrating a portion K in FIG. 14A as a representative. More strictly speaking, the eddy currents E// have portions canceling out each other, and portions enhancing each other between adjacent eddy currents, and sums E1 and E2 of the eddy currents indicated by dotted arrows become dominant in the end. Hereinafter, the currents E1 and E2 will be referred to as “skin currents”. When these skin currents E1 and E2 are produced in the circumferential direction, Joule heat is generated proportionally to a skin resistance of the conductive layer 1a of the fixing roller 1. These currents are also repeatedly produced and vanished, and have directions repeatedly reversing in synchronization with the high-frequency current.
Generally, this heat generation by the eddy currents E//, or the heat generation by the skin currents E1 and E2 is referred to as an “iron loss”, and is expressed by the following expression (11-1).




  
    
      
        
          P
          e
        
        =
        
          
            k
            e
          
          ⁢
          
            
              
                (
                
                  tfB
                  m
                
                )
              
              2
            
            ρ
          
        
      
    
    
      
        (
        
          11
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




    Pe: the heat generation amount generated by the eddy-current loss
    t: the thickness of the fixing roller 1
    f: the frequency
    Bm: a maximum magnetic flux density
    ρ: the resistivity
    ke: a proportional constant


The iron loss is proportional to the square of the thickness t, whereby a reduction in the thickness of the conductive layer 1a of the fixing roller 1 leads to a reduction in the iron loss that is proportional to the square of the thickness t. As indicated by the expression (11-1), the heat generation amount Pe is proportional to the square of the “Bm: the maximum magnetic flux density within the material”, whereby it is desirable to select a ferromagnetic material such as iron, cobalt, nickel, and an alloy thereof as the material of the conductive layer 1. On the other hand, use of a weakly magnetic material or a diamagnetic material results in a reduction in heat generation efficiency. Further, the heat generation amount Pe is also proportional to the square of the thickness t, whereby reducing the thickness to 200 μm or thinner results in a reduction in the heat generation efficiency. There is such a problem that a material having a high resistivity p is also disadvantageous. Therefore, it is difficult to realize the design according to the table 6, which is provided as the specific examples of the numerical values that satisfy “1-8. CONDITION REQUIRED FOR CIRCUMFERENTIAL RESISTANCE OF SLEEVE”. Then, because this configuration corresponds to the mechanism that generates heat by the skin current, the calculation of the circumferential resistance described in “1-6. METHOD FOR CALCULATING POWER ACCORDING TO TRANSFORMER MODEL” and illustrated in FIG. 8 cannot be applied thereto. This is because the current does not flow through the entire sleeve material but is concentrated in around the skin portion of the material. Therefore, the resistance value tends to become significantly larger, and it is easy to reduce the number of turns of the coil 6. On the other hand, the thickness of the sleeve 1 cannot be reduced.
2-2. Guideline for Designing State in which more Perpendicular Components of Lines of Magnetic Force Pass Through
2-2-1. Relationship between Rate of Magnetic Flux Passing through Outside Conductive Layer and Power Conversion Efficiency
The magnetic core 7 illustrated in FIG. 13A is shaped so as to have ends without forming a loop. In a fixing apparatus configured in such a manner that the magnetic core 7 forms a loop outside the conductive layer 1a as illustrated in FIG. 13C, lines of magnetic force exit from the inside to the outside of the conductive layer 1a and return to the inside of the conductive layer 1a by being guided by the magnetic core 7. However, if the magnetic core 7 is configured so as not to form a loop outside the fixing sleeve 1 (the conductive layer 1a), like the present exemplary embodiment, there is nothing to guide the lines of magnetic force that exit from an end of the magnetic core 7. Therefore, there is a possibility that a path of the lines of magnetic force returning to the other end of the magnetic core 7 after exiting from one end of the magnetic core 7 (from N to S) may extend through both an external route passing through outside the conductive layer 1a, and an internal route passing through inside the conductive layer 1a. Hereinafter, the term “external route” will be used to refer to the route going from N to S of the magnetic core 7 while passing through outside the conductive layer 1a, and the term “internal route” will be used to refer to the route going from N to S of the magnetic core 7 while passing through inside the conductive layer 1a. 
A rate of lines of magnetic force passing through the external route among these lines of magnetic force exiting from the one end of the magnetic core 7 is correlated to the power consumed by the heat generation of the conductive layer 1a in the power supplied to the coil 6 (power conversion efficiency), and is an important parameter. As the rate of the lines of magnetic force passing through the external route increases, a rate of the power consumed by the heat generation of the conductive layer 1a with respect to the power supplied to the coil 6 (the power conversion efficiency) increases. A principle of this reason is similar to such a principle that the power conversion efficiency increases, if a leakage flux is sufficiently little in the transformer, and the number of lines of magnetic force passing through the secondary winding of the transformer is equal to the number of lines of magnetic force passing through the primary winding of the transformer. In other words, in the present exemplary embodiment, as the number of lines of magnetic force passing through the external route gets closer to the number of lines of magnetic force passing through within the magnetic core 7, the power conversion efficiency increases, and the high-frequency current supplied to the coil 6 can be more efficiently used for electromagnetic induction as the circumferential current around the conductive layer 1a. 
As understood from the above description, it is important to manage the rate of the lines of magnetic force passing through the external route to acquire the required power conversion efficiency for the fixing apparatus according to the present exemplary embodiment.
2-2-2. Index Indicating Rate of Magnetic Flux Passing through Outside Conductive Layer
Therefore, the rate of the lines of magnetic force passing through the external route in the fixing apparatus is expressed with use of an index called a permeance, which indicates how easily a line of magnetic force can pass through. First, a common idea about a magnetic circuit will be described. A circuit of a magnetic path which a line of magnetic force passes through is referred to as a magnetic circuit, while a circuit of an electric current is referred to as an electric circuit. A magnetic flux in the magnetic circuit can be calculated corresponding to a calculation of the current in the electric circuit. Ohm's law regarding the electric circuit can be employed for the magnetic circuit. The following expression (501) can be established, assuming that Φ represents the magnetic flux corresponding to the current in the electric circuit, V represents a magnetomotive force corresponding to an electromotive force, and R represents a magnetic resistance corresponding to an electric resistance.

Φ=V/R  (501)

However, the principle will be described here with use of a permeance P, which is an inverse of the magnetic resistance R, to facilitate better understanding of the principle. Use of the permeance P allows the above-described expression (501) to be expressed by the following expression (502).

Φ=V×P  (502)

Further, this permeance P can be expressed by the following expression (503), assuming that B represents a length of the magnetic path, S represents a cross-sectional area of the magnetic path, and μ represents a magnetic permeability of the magnetic path.

P=μ×S/B  (503)

The permeance P is proportional to the cross-sectional area S and the magnetic permeability μ, and is inversely proportional to the magnetic path length B.

FIG. 15A illustrates the conductive layer 1a containing therein the magnetic core 7 having a radius a1 [m], the length B [m], and a relative magnetic permeability μ1 with the coil 6 wound around the magnetic core 7 by N turns [turns] in such a manner that the axis of the helix extends substantially in parallel with the generatrix direction of the conductive layer 1a. In the example illustrated in FIG. 15A, the conductive layer 1a is a conductive body having the length B [m], an inner diameter a2 [m], an outer diameter a3 [m], and a relative magnetic permeability μ2. A vacuum space inside and outside the conductive layer 1a has a magnetic permeability μ0 [H/m]. A magnetic flux φc(x) indicates a magnetic flux 8 that is produced per unit length of the magnetic core 7 when a current I [A] is supplied to the coil 6. FIG. 15B is a cross-sectional view perpendicular to the longitudinal direction of the magnetic core 7. Arrows illustrated in FIG. 15B indicate magnetic fluxes that pass through the body of the magnetic core 7, inside the conductive layer 1a, and outside the conductive layer 1a in parallel with the longitudinal direction of the magnetic core 7 when the current I is supplied to the coil 6. A magnetic flux φc(=φc(x)) passes through the body of the magnetic core 7. A magnetic flux φa_in passes through inside the conductive layer 1a (passes through a region between the conductive layer 1a and the magnetic core 7). A magnetic flux φs passes through the conductive layer 1a itself. A magnetic flux φa_out passes through outside the conductive layer 1a. 
FIG. 16A illustrates a magnetic equivalent circuit of a space containing the core 7, the coil 6, and the conductive layer 1a per unit length illustrated in FIG. 13A. Assume that Vm represents the magnetomotive force produced by the magnetic flux φc passing through the magnetic core 7, Pc represents a permeance of the magnetic core 7, Pa_in represents a permeance inside the conductive layer 1a, Ps represents a permeance of the body of the film conductive layer 1a itself, and Pa_out represents a permeance outside the conductive layer 1a. 
If the permeance Pc is sufficiently large compared to the permeances Pa_in and Ps, the magnetic flux passing through the body of the magnetic core 7 and exiting from the one end of the magnetic core 7 is considered to return to the other end of the magnetic core 7 by passing through any of the magnetic fluxes φa_in, φs, and φa_out. Therefore, the following relational expression, an expression (504) is established.

φc=φa_in+φs+φa_out  (504)

Further, the magnetic fluxes φc, φa_in, φs, and φa_out are expressed by the following expressions, expressions (505) to (508), respectively.

φc=Pc×Vm  (505)

φs=Ps×Vm  (506)

φa_in=Pa_in×Vm  (507)

φa_out=Pa_out·Vm  (508)

Therefore, if the expressions (505) to (508) are substituted into the expression (504), the permeance Pa_out is expressed by the following expression, an expression (509).

Pc×Vm=Pa_in×Vm+Ps×Vm+Pa_out×Vm=(Pa_in+Ps+Pa_out)×Vm∴ Pa_out=Pc−Pa_in−Ps  (509)

The permeances can be expressed as “magnetic permeability×cross-sectional area”, and therefore can be expressed by the following expressions from the illustration of FIG. 15B, assuming that Sc represents a cross-sectional area of the magnetic core 7, Sa_in represents a cross-sectional area inside the conductive layer 1a, and Ss represents a cross-sectional area of the conductive layer 1a itself. The unit is [H·m].

Pc=μ1·Sc=μ1·π(a1)2  (510)

Pa_in=μ0·Sa_in=μ0·π·((a2)2−(a1)2)  (511)

Ps=μ2·Ss=μ2·π·((a3)2−(a2)2)  (512)

Substituting these expressions (510) to (512) into the expression (509) allows the permeance Pa_out to be expressed by an expression (513).

Pa_out=Pc−Pa_in−Ps=μ1·Sc−μ0·Sa_in−μ2·Ss=π·μ1·(a1)2−π·μ0·((a2)2−(a1)2)−π·μ2·((a3)2−(a2)2)  (513)

Use of the above-described expression (513) allows Pa_out/Pc, which is the rate of the lines of magnetic force passing through outside the conductive layer 1a, to be calculated.

The magnetic resistance R may be used instead of the permeance P. If the rate of the lines of magnetic force passing through outside the conductive layer 1a is discussed with use of the magnetic resistance R, the magnetic resistance R is simply an inverse of the permeance P, so that the magnetic resistance R per unit length can be expressed as “1/(magnetic permeability x cross-sectional area)”. The unit is “1/(H·m)”.
The following table 10A indicates a result of a specific calculation with use of parameters of the apparatus according to the present exemplary embodiment.














TABLE 10










INSIDE

OUTSIDE




MAGNETIC
FILM
CONDUCTIVE
CONDUCTIVE
CONDUCTIVE



UNIT
CORE
GUIDE
LAYER
LAYER
LAYER






















CROSS-
m{circumflex over ( )}2
2.6E−05
1.0E−04
5.8E−04
3.3E−06



SECTIONAL


AREA


RELATIVE

1800
1
1
1


MAGNETIC


PERMEABILITY


MAGNETIC
H/m
2.3E−3
1.3E−6
1.3E−6
1.3E−6


PERMEABILITY


PERMEANCE
H · m
5.9E−08
1.3E−10
7.3E−10
4.1E−12
5.8E−08


PER UNIT


LENGTH


MAGNETIC
1/(H · m)
1.7E+07
8.0E+09
1.4E+09
2.4E+11
1.7E+07


RESISTANCE


PER


UNIT


LENGTH


RATE OF
%
100.0%
0.2%
1.2%
0.0%
98.5%


MAGNETIC


FLUX









The magnetic core 7 is made from ferrite (having a relative magnetic permeability of 1800), and has the diameter of 14 [mm] and a cross-sectional area of 2.6×10−5 [m2]. A film guide is made from Polyphenylenesulfide (PPS) (having a relative magnetic permeability of 1.0), and has a cross-sectional area of 1.0×10−4 [m2]. The conductive layer 1a is made from stainless steel (having a relative magnetic permeability of 1.0), and has a diameter of 30 [mm], a thickness of 35 [μm], and a cross-sectional area of 3.3×10−6 [m2].
The cross-sectional area of the region between the conductive layer 1a and the magnetic core 7 is calculated by subtracting the cross-sectional area of the magnetic core 7 and the cross-sectional area of the film guide from a cross-sectional area of the hollow portion inside the conductive layer 1a having the diameter of 30 [mm]. The elastic layer 1b and the front layer 1c are disposed on an outer side with respect to the conductive layer 1a, and do not contribute to the heat generation. Therefore, they can be considered as an air layer outside the conductive layer 1a in the magnetic circuit model for calculating the permeance, and therefore do not have to be included in the calculation.
According to the table 10, the permeances Pc, Pa_in, and Ps have the following values.

Pc=5.9×10−8 [H·m]

Pa_in=1.3×10−10+7.3×10−10 [H·m]

Ps=4.1×10−12 [H·m]

The rate Pa_out/Pc can be calculated with use of these values according to the following expression, an expression (514).

Pa_out/Pc=(Pc−Pa_in−Ps)/Pc=0.985(98.5%)  (514)

The magnetic core 7 may be divided into a plurality of pieces in the longitudinal direction, and a space (a gap) may be provided between the respective divided magnetic cores In this case, if this space is filled with air, a material having a relative magnetic permeability that can be regarded as 1.0, or a material having a far lower relative magnetic permeability than the magnetic core 7, this leads to an increase in the magnetic resistance R of the entire magnetic core 7, resulting in significant deterioration of the function of guiding the lines of magnetic force.
The permeance of the magnetic core 7 divided in this manner should be calculated with use of a complicated calculation method. In the following description, for a configuration in which the magnetic core 7 is divided into a plurality of pieces, and the divided pieces are arranged at even intervals with a space or a sheet-like non-magnetic body sandwiched between adjacent pieces, a method for calculating the permeance of the entire magnetic core 7 will be described. In this case, a magnetic resistance per unit length should be acquired by calculating a magnetic resistance of the entire magnetic core 7 in the longitudinal direction, and then dividing the calculated magnetic resistance by the entire length. Then, a permeance per unit length should be acquired by calculating an inverse of the magnetic resistance per unit length.
First, FIG. 17 illustrates a configuration of the magnetic core 7 in the longitudinal direction. Magnetic cores c1 to c10 each have the cross-sectional area Sc, the magnetic permeability μc, and a width Lc per divided magnetic core. Gaps g1 to g9 each have a cross-sectional area Sg, a magnetic permeability μg, and a width Lg per gap. In this case, a magnetic resistance Rm_all an of the entire magnetic core 7 in the longitudinal direction is expressed by the following expression, an expression (515).

Rm_all=(Rm_c1+Rm_c2+ . . . +Rm_c10)+(Rm_g1+Rm_g2+ . . . +Rm_g9)   (515)

According to the present configuration, the magnetic cores c1 to c10 have the same shapes and are made from the same materials, and the gaps g1 to g9 have equal widths. Therefore, the magnetic resistances can be expressed by the following expressions (516) to (518), in which a sum of the magnetic resistances Rm_c is indicated as ΣRm_c and a sum of the magnetic resistances Rm_g is indicated as ΣRm_g.

Rm_all=(ΣRm_c)+(ΣRm_g)  (516)

Rm_c=Lc/(μc·Sc)  (517)

Rm_g=Lg/(μg·Sg)  (518)

Substituting the expressions (517) and (518) into the expression (516) allows the magnetic resistance Rm_all of the entire magnetic core 7 in the longitudinal direction to be expressed by the following expression, an expression (519).

Rm_all=(ΣRm_c)+(ΣRm_g)=(Lc/(μc·Sc))×10+(Lg/(μg·Sg))×9  (519)

Then, the magnetic resistance Rm per unit length is expressed by the following expression, an expression (520), in which a sum of the widths Lc is indicated as ΣLc and a sum of the widths Lg is indicated as ΣLg.

Rm=Rm_all/(ΣLc+ΣLg)=Rm_all/(Lc×10+Lg×9)  (520)

From these expressions, the permeance Pm per unit length can be acquired from the following expression (521).

Pm=1/Rm=(ΣLc+ΣLg)/Rmall=(ΣLc+ΣLg)/[{ΣLc/(μc×Sc)}+{ΣLg/(μg×Sg)}]  (521)

An increase in the gap Lg leads to an increase in the magnetic resistance of the magnetic core 7 (a reduction in the permeance). It is desirable to design the fixing apparatus in such a manner that the magnetic core 7 has a low magnetic resistance (a high permeance) in light of the heat generation principle when configuring the fixing apparatus according to the present exemplary embodiment, whereby it is undesirable to form a gap. However, the magnetic core 7 may be divided into a plurality of pieces with a gap formed therebetween to prevent the magnetic core 7 from being broken.
In this manner, the calculation described above has revealed that the rate of the lines of magnetic force passing through the external route can be also expressed with use of the permeance or the magnetic resistance.
<One Specific Example of Calculation of Magnetic Permeance>
A case example of the calculation for the configuration in which a space is provided between the divided cores with use of the above-described calculation method will be described. As illustrated in FIG. 17, each of all of magnetic cores c1 to c10 is ferrite having a relative magnetic permeability of 1800 and a saturation magnetic flux density of 500 mT, and is shaped into a columnar shape having a diameter of 11 mm and a length B of 20 mm. Ten magnetic cores are arranged at even intervals with a gap of G=0.5 mm formed between adjacent ones. A member made of a nickel (having a relative magnetic permeability of 600) having a diameter of 40 mm and a thickness of 0.5 mm is used as the fixing roller as the cylindrical member. The magnetic permeance per unit length can be calculated by the above-described method, and has a value as indicated in the following table 11.










TABLE 11







NUMERICAL



CALCULATION EXAMPLE
SYMBOL
VALUE
UNIT


















LONGITUDINAL LENGTH OF
Lc
0.022
m


MAGNETIC CORE


MAGNETIC PERMEABILITY OF
μc
2.3E−03
H (A/m)


MAGNETIC CORE


CROSS-SECTIONAL AREA OF
Sc
9.5E−05
m{circumflex over ( )}2


MAGNETIC CORE


MAGNETIC RESISTANCE OF
Rm—c
1.0E+05
1/A


MAGNETIC CORE


LONGITUDINAL LENGTH OF GAP
Lg
0.0005
m


MAGNETIC PERMEABILITY OF GAP
μg
1.3E−06
H (A/m)


CROSS-SECTIONAL AREA OF GAP
Sg
9.5E−05
m{circumflex over ( )}2


MAGNETIC RESISTANCE OF GAP
Rm—g
4.2E+06
1/A


MAGNETIC RESISTANCE OF
Rm—all
4.3E+07
1/A


ENTIRE MAGNETIC CORE


MAGNETIC RESISTANCE PER UNIT LENGTH
Rm
1.7E+08
1/(A · m)


PERMEANCE PER UNIT LENGTH
Pm
5.7E−09
H/m









The magnetic resistance of the gap has a value several times larger than the magnetic resistance of the magnetic core. From the above calculation, 5.7×10−9 [H/m] is acquired as the magnetic permeance of the magnetic core per unit length. Then, calculating the ratio of the magnetic flux passing through each region based on this magnetic permeance produces a result as indicated in the following table 12.












TABLE 12









INSIDE





MAGNETIC
FILM
CONDUCTIVE
CONDUCTIVE


ITEM
UNIT
CORE
GUIDE
LAYER
LAYER




















CROSS-
m{circumflex over ( )}2
9.5E−05
1.0E−04
1.0E−03
6.2E−05


SECTIONAL


AREA


RELATIVE


1
1
600


MAGNETIC


PERMEABILITY


MAGNETIC
H/m

1.3E−6
1.3E−6
7.5E−04


PERMEABILITY


PERMEANCE
H · m
5.7E−09
1.3E−10
1.3E−09
4.7E−08


PER UNIT


LENGTH


MAGNETIC
1/(H · m)
1.7E+08
8.0E+09
8.0E+08
2.1E+07


RESISTANCE


PER UNIT


LENGTH









As the ratio of the magnetic permeances according to the present configuration, the magnetic permeance of the conductive layer is eight times larger than the magnetic permeance of the magnetic core. Therefore, the air outside the cylindrical member is not used as the magnetic path, whereby the rate of the magnetic flux outside the cylindrical member is 0%. Therefore, the magnetic flux does not pass through outside the cylindrical member, and is guided into the body of the heat generation rotatable member. In this configuration, the lines of magnetic force are shaped as illustrated in FIG. 13B.
2-2-3. Power Conversion Efficiency Required for Fixing Apparatus
Next, the power conversion efficiency required for the fixing apparatus according to the present exemplary embodiment will be described. For example, if the power conversion efficiency is 80%, power of remaining 20% is converted into heat energy and is consumed by the coil 6, the core 7, and the like other than the conductive layer 1a. If the power conversion efficiency is low, the members that should not generate heat, such as the magnetic core 7 and the coil 6, may generate heat to thereby raise the necessity of taking a measure for cooling down these members.
In the present exemplary embodiment, to cause the conductive layer 1a to generate heat, a high-frequency alternating current is supplied to the exciting coil 6 to produce an alternating magnetic field. This alternating magnetic field induces a current in the conductive layer 1a. As a physical model, this mechanism highly resembles the magnetic coupling of the transformer. Therefore, an equivalent circuit to the magnetic coupling of the transformer can be used to consider the power conversion efficiency. The exciting coil 6 and the conductive layer 1a are magnetically coupled to each other due to this alternating magnetic field, and the power supplied to the exciting coil 6 is transmitted to the conductive layer 1a. The “power conversion efficiency” described here means a ratio between the power supplied to the exciting coil 6, which is a magnetic field generation unit, and the power consumed by the conductive layer 1a. In the present exemplary embodiment, the power conversion efficiency means a ratio of the power supplied to a high-frequency converter 13 for the exciting coil 6 and the power consumed by the conductive layer 1a. This power conversion efficiency can be expressed by the following expression, an expression (522).

POWER CONVERSION EFFICIENCY=POWER CONSUMED BY CONDUCTIVE LAYER/POWER SUPPLIED TO EXCITING COIL  (522)

Power supplied to the exciting coil 6 and consumed by other members than the conductive layer 1a includes a loss due to a resistance of this exciting coil 6, a loss due to a magnetic characteristic of the material of the magnetic core 7, and the like.
FIGS. 18A and 18B illustrate the efficiency of the circuit. FIG. 18A illustrates the conductive layer 1a, the magnetic core 7, and the exciting coil 6. FIG. 18B illustrates an equivalent circuit.
The equivalent circuit illustrated in FIG. 18B includes a loss R1 due to the exciting coil 6 and the magnetic core 7, an inductance L1 of the exciting coil 6 wound around the magnetic core 7, a mutual inductance M of the winding and the conductive layer 1a, an inductance L2 of the conductive layer 1a, and a resistance R2 of the conductive layer 1a. FIG. 19A illustrates an equivalent circuit when the conductive layer 1a is not mounted. Measuring the series equivalent resistance R1 from both ends of the exciting coil 6 and the equivalent inductance L1 with use of an apparatus such as an impedance analyzer and an LCR meter allows an impedance ZA as viewed from the both ends of the exciting coil 6 to be expressed by an expression (523).

ZA=R1+jωL1  (523)

A current flowing through this circuit incurs a loss due to the resistance R1. In other words, the resistance R1 indicates the loss derived from the coil 6 and the magnetic core 7.

FIG. 19B illustrates an equivalent circuit when the conductive layer 1a is mounted. A relational expression (524) can be acquired by measuring a series equivalent resistance Rx (525) and an inductance Lx (526) when the conductive layer 1a is mounted, and performing equivalent conversion illustrated in FIG. 19C.




  
    
      
        
          
            
              Z
              =
              
              ⁢
              
                
                  R
                  1
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ω
                    ⁡
                    
                      (
                      
                        
                          L
                          1
                        
                        -
                        M
                      
                      )
                    
                  
                
                +
                
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      ⁡
                      
                        (
                        
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              ⁡
                              
                                (
                                
                                  
                                    L
                                    2
                                  
                                  -
                                  M
                                
                                )
                              
                            
                          
                          +
                          
                            R
                            2
                          
                        
                        )
                      
                    
                  
                  
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      M
                    
                    +
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ω
                        ⁡
                        
                          (
                          
                            
                              L
                              2
                            
                            -
                            M
                          
                          )
                        
                      
                    
                    +
                    
                      R
                      2
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  R
                  1
                
                +
                
                  
                    
                      ω
                      2
                    
                    ⁢
                    
                      M
                      2
                    
                    ⁢
                    
                      R
                      2
                    
                  
                  
                    
                      R
                      2
                      2
                    
                    +
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        L
                        2
                        2
                      
                    
                  
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ω
                    ⁡
                    
                      (
                      
                        
                          L
                          1
                        
                        -
                        M
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    M
                    ·
                    
                      R
                      2
                      2
                    
                  
                  +
                  
                    
                      ω
                      2
                    
                    ⁢
                    
                      
                        ML
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            L
                            2
                          
                          -
                          M
                        
                        )
                      
                    
                  
                
                
                  
                    R
                    2
                    2
                  
                  +
                  
                    
                      ω
                      2
                    
                    ⁢
                    
                      L
                      2
                      2
                    
                  
                
              
            
          
        
      
    
    
      
        (
        524
        )
      
    
  
  
    
      
        
          R
          x
        
        =
        
          
            R
            1
          
          +
          
            
              
                ω
                2
              
              ⁢
              
                M
                2
              
              ⁢
              
                R
                2
              
            
            
              
                R
                2
                2
              
              +
              
                
                  ω
                  2
                
                ⁢
                
                  L
                  2
                  2
                
              
            
          
        
      
    
    
      
        (
        525
        )
      
    
  
  
    
      
        
          L
          x
        
        =
        
          
            ω
            ⁡
            
              (
              
                
                  L
                  1
                
                -
                M
              
              )
            
          
          +
          
            
              
                M
                ·
                
                  R
                  2
                  2
                
              
              +
              
                
                  ω
                  2
                
                ⁢
                
                  
                    ML
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        L
                        2
                      
                      -
                      M
                    
                    )
                  
                
              
            
            
              
                R
                2
                2
              
              +
              
                
                  ω
                  2
                
                ⁢
                
                  L
                  2
                  2
                
              
            
          
        
      
    
    
      
        (
        526
        )
      
    
  




In these expressions, M represents the mutual inductance of the exciting coil 6 and the conductive layer 1a. 

As illustrated in FIG. 19C, an expression (527) is established, assuming that I1 represents a current flowing through the resistance R1, and I2 represents a current flowing through the resistance R2.

jωM(I1−I2)=(R2+jω(L2−M))I2  (527)

Further, an expression (528) can be acquired from the expression (527).




  
    
      
        
          I
          1
        
        =
        
          
            
              
                R
                2
              
              +
              
                j
                ⁢
                
                  
                
                ⁢
                ω
                ⁢
                
                  
                
                ⁢
                
                  L
                  2
                
              
            
            
              j
              ⁢
              
                
              
              ⁢
              ω
              ⁢
              
                
              
              ⁢
              M
            
          
          ⁢
          
            I
            2
          
        
      
    
    
      
        (
        528
        )
      
    
  




The efficiency (power conversion efficiency) is expressed as (power consumed by the resistance R2)/(power consumed by the resistance R1+power consumed by the resistance R2), and therefore can be expressed by an expression (529).




  
    
      
        
          
            
              
                POWER
                ⁢
                
                  
                
                ⁢
                CONVERSION
                ⁢
                
                  
                
                ⁢
                EFFICIENCY
              
              =
              
              ⁢
              
                
                  
                    R
                    2
                  
                  ×
                  
                    
                    
                      I
                      2
                      2
                    
                    
                  
                
                
                  
                    
                      R
                      1
                    
                    ×
                    
                      
                      
                        I
                        1
                        2
                      
                      
                    
                  
                  +
                  
                    
                      R
                      2
                    
                    ×
                    
                      
                      
                        I
                        2
                        2
                      
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    M
                    2
                  
                  ⁢
                  
                    R
                    2
                  
                
                
                  
                    
                      
                        
                          
                            ω
                            2
                          
                          ⁢
                          
                            L
                            2
                            2
                          
                          ⁢
                          
                            R
                            1
                          
                        
                        +
                        
                          
                            R
                            1
                          
                          ⁢
                          
                            R
                            2
                            2
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          M
                          2
                        
                        ⁢
                        
                          R
                          2
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    R
                    X
                  
                  -
                  
                    R
                    1
                  
                
                
                  R
                  X
                
              
            
          
        
      
    
    
      
        (
        529
        )
      
    
  




The power conversion efficiency, which indicates how much power is consumed by the conducive layer 1a with respect to the power supplied to the exciting coil 6, can be acquired by measuring the series equivalent resistance R1 before the conductive layer 1a is mounted and the series equivalent resistance Rx after the conductive layer 1a is mounted. In the present exemplary embodiment, Impedance Analyzer 429A manufactured by Agilent Technologies, Inc. was used to measure the power conversion efficiency. First, the series equivalent resistance R1 from the both ends of the winding was measured without the fixing film mounted. Next, the series equivalent resistance Rx from the both ends of the winding was measured with the magnetic core 7 inserted in the fixing film. The measurement result was R1=103 mΩ and Rx=2.2Ω, so that 95.3% could be acquired as the power conversion efficiency at this time according to the expression (529). Hereinafter, the performance of a fixing apparatus will be evaluated with use of this power conversion efficiency.
Now, the power conversion efficiency required for the apparatus will be determined. The power conversion efficiency will be evaluated by acquiring the rate of the magnetic flux passing through the external route of the conductive layer 1a. FIG. 20 illustrates an experiment apparatus for use in an experiment of measuring the power conversion efficiency. A metallic sheet 1S is an aluminum sheet having a width of 230 mm, a length of 600 mm, and a thickness of 20 μm. This metallic sheet 1S is cylindrically rolled so as to surround the magnetic core 7 and the coil 6, and conductivity is established at a portion indicated by a thick line 1ST, by which this metallic sheet 1S is configured as the conductive layer. The magnetic core 7 is made from ferrite having a relative magnetic permeability of 1800 and a saturation magnetic flux density of 500 mT, and is shaped into a columnar shape having a cross-sectional area of 26 mm2 and a length of 230 mm. The magnetic core 7 is disposed at a substantially central position of the cylinder formed from the aluminum sheet 1S with use of a not-illustrated fixing unit. The coil 6 is helically wound around the magnetic core 7 by twenty-five turns. A diameter 1SD of the conductive layer can be adjusted within a range of 18 to 191 mm by pulling an edge of the metallic sheet 1S in a direction indicated by an arrow 1SZ.
FIG. 21 is a graph in which the rate [%] of the magnetic flux passing through the external route of the conductive layer is set to a horizontal axis, and the power conversion efficiency with a frequency of 21 kHz is set to a vertical axis.
The power conversion efficiency drastically increases after a plotted point P1 in the graph of FIG. 21 to then exceed 70%, and is maintained at 70% or higher in a range R1 indicated by an arrow. The power conversion efficiency drastically increases again at around a plotted point P3, and is maintained at 80% or higher in a range R2. The power conversion efficiency is stabilized at a high value of 94% or higher in a range R3 after a plotted point P4. A start of this drastic increase in the power conversion efficiency is due to a start of an efficient flow of the circumferential current around the conductive layer.
The following table 13 indicates a result of an experiment in which configurations corresponding to the plotted points P1 to P4 illustrated in FIG. 21 were actually designed as fixing apparatuses, and were evaluated.












TABLE 13








RATE OF







MAGNETIC

EVALUATION





FLUX

RESULT




DIAMETER
PASSING

(PROVIDED




OF
THROUGH

THAT FIXING




CONDUCTIVE
OUTSIDE
CONVERSION
APPARATUS




LAYER
CONDUCTIVE
EFFICIENCY
IS HIGH-


NUMBER
REGION
[mm]
LAYER
[%]
SPEC)




















P1
—
143.2
64.0
54.4
POWER MAY







BE







INSUFFICIENT


P2
R1
127.3
71.2
70.8
PROVISION







OF COOLING







UNIT IS







DESIRABLE


P3
R2
63.7
91.7
83.9
OPTIMIZATION







OF







THERMALLY-







RESISTANT







DESIGN IS







DESIRABLE


P4
R3
47.7
94.7
94.7
OPTIMUM







CONFIGURATION







FOR







FLEXIBLE







FILM









<Fixing Apparatus P1>

According to this configuration, the magnetic core 7 had a cross-sectional area of 26.5 mm2 (5.75 mm×4.5 mm). The conductive layer had a diameter of 143. 2 mm. The rate of the magnetic flux passing through the external route was 64%. The power conversion efficiency of this apparatus was measured by the impedance analyzer, and the result was 54.4%. The power conversion efficiency is a parameter that indicates the power having contributed to the heat generation of the conductive layer with respect to the power supplied to the fixing apparatus. Therefore, even if the fixing apparatus P1 is designed as a fixing apparatus capable of outputting 1000 W at most, approximately 450 W becomes a loss, and this loss is turned into heat generation of the coil 6 and the magnetic core 7.
According to this configuration, when the apparatus is powered on, a temperature of the coil 6 may exceed 200° C. only by supplying 1000 W for several seconds. The loss of 45% makes it difficult to maintain temperatures of the members such as the exciting coil 6 under upper temperature limits, in consideration of the facts that an upper limit temperature of an insulating body of the coil 6 is in the high 200° C., and a Curie point of the magnetic core 7 made from ferrite is normally approximately 200° C. to 250° C. Further, if a temperature of the magnetic core 7 exceeds the Curie point, the inductance of the coil 6 drastically decreases, leading to a load change.
Since approximately 45% of the power supplied to the fixing apparatus P1 is not used for the heat generation of the conductive layer, power of approximately 1636 W should be supplied to realize supply of power of 900 W (assuming that 90% of 1000 W should be satisfied) to the conductive layer. This means a power source consuming 16.36 A when 100 V is input. This may exceed an allowable current that can be supplied from an attachment plug for the commercial alternating current. Therefore, the fixing apparatus P1 corresponding to the power conversion efficiency of 54.4% may lead to insufficiency of the power supplied to the fixing apparatus P1.
<Fixing Apparatus P2>
According to this configuration, the magnetic core 7 had an equal cross-sectional area to the fixing apparatus P1. The conductive layer had a diameter of 127.3 mm. The rate of the magnetic flux passing through the external route was 71.2%. The power conversion efficiency of this apparatus was measured by the impedance analyzer, and the result was 70.8%. Temperature increases of the coil 6 and the core 7 may become a problem depending on the specification of the fixing apparatus P2. If the fixing apparatus P2 according to the present configuration is configured as a high-spec fixing apparatus capable of performing a printing operation corresponding to 60 pages per minute, the conductive layer rotates at a speed of 330 mm/sec, and a temperature of the conductive layer should be maintained at 180° C. Maintaining the temperature of the conductive layer at 180° C. may lead to exceedance of the temperature of the magnetic core 7 over 240° C. in twenty seconds. Since the Curie point of the ferrite used as the magnetic core 7 is normally approximately 200° C. to 250° C., the ferrite may exceed the Curie point, so that the magnetic permeability of the magnetic core 7 may drastically decrease, which may make it impossible for the magnetic core 7 to appropriately guide the lines of magnetic force. As a result, it may become difficult to induce the circumferential current to allow the conductive layer to generate heat.
Therefore, if the fixing apparatus having the rate of the magnetic flux passing through the external route within the range R1 is configured as the above-described high-spec fixing apparatus, it is desirable to provide a cooling unit for reducing the temperature of the ferrite core. An air-cooling fan, a water-cooling unit, a heat sink, a radiating fin, a heat pipe, a Peltier device, or the like can be used as the cooling unit. It is apparent that the cooling unit is unnecessary if the present configuration does not have to be so much high-spec.
<Fixing Apparatus P3>
According to this configuration, the magnetic core 7 had an equal cross-sectional area to that of the fixing apparatus P1. The conductive layer had a diameter of 63.7 mm. The power conversion efficiency of this apparatus was measured by the impedance analyzer, and the result was 83.9%. Although a heat amount is invariably generated at the magnetic core 7, the coil 6, and the like, this heat generation does not reach a level that necessitates the cooling unit. If the fixing apparatus P3 according to the present configuration is configured as the high-spec fixing apparatus capable of performing the printing operation corresponding to 60 pages per minute, the conductive layer rotates at the speed of 330 mm/sec, and the surface temperature of the conductive layer should be maintained at 180° C. However, the temperature of the magnetic core 7 (ferrite) does not increase to 220° C. or higher. Therefore, if the fixing apparatus P3 according to the present configuration is configured as the above-described high-spec fixing apparatus, it is desirable to use ferrite having a Curie point of 220° C. or higher.
As understood from the above description, if the fixing apparatus having the rate of the magnetic flux passing through the external route within the range R2 is configured as the high-spec fixing apparatus, it is desirable to optimize a thermally-resistant design of ferrite and the like. On the other hand, such a thermally-resistant design is unnecessary if the fixing apparatus does not have to be high-spec.
<Fixing Apparatus P4>
According to this configuration, the magnetic core 7 had an equal cross-sectional area to that of the fixing apparatus P1. The cylindrical member had a diameter of 47.7 mm. The power conversion efficiency of this apparatus was measured by the impedance analyzer, and the result was 94.7%. Even if the fixing apparatus P4 according to the present configuration is configured as the high-spec fixing apparatus capable of performing the printing operation corresponding to 60 pages per minute (the conductive layer rotates at the speed of 330 mm/sec) so that the surface temperature of the conductive layer should be maintained at 180° C., the temperatures of the exciting coil 6, the core 7, and the like do not reach 180° C. or higher. Therefore, this configuration does not require the cooling unit for cooling down the magnetic core 7, the coil 6, and the like, and the special thermally-resistant design.
As understood from the above description, if the fixing apparatus has the rate of the magnetic flux passing through the external route within the range R3, which is 94.7% or higher, the power conversion efficiency reaches 94.7% or higher and therefore is sufficiently high. Accordingly, even if this configuration is used as a further high-spec fixing apparatus, the cooling unit is unnecessary.
Further, within the range R3 where the power conversion efficiency is stabilized at a high value, even when a slight change occurs in an amount of the magnetic flux passing through inside the conductive layer per unit time due to a change in the positional relationship between the conductive layer and the magnetic core 7, the power conversion efficiency changes only by a small amount, so that the conductive layer can generate heat by a stabilized quantity. A huge merit is brought out by using this region R3 where the power conversion efficiency is stabilized at a high value for a fixing apparatus prone to a change in the distance between the conductive layer and the magnetic core 7, like a flexible film.
From the above description, it can be understood that the fixing apparatus according to the present exemplary embodiment should have 72% or higher as the rate of the magnetic flux passing through the external route to at least satisfy the required power conversion efficiency (the table 13 indicates 71.2%, but this is rounded to 72% in consideration of a measurement error and the like).
2-2-4. Relational Expression among Permeances or Magnetic Resistances that Apparatus should Satisfy
Having 72% or higher as the rate of the magnetic flux passing through the external route of the conductive layer is equivalent to a sum of the permeance of the conductive layer and the permeance inside the conductive layer (the region between the conductive layer and the magnetic core 7) being 28% or lower of the permeance of the magnetic core 7. Therefore, one of characteristic features of the present exemplary embodiment is satisfaction of the following expression (530), assuming that Pc represents the permeance of the magnetic core 7, Pa represents the permeance inside the conductive layer 1a, and Ps represents the permeance of the conductive layer 1a. 

0.28×Pc≧Ps+Pa  (530)

Further, if the relational expression among the permeances is expressed with the permeances replaced with the magnetic resistances, this expression is converted into the following expression, an expression (531).




  
    
      
        
          
            0.28
            ×
            
              P
              c
            
          
          ≥
          
            
              P
              s
            
            +
            
              P
              a
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            0.28
            ×
            
              1
              
                R
                c
              
            
          
          ≥
          
            
              1
              
                R
                s
              
            
            +
            
              1
              
                R
                a
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            0.28
            ×
            
              1
              
                R
                c
              
            
          
          ≥
          
            1
            
              R
              sa
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            0.28
            ×
            
              R
              sa
            
          
          ≥
          
            R
            c
          
        
      
    
    
      
        (
        531
        )
      
    
  




Then, a combined magnetic resistance Rsa, which is a combination of the resistances Rs and Ra, is calculated according to the following expression, an expression (532).




  
    
      
        
          
            1
            
              R
              sa
            
          
          =
          
            
              1
              
                R
                s
              
            
            +
            
              1
              
                R
                a
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            sa
          
          =
          
            
              
                R
                a
              
              ×
              
                R
                s
              
            
            
              
                R
                a
              
              +
              
                R
                s
              
            
          
        
      
    
    
      
        (
        532
        )
      
    
  




    Rc: the magnetic resistance of the magnetic core 7
    Rs: the magnetic resistance of the conductive layer 1a 
    Ra: the magnetic resistance of the region between the conductive layer 1a and the magnetic core 7
    Rsa: the combined magnetic resistance of the magnetic resistances Rs and Ra 


It is desirable that the above-described relational expression among the permeances or the magnetic resistances is satisfied over a whole extent of a maximum region of the fixing apparatus which the recording material is conveyed through (a maximum region which the image passes through), in cross-section perpendicular to the generatrix direction of the cylindrical rotatable member. Similarly, the fixing apparatus within the range R2 according to the present exemplary embodiment has 92% or higher as the rate of the magnetic flux passing through the external route of the conductive layer (the numerical value indicated in the table 13 is 91.7%, but this is rounded to 92% in consideration of a measurement error and the like). Having 92% or higher as the rate of the magnetic flux passing through the external route of the conductive layer is equivalent to the sum of the permeance of the conductive layer and the permeance inside the conductive layer (the region between the conductive layer and the magnetic core 7) being 8% or lower of the permeance of the magnetic core 7. Therefore, the following expression, an expression (533) is acquired as a relational expression among the permeances.

0.08×Pc≧Ps+Pa  (533)

The following expression (534) is acquired by converting the above-described relational expression among the permeances into a relational expression among the magnetic resistances.

0.08×Pc≧Ps+Pa 

0.08×Rsa≧Rc  (534)

Further, the fixing apparatus within the range R3 according to the present exemplary embodiment has 95% or higher as the rate of the magnetic flux passing through the external route of the conductive layer (the table 13 indicates 94.7%, but this is rounded to 95% in consideration of a measurement error and the like).
Having 95% or higher as the rate of the magnetic flux passing through the external route of the conductive layer is equivalent to the sum of the permeance of the conductive layer and the permeance inside the conductive layer (the region between the conductive layer and the magnetic core 7) being 5% or lower of the permeance of the magnetic core 7.
Therefore, the following expression (535) is acquired as a relational expression among the permeances.

0.05×Pc≧Ps+Pa  (535)

The following expression, an expression (536) is acquired by converting this relational expression (535) among the permeances into a relational expression among the magnetic resistances.

0.05×Pc≧Ps+Pa 

0.05×Rsa≧Rc  (536)

The relational expressions among the permeances and the magnetic resistances have been described for the fixing apparatus in which the members and the like in a maximum image region of the fixing apparatus have an even cross-sectional configuration in the longitudinal direction. Next, a fixing apparatus in which the members included in the fixing apparatus have an uneven cross-sectional configuration in the longitudinal direction will be described. FIG. 22 illustrates a fixing apparatus including a temperature detection member 240 inside the conductive layer (in the region between the magnetic core 7 and the conductive layer). Other than that, the fixing apparatus illustrated in FIG. 22 is configured similarly to the second exemplary embodiment, and includes a film 1 having the conductive layer, the magnetic core 7, and a nip portion formation member (a film guide) 9.
Assuming that an X axis direction corresponds to the longitudinal direction of the magnetic core 7, a maximum image formation region is a range of 0 to Lp on the X axis. For example, for an image forming apparatus in which the maximum conveyance region for the recording material is 215.9 mm that is a letter (LTR) size, Lp can be set to 215.9 mm. The temperature detection member 240 is made of a non-magnetic body having a relative magnetic permeability of 1, and has a cross-sectional area of 5 mm×5 mm in a direction perpendicular to the X axis, and a length of 10 mm in a direction in parallel with the X axis. The temperature detection member 240 is disposed at a position from L1 (102.95 mm) to L2 (112.95 mm) on the X axis. A region from 0 to L1 as X coordinates is referred to as a region 1. A region from L1 to L2, where the temperature detection member 240 exists, is referred to as a region 2. A region from L2 to Lp is referred to as a region 3. FIG. 23A illustrates a cross-sectional configuration in the region 1, and FIG. 23B illustrates a cross-sectional configuration in the region 2. As illustrated in FIG. 23B, the temperature detection member 240 is contained in the film 1, and therefore is included in the magnetic resistance calculation. The following procedure is performed to strictly calculate the magnetic resistance. A “magnetic resistance per unit length” is calculated separately for each of the regions 1, 2, and 3. An integration calculation is performed according to a length of each region. Then, a combined magnetic resistance is calculated by adding up them. First, the following table 14 indicates the magnetic resistances of the respective members per unit length in the region 1 or 3.












TABLE 14









INSIDE





MAGNETIC
FILM
CONDUCTIVE
CONDUCTIVE


ITEM
UNIT
CORE
GUIDE
LAYER
LAYER







CROSS-
m{circumflex over ( )}2
1.5E−04
1.0E−04
2.0E−04
1.5E−06


SECTIONAL


AREA


RELATIVE

1800
1
1
1


MAGNETIC


PERMEABILITY


MAGNETIC
H/m
2.3E−03
1.3E−06
1.3E−06
1.3E−06


PERMEABILITY


PERMEANCE
H · m
3.5E−07
1.3E−10
2.5E−10
1.9E−12


PER UNIT


LENGTH


MAGNETIC
1/(H · m)
2.9E+06
8.0E+09
4.0E+09
5.3E+11


RESISTANCE


PER UNIT


LENGTH









A magnetic resistance rc1 of the magnetic core 7 per unit length in the region 1 has the following value.

rc1=2.9×106 [1/(H·m)]

A magnetic resistance ra of the region between the conductive layer and the magnetic core 7 per unit length is a combined magnetic resistance that is a combination of a magnetic resistance rf of the film guide per unit length, and a magnetic resistance rair inside the conductive layer per unit length. Therefore, the magnetic resistance ra can be calculated with use of the following expression, an expression (537).




  
    
      
        
          1
          
            r
            a
          
        
        =
        
          
            1
            
              r
              f
            
          
          +
          
            1
            
              r
              air
            
          
        
      
    
    
      
        (
        537
        )
      
    
  




As a result of the calculation, a magnetic resistance ra1 in the region 1 and a magnetic resistance rs1 in the region 1 have the following values.

ra1=2.7×109 [1/(H·m)]

rs1=5.3×1011 [1/(H·m)]

Further, the region 3 is configured similarly to the region 1, whereby the respective magnetic resistances therein have the following values.

rc3=2.9×106 [1/(H·m)]

ra3=2.7×109 [1/(H·m)]

rs3=5.3×1011 [1/(H·m)]

Next, the following table 15 indicates the magnetic resistances of the respective members per unit length in the region 2.













TABLE 15










INSIDE





MAGNETIC
FILM

CONDUCTIVE
CONDUCTIVE


ITEM
UNIT
CORE c
GUIDE
THERMISTOR
LAYER
LAYER







CROSS-
m{circumflex over ( )}2
1.5E−04
1.0E−04
2.5E−05
1.72E−04
1.5E−06


SECTIONAL


AREA


RELATIVE

1800
1
1
1
1


MAGNETIC


PERMEABILITY


MAGNETIC
H/m
2.3E−03
1.3E−06
1.3E−06
1.3E−06
1.3E−06


PERMEABILITY


PERMEANCE
H · m
3.5E−07
1.3E−10
3.1E−11
2.2E−10
1.9E−12


PER UNIT


LENGTH


MAGNETIC
1/(H · m)
2.9E+06
8.0E+09
3.2E+10
4.6E+09
5.3E+11


RESISTANCE


PER UNIT


LENGTH









A magnetic resistance rc2 of the magnetic core 7 per unit length in the region 2 has the following value.

rc2=2.9×106 [1/(H·m)]

The magnetic resistance ra of the region between the conductive layer and the magnetic core 7 per unit length is a combined magnetic resistance that is a combination of the magnetic resistance rf of the film guide per unit length, a magnetic resistance rt of the thermistor 240 per unit length, and the magnetic resistance rair of the air inside the conductive layer per unit length. Therefore, the magnetic resistance ra can be calculated with use of the following expression, an expression (538).




  
    
      
        
          1
          
            r
            a
          
        
        =
        
          
            1
            
              r
              t
            
          
          +
          
            1
            
              r
              f
            
          
          +
          
            1
            
              r
              air
            
          
        
      
    
    
      
        (
        538
        )
      
    
  




As a result of the calculation, a magnetic resistance ra2 per unit length in the region 2 and a magnetic resistance ra2 per unit length in the region 2 have the following values.

ra2=2.7×109 [1/(H·m)]

rs2=5.3×1011 [1/(H·m)]

A calculation method for the region 3 is similar to the region 1, and therefore a description thereof is omitted here.

A reason why ra1=ra2=ra3 is established regarding the magnetic resistance ra of the region between the conductive layer and the magnetic core 7 per unit length will be described now. In the magnetic resistance calculation for the region 2, the cross-sectional area of the thermistor 240 increases and the cross-sectional area of the air inside the conductive layer decreases. However, both of them have a relative magnetic permeability of 1, whereby the magnetic resistance does not change in the end regardless of whether the thermistor 240 exists. In other words, when only a non-magnetic body is disposed in the region between the conductive layer and the magnetic core 7, the calculation can maintain sufficient accuracy even when this non-magnetic body is handled in a similar manner to the air in the magnetic resistance calculation. This is because the non-magnetic body has a relative magnetic permeability almost close to 1. Conversely, if a magnetic body (nickel, iron, silicon steel, or the like) is disposed, the region where there is the magnetic body had better be calculated separately from other regions.
An integration of the magnetic resistance R [A/Wb(1/H)] as the combined magnetic resistance in the generatrix direction of the conductive layer can be calculated with respect to the magnetic resistances r1, r2, and r3 [1/(H·m)] in the respective regions 1, 2, and 3, according to the following expression, an expression (539).

R=∫0L1r1dl+∫L1L2r2dl+∫L2Lpr3dl=r1(L1−0)+r2(L2−L1)+r3(Lp−L2)   (539)

Therefore, the magnetic resistance Rc [H] of the core 7 in a section from one end to the other end of the maximum conveyance region for the recording material can be calculated according to the following expression, an expression (540).

Rc=∫0L1rc1dl+∫L1L2rc2dl+∫L2Lprc3dl=rc1(L1−0)+rc2(L2−L1)+rc3(Lp−L2)   (540)

Further, the combined magnetic resistance Ra [H] of the region between the conductive layer and the magnetic core 7 in the section from the one end to the other end of the maximum conveyance region for the recording material can be calculated according to the following expression (541).

Ra=∫0L1ra1dl+∫L1L2ra2dl+∫L2Lpra3dl=ra1(L1−0)+ra2(L2−L1)+ra3(Lp−L2)   (541)

The combined magnetic resistance Rs [H] of the conductive layer in the section from the one end to the other end of the maximum conveyance region for the recording material can be calculated according to the following expression, an expression (542). The maximum conveyance region for the recording material may be the maximum region which the image passes through.

Rs=∫0L1rs1dl+∫L1L2rs2dl+∫L2Lprs3dl=rs1(L1−0)+rs2(L2−L1)+rs3(Lp−L2)   (542)

The following table 16 indicates results of the above-described calculations performed for the respective regions.












TABLE 16










COMBINED






MAGNETIC



REGION 1
REGION 2
REGION 3
RESISTANCE




















START POINT OF
0
102.95
112.95



INTEGRATION [mm]


END POINT OF
102.95
112.95
215.9


INTEGRATION [mm]


DISTANCE [mm]
102.95
10
102.95


PERMEANCE pc PER
3.5E−07
3.5E−07
3.5E−07


UNIT LENGTH [H · m]


MAGNETIC
2.9E+06
2.9E+06
2.9E+06


RESISTANCE rc PER


UNIT LENGTH [1/(H · m)]


INTEGRATION OF
3.0E+08
2.9E+07
3.0E+08
6.2E+08


MAGNETIC


RESISTANCE rc


[A/Wb(1/H)]


PERMEANCE pa PER
3.7E−10
3.7E−10
3.7E−10


UNIT LENGTH [H · m]


MAGNETIC
2.7E+09
2.7E+09
2.7E+09


RESISTANCE ra PER


UNIT LENGTH


[1/(H · m)]


INTEGRATION OF
2.8E+11
2.7E+10
2.8E+11
5.8E+11


MAGNETIC


RESISTANCE ra


[A/Wb(1/H)]


PERMEANCE ps PER
1.9E−12
1.9E−12
1.9E−12


UNIT LENGTH [H · m]


MAGNETIC
5.3E+11
5.3E+11
5.3E+11


RESISTANCE rs PER


UNIT LENGTH


[1/(H · m)]


INTEGRATION OF
5.4E+13
5.3E+12
5.4E+13
1.1E+14


MAGNETIC


RESISTANCE rs


[A/Wb(1/H)]









According to the table 16 provided above, the magnetic resistances Rc, Ra, and Rs have the following values.

Rc=6.2×108 [1/H]

Ra=5.8×1011 [1/H]

Rs=1.1×1014 [1/H]

The combined magnetic resistance Rsa as the combination of the magnetic resistances Rs and Ra can be calculated according to the following expression, an expression (543).




  
    
      
        
          
            1
            
              R
              sa
            
          
          =
          
            
              1
              
                R
                s
              
            
            +
            
              1
              
                R
                a
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            sa
          
          =
          
            
              
                R
                a
              
              ×
              
                R
                s
              
            
            
              
                R
                a
              
              +
              
                R
                s
              
            
          
        
      
    
    
      
        (
        543
        )
      
    
  




From the above-described calculation, Rsa=5.8×1011 [1/H] is acquired as the combined magnetic resistance Rsa, and therefore the following expression, an expression (544) is satisfied.

0.28×Rsa≧Rc  (544)

In this manner, for the fixing apparatus having an uneven cross-sectional shape in the generatrix direction of the conductive layer, the permeance or the magnetic resistance can be calculated by dividing the fixing apparatus into a plurality of regions in the generatrix direction of the conductive layer, calculating the permeance or the magnetic resistance for each of the regions, and lastly calculating the combined permeance or the combined magnetic resistance as a combination of them. However, if a target member is a non-magnetic body, the permeance or the magnetic resistance may be calculated while handling the non-magnetic body as air, since the magnetic permeability of the non-magnetic body is substantially equal to the magnetic permeability of air. Next, a member that should be included in the above-described calculation will be described. It is desirable to calculate the permeance or the magnetic resistance for a member located in the region between the conductive layer and the magnetic core 7 and having at least a part thereof located within the maximum conveyance region (0 to Lp) for the recording medium. Conversely, the permeance or the magnetic resistance does not have to be calculated for a member located outside the conductive layer. This is because the induced electromotive force is proportional to a temporal change in the magnetic flux perpendicularly penetrating through the circuit according to Faraday's law as described above, and is unrelated to the magnetic flux outside the conductive layer. Further, a member disposed outside the maximum conveyance region for the recording material in the generatrix direction of the conductive layer does not affect the heat generation of the conductive layer, and therefore does not have to be included in the calculation.
In this manner, the “guideline for designing the state in which more perpendicular components of lines of magnetic force pass through” has been described.
2-3. Result of Comparison
Compared to the configuration according to the first exemplary embodiment, the configuration according to the second exemplary embodiment has such a merit that this configuration can be constructed with a reduced number of components and allows the entire apparatus to be designed as a compact structure, because this configuration does not require formation of the closed magnetic path. Further, the configuration according to the second exemplary embodiment has such a merit that this configuration can reduce the loss due to the core, because the core can be designed so as to have a reduced volume.
While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions.
This application claims the benefit of Japanese Patent Application No. 2013-261520 filed Dec. 18, 2013, which is hereby incorporated by reference herein in its entirety.

## Claims

1. An image heating apparatus configured to heat an image formed on a recording material, the image heating apparatus comprising:
a cylindrical rotatable member including a conductive layer;
a magnetic core provided in a hollow portion of the rotatable member;
a coil helically wound around an outer side of the magnetic core in the hollow portion of the rotatable member, a helical axis of the coil extending along a generatrix direction of the rotatable member; and
an inverter configured to supply an alternating current to the coil, in which a frequency of the alternating current supplied from the inverter is within a range of 20.5 to 100 kHz,
wherein the conductive layer generates heat by electromagnetic induction due to an alternating magnetic field produced from the alternating current supplied to the coil,
wherein the coil is wound at an interval of 1 mm or longer, and
wherein a magnetic resistance of the magnetic core is 28% or less of a combined magnetic resistance that is a combination of a magnetic resistance of the conductive layer and a magnetic resistance of a region between the conductive layer and the magnetic core, in a section from one end to the other end of a maximum region which the image passes through with respect to the generatrix direction of the rotatable member.

2. The image heating apparatus according to claim 1, wherein an induced current flows in a circumferential direction of the rotatable member, by which the conductive layer generates the heat, the induced current flowing in an entire circumference of the conductive layer.

3. The image heating apparatus according to claim 1, wherein the magnetic core is shaped not to form a loop outside the rotatable member.

4. The image heating apparatus according to claim 1, wherein magnetic fluxes, which generate according to the alternating magnetic field, pass through the magnetic core in a longitudinal direction of the rotatable member.