# Patent

## Title
Method and numerical tool for optimizing light emitting diode systems

## Abstract

Methods and numerical tools for designing and optimizing LED systems are provided to achieve a desired luminous performance and to increase reliability and operating lifetime. In addition, methods for designing LED illumination systems are also disclosed to determine an optimum operating power for a desired output luminous flux, given the condition of the rated power of the LED and the heatsink. By the invention, LED illumination systems can be designed with a suitable choice of LED and/or heatsink.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a U.S. national stage filing of International Patent Application No. PCT/IB2010/000272 filed on Feb. 12, 2010, which is a continuation-in-part of U.S. patent application Ser. No. 12/370,101, filed on Feb. 12, 2009, and which issued as U.S. Pat. No. 8,086,434 on Dec. 27, 2011.


FIELD OF THE DISCLOSURE
This invention relates to methods and numerical tools for forming, and optimizing the performance of, light emitting diode (LED) systems having single or multiple LEDs so that the luminous flux output from a given thermal design of the LED systems is maximized.
BACKGROUND OF THE DISCLOSURE
A light emitting diode (LED) is a semiconductor device that emits light when a current is passed through it in a forward direction. Many types of LEDs are known that emit light in various wavelengths including infra-red, visible and ultra-violet regions. Many applications for LEDs are known including as indicator lights of various colours, for use in advertising displays, and in video displays.
In the past LEDs have tended to be lower power devices that produce relatively low power outputs and have not been used for general illumination purposes. More recently, however, high-power LED devices have become known that can provide an alternative to incandescent and fluorescent light sources. LED devices produce more light per watt than incandescent light sources and may therefore be useful as energy efficient light sources, while they have a number of advantages over fluorescent light sources including being easier to dim and not requiring the use of potentially toxic and polluting elements such as mercury to create the plasma that is the source of fluorescent light.
Light emitting diodes (LEDs) have therefore emerged as promising lighting devices for the future. However, LEDs are still primarily restricted to decorative, display and signaling applications so far and have not yet entered the market for general illumination to any great extent.
In photometry, one important factor that is commonly used for comparing different lighting devices is the luminous efficacy (lumen per Watt). One major hindrance to the widespread use of LEDs in general illumination applications is that the luminous flux of LEDs decreases with the junction temperature of the LEDs. The luminous efficacy of various LEDs typically decreases by approximately 0.2% to 1% per degree Celsius rise in temperature. Due to the ageing effect, the actual degradation of luminous efficacy could be higher than this quoted figures. Accelerated aging tests show that the light output can drop by a further 45%. For aged LEDs, the efficacy degradation rate could be up to 1% per ° C. In some applications such as automobile headlights and compact lamps, the ambient temperature could be very high and the size of the heatsink is limited. The drop in luminous efficacy due to thermal problem would be serious, resulting in reduction of luminous output.
In a recent article on lighting titled “Let there be light” (Harris, Mark, IET Engineering & Technology (E&T) Magazine, Vol. 4, Issue 20, 21 Nov.-4 Dec. 2009, pp. 18-21), it was commented that, in respect of LED products, “the majority of LED A-type replacement lamps do not meet manufacturer performance claims” and that “testing reveals that these lamps produce only 10 to 60 percent of their claimed light output”. Besides the quality issues, one possible reason for such mismatch in the claimed and actual luminous performance is the understanding of the luminous efficacy figures of LED devices. LED device manufacturers usually cite high luminous efficacy figures which are only correct at a junction temperature of 25° C. In practice, luminous efficacy will decrease significantly with increasing LED junction temperature. At a junction temperature under normal operation, it is not unusual that the luminous efficacy could drop by 25% or more.
FIG. 1 shows a conventional LED. At the heart of the LED device is a light emitting semiconductor material such as InGaN though other materials will be known to those skilled in the art. In the example of FIG. 1 a light-emitting InGaN chip 1 is mounted on a silicon substrate 2 and is connected to electrodes such as cathode 3 through gold wires 4 and solder connection 5. The light-emitting chip 1 is covered by a silicone encapsulant 6 and a plastic lens 7.
When a LED of the type shown in FIG. 1 is used to generate light a substantial amount of heat is generated that will damage the light-emitting chip if not removed. Therefore a heat sink must be provided and beneath the light-emitting chip 1 is a heatsink slug 2. In practice when used to provide a source of light for illumination, conventionally multiple LEDs are provided to form a LED system as shown in FIG. 2 where multiple LEDs 10 are provided on a single heatsink 11.
SUMMARY OF THE DISCLOSURE
The present invention provides, in a first aspect, a method of forming or optimizing an LED system to achieve a desired luminous performance over a time period, the LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, the method comprising the steps of: (a) calculating a predicted luminous performance over the time period with a computer-implemented dynamic model; and (b) modifying the LED system such that the predicted luminous performance substantially matches the desired luminous performance.
In a second aspect, the present invention provides a method of optimizing an LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, the method comprising the step of calculating junction temperature over a time period with a computer-implemented dynamic model, thereby allowing modification of the LED system if a calculated peak junction temperature exceeds a maximum rated junction temperature, and thereby optimizing the LED system.
In both the first and second aspects above, the computer-implemented dynamic model, in one embodiment, is in the form of a software tool on a computer accessible through an internet website. In another embodiment, the computer-implemented dynamic model is in the form of a software tool stored on a portable computer-readable medium.
The present invention also provides, in a third aspect, a computer-implemented numerical tool that calculates a predicted luminous performance over a time period for an LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, thereby allowing modification of the LED system such that the predicted luminous performance substantially matches a desired luminous performance in order to form or optimize the LED system to achieve the desired luminous performance.
In a fourth aspect, the present invention provides a computer-implemented numerical tool for use with an LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, the computer-implemented numerical tool calculating junction temperature over a time period, thereby allowing modification of the LED system if a calculated peak junction temperature exceeds a maximum rated junction temperature, and thereby optimizing the LED system.
In both the third and fourth aspects above, the computer-implemented numerical tool, in one embodiment, is in the form of a software tool on a computer accessible through an internet website. In another embodiment, the computer-implemented numerical tool is in the form of a software tool stored on a portable computer-readable medium.
Preferably, in both the first and third aspects above, luminous performance is one or more of luminous flux of the LED system, luminous efficacy of the LED system, heatsink temperature and junction temperature over the time period.
In a fifth aspect, the present invention provides a method of forming an LED illumination system comprising a single or a plurality of LEDs on a heatsink with a desired output flux, comprising the steps of: (a) modeling on a computer the luminous flux emitted by said LED system as a function of the thermal resistance of said heatsink and the power applied to each LED, and (b) selecting an LED system such that the maximum luminous flux is emitted at a power equal to or below a rated power of said LED system provided that said maximum luminous flux is equal to or greater than the desired output flux, or (c) selecting an LED such that the rated power of said LED system is below the power at which the maximum luminous flux is emitted, provided that the flux emitted by said LED system at said rated power is equal to or greater than the desired output flux.
Preferably, in option (c) the rated power is at between 80% and 96% of the power at which maximum flux would be output.
In a sixth aspect, the present invention provides a method of forming an LED illumination system comprising a single or a plurality of LEDs on a heatsink with a desired output flux, comprising the steps of: (a) modeling on a computer the luminous flux emitted by said LED system as a function of the thermal resistance of said heatsink and the power applied to each LED, and (b) selecting a heatsink having a thermal resistance such that the maximum luminous flux is emitted at a power equal to or below a rated power of said LEDs, or (c) selecting a heatsink having a thermal resistance such that the rated power of said LED system is below the power at which the maximum luminous flux is emitted, provided that the flux emitted by said LED system at said rated power is equal to or greater than the desired output flux.
Preferably in step (c) the rated power is at between 80% and 96% of the power at which maximum flux would be output.
In a seventh aspect, the present invention provides an LED illumination system comprising a plurality of LEDs on a heatsink, wherein said heatsink has a thermal resistance such that the maximum luminous flux is emitted at a power below a rated power of said LEDs.
The present invention also provides, in an eighth aspect, a method of forming an LED illumination system comprising a plurality of LEDs on a heatsink with a desired output flux, comprising the steps of: (a) selecting an LED system such that the maximum luminous flux is emitted at a power below a rated power of said LED system provided that said maximum luminous flux is equal to or greater than the desired output flux, or (b) selecting an LED such that the rated power of said LED system is below the power at which the maximum luminous flux is emitted, provided that the flux emitted by said LED system at said rated power is equal to or greater than the desired output flux.
In a ninth aspect, the present invention provides a method of forming an LED illumination system comprising a plurality of LEDs on a heatsink with a desired output flux, comprising the steps of: (a) selecting a heatsink having a thermal resistance such that the maximum luminous flux is emitted at a power below a rated power of said LEDs, or (b) selecting a heatsink having a thermal resistance such that the rated power of said LED system is below the power at which the maximum luminous flux is emitted, provided that the flux emitted by said LED system at said rated power is equal to or greater than the desired output flux.



BRIEF DESCRIPTION OF THE DRAWINGS
Some embodiments of the present invention will now be described by way of example and with reference to the accompanying drawings, in which:—
FIG. 1 shows schematically the structure of a conventional LED;
FIG. 2 shows a conventional system with multiple individual LEDs mounted on a single heatsink;
FIG. 3 is a plot of the typical relationship between efficacy of LEDs versus junction temperature;
FIG. 4 is a plot of the typical relationship between heatsink temperature and power dissipation;
FIGS. 5(a) and (b) show (a) simplified dynamic thermal equivalent circuit of N LEDs mounted on the same heatsink, and (b) a simplified steady-state thermal equivalent circuit with N LEDs mounted on the same heatsink;
FIG. 6 shows an assumed linear function of junction-to-case thermal resistance Rjc;
FIG. 7 shows calculated and measured total luminous flux versus lamp power for eight 3W LEDs mounted on a heatsink with thermal resistance of 6.3° C./W;
FIG. 8 shows calculated and measured total luminous efficacy versus lamp power for eight 3W LEDs mounted on a heatsink with thermal resistance of 6.3° C./W;
FIG. 9 shows calculated and measured total luminous flux versus lamp power for eight 3W LEDs mounted on a heatsink with thermal resistance of 4.5° C./W;
FIG. 10 shows calculated and measured total luminous efficacy versus lamp power for eight 3W LEDs mounted on a heatsink with thermal resistance of 4.5° C./W;
FIG. 11 shows calculated and measured total luminous flux versus lamp power for eight 3W LEDs mounted on heatsink with thermal resistance of 2.2° C./W;
FIG. 12 shows calculated and measured total luminous efficacy versus lamp power for eight 3W LEDs mounted on a heatsink with thermal resistance of 2.2° C./W;
FIG. 13 shows calculated and measured total luminous flux versus lamp power for two 5W LEDs mounted on heatsink with thermal resistance of 10° C./W;
FIG. 14 shows calculated and measured total luminous efficacy versus lamp power for two 5W LEDs mounted on a heatsink with thermal resistance of 10° C./W;
FIG. 15 shows calculated and measured total luminous flux versus lamp power for two 5W LEDs mounted on heatsink with thermal resistance of 6.8° C./W;
FIG. 16 shows calculated and measured total luminous efficacy versus lamp power for two 5W LEDs mounted on a heatsink with thermal resistance of 6.8° C./W;
FIG. 17 plots total luminous flux emitted as a function of LED power;
FIG. 18 illustrates in more detail one embodiment of the invention where an LED is operated below the peak value of emitted luminous flux;
FIG. 19 illustrates the operating principles in the context of forced cooling;
FIG. 20 shows an apparatus that may implement embodiments of the invention;
FIG. 21 shows a flowchart illustrating the use of an embodiment of the invention;
FIG. 22 shows a further flowchart illustrating the use of an embodiment of the invention;
FIG. 23 shows another simplified thermal equivalent circuit of N LEDs mounted on a heatsink;
FIG. 24 shows a flowchart illustrating the use of another embodiment of the invention;
FIG. 25 shows a flowchart illustrating the use of a further embodiment of the invention;
FIG. 26 shows a chart of the measured luminous flux for an LED system having eight LEDs mounted on a heatsink with thermal resistance of 1.8° C./W;
FIG. 27 shows a chart of the calculated luminous flux for the LED system referred to in FIG. 26, the luminous flux calculated in accordance with an embodiment of the invention;
FIG. 28 shows a chart of the measured luminous flux for an LED system having eight LEDs mounted on a heatsink with thermal resistance of 2.9° C./W;
FIG. 29 shows a chart of the calculated luminous flux for the LED system referred to in FIG. 28, the luminous flux calculated in accordance with an embodiment of the invention;
FIG. 30 shows a chart depicting the variation with time of the measured and calculated luminous flux for the two LED systems referred to in FIGS. 26 and 28;
FIG. 31 shows a chart of the junction temperature over time of an offline LED mounted on a heatsink with thermal resistance of 4.1° C./W, the junction temperature calculated in accordance with an embodiment of the invention; and
FIG. 32 shows a chart of the junction temperature over time of an offline LED mounted on a heatsink with thermal resistance of 2.2° C./W, the junction temperature calculated in accordance with an embodiment of the invention.



DETAILED DESCRIPTION OF THE DISCLOSURE
To increase the light emitted from a conventional LED system it is necessary to increase the current applied to the LED. Continuing to increase the LED power will have diminishing returns, however, as the increasing temperature of the LED will reduce its efficiency and potentially damage the LED. The heatsink is therefore important as it is essential for heat to be carried away from the LED so as not to cause it any damage. The light emitted by a LED will increase with applied current provided that the heat produced can be removed, but for any individual LED design there will come a point where increasing power applied to the LED will not result in greater light being emitted because heat is no longer being effectively removed. Identifying the relationship between power applied and light emitted is an important aspect of the present invention.
The following is a photo-electro-thermal (PET) theory that characterizes this relationship.
Let φν be the total luminous flux of an LED system consisting of N LED devices.

φν=N×E×Pd  (1)

where E is efficacy (lumen/Watt) and Pd is the real power of one LED (W)

It is well known that the efficacy (E) of LEDs will decrease with increasing junction temperature of the LEDs. FIG. 3 shows a typical relationship provided by an LED manufacturer. It can be seen that:

E=Eo└1+ke(Tj−To)┘ for Tj≧To and E≧0  (2)

where Eo, is the rated efficacy at the rated temperature To (typically 25° C.) and ke is the relative reduction of efficacy with increase in temperature. For example, if E reduces by 20% over a temperature increase of 100° C., then ke=0.002.

In general, the LED power can be defined as Pd=Vd×Id, where Vd and Id are the diode voltage and current respectively. But only part of the power will be dissipated as heat. Thus, the heat generated in one LED is defined as:

Pheat=khPd=khVdId  (3)

where kh is a constant less than 1.

Now consider a typical relationship of the heatsink temperature and the heat generated in the LED system as shown in FIG. 4. A simplified dynamic thermal equivalent circuit of a LED system is shown in FIG. 5(a), assuming that (i) the N LEDs are placed on the same heatsink with thermal resistance of Rhs, (ii) each LED has a junction to case thermal resistance Rjc and (iii) a thermal conductor with electrical isolation (such as heatsink compound) is used to isolate the LEDs from the heatsink and which has a thermal resistance of Rins. A distributed thermal model is used for the heatsink due to its relatively large size. The corresponding thermal capacitances are needed if dynamic response is to be studied.
Under steady-state conditions, the thermal model can be further simplified into a steady-state model as shown in FIG. 5(b). In practice, a heat sink compound or equivalent may be used between the LEDs and the heat sink to ensure good thermal contact. The thermal resistance of such thermal conductor/electric insulator is relatively small when compared with Rjc of LEDs and is neglected in FIG. 5(b) and the following equations.
Based on the model in FIG. 5(b), the steady-state heatsink temperature can be expressed as:

Ths=Ta+Rhs(NPheat)=′Ta+Rhs(NkhPd)  (4)

where Ta=ambient temperature.

From FIG. 5(b) and (4), the junction of each LED is therefore:

Tj=Ths+Rjc(Pheat)=Ths+Rjc(khPd)  (5a)

Tj=Ta+(Rjc+NRhs)khPd  (5b)

Now, Tj obtained in (5) can be used in (2):

E=Eo└1+ke(Tj−To)┘

E=Eo{1+ke└Ta+(Rjc+NRhs)khPd−To┘}

E=Eo└1+ke(Ta−To)+kekh(Rjc+NRhs)Pd┘  (6)

So, the total luminous flux φν is:

φν=NEPd

φ=N{Eo└1+ke(Ta−To)+kekh(Rjc+NRhs)Pd┘}Pd

φν=NEo{[1+ke(Ta−To)]Pd+kekh(Rjc+NRhs)Pd2}  (7a)

Equation (7a) can also be expressed as follows:

φν=NEo{Pd+[ke(Ta−To)]Pd+kekh(Rjc+NRhs)Pd2}  (7b)

Several important observations can be made from equations (7a) and (7b).

    
    
        1. Equation (7) relates the luminous flux (φν) to the electrical power of the LED (Pd) and the thermal resistance of the heatsink (Rhs) and the LED junction (Rjc) together. It is an equation that integrates the photometric, electrical and thermal aspects of the LED system together.
        2. For a given heatsink (that may be restricted in size by a specific application), the operating point Pd* at which maximum φν occurs can be determined. Alternatively where there is flexibility in designing the heatsink, the equations can be used for thermal design to optimize the size of the heatsink (Rhs) for a given LED array.
        3. Because ke is negative and less than 1, (7) is in the form of φν=α1Pd−α2Pd2 where a1 and a2 are two positive coefficients. As Pd is increased from zero, φν increases almost linearly because the second term is negligible when Pd is small. As Pd increases, the second negative term which is proportional to the square of Pd will reduce φν significantly. After reaching the maximum point, the φν will drop faster as Pd and Rjc increase (due to the increasing significance of the negative terms in (7b)). This means that the parabola of φν is not symmetrical. Since the luminous flux function is a parabola and therefore has a maximum value, this maximum point can be obtained from
    
    






  
    
      d
      
        φ
        v
      
    
    
      d
      
        P
        d
      
    
  
  =
  0.




By differentiating (7) with respect to Pd,




  
    
      
        
          
            d
            
              φ
              v
            
          
          
            d
            
              P
              d
            
          
        
        =
        
          
            NE
            o
          
          ⁢
          
            {
            
              
                [
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                
                ]
              
              +
              
                2
                ⁢
                
                  k
                  e
                
                ⁢
                
                  
                    k
                    h
                  
                  ⁡
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                
                ⁢
                
                  P
                  d
                
              
              +
              
                
                  [
                  
                    
                      
                        (
                        
                          
                            T
                            a
                          
                          -
                          
                            T
                            o
                          
                        
                        )
                      
                      ⁢
                      
                        P
                        d
                      
                    
                    +
                    
                      
                        
                          k
                          h
                        
                        ⁡
                        
                          (
                          
                            
                              R
                              jc
                            
                            +
                            
                              NR
                              hs
                            
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        d
                        2
                      
                    
                  
                  ]
                
                ⁢
                
                  
                    d
                    
                      k
                      e
                    
                  
                  
                    d
                    
                      P
                      d
                    
                  
                
              
              +
              
                
                  [
                  
                    
                      
                        k
                        e
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  ]
                
                ⁢
                
                  
                    d
                    
                      k
                      h
                    
                  
                  
                    d
                    
                      P
                      d
                    
                  
                
              
              +
              
                
                  (
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  )
                
                ⁢
                
                  
                    d
                    
                      R
                      jc
                    
                  
                  
                    d
                    
                      P
                      d
                    
                  
                
              
            
            }
          
        
      
    
    
      
        (
        8
        )
      
    
  




It should be noted that the first two terms on the right hand side of (8) do not have derivatives, while the remaining three terms do. Strictly speaking, ke, kh and Rjc are not constant. It must be noted that Rjc will indeed increase significantly with lamp power.
The above equations can usefully be simplified for practical applications. As a first approximation, it is assumed that ke, kh, and Rjc are constant for the time being. It is known that kh will reduce slightly for a few percent under dimming conditions. From LED manufacturer data sheets the degradation of the efficacy with junction temperature is usually assumed to be linear and thus ke is assumed to be constant. This assumption is acceptable for ke and kh, and will be relaxed to accommodate the changing nature of Rjc in the analysis later. Based on this assumption, (8) can be simplified as:




  
    
      
        
          
            d
            
              φ
              v
            
          
          
            d
            
              P
              d
            
          
        
        =
        
          
            NE
            o
          
          ⁢
          
            {
            
              
                [
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                
                ]
              
              +
              
                
                  2
                  ⁡
                  
                    [
                    
                      
                        k
                        e
                      
                      ⁢
                      
                        
                          k
                          h
                        
                        ⁡
                        
                          (
                          
                            
                              R
                              jc
                            
                            +
                            
                              NR
                              hs
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  P
                  d
                
              
            
            }
          
        
      
    
    
      
        (
        9
        )
      
    
  




Therefore, maximum-φν point can be obtained by putting




  
    
      
        
          
            
              d
              
                φ
                v
              
            
            
              d
              
                P
                d
              
            
          
          =
          
            0
            ⁢
            
              
            
            ⁢
            and
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            d
            *
          
          =
          
            -
            
              
                [
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                
                ]
              
              
                2
                ⁢
                
                  k
                  e
                
                ⁢
                
                  
                    k
                    h
                  
                  ⁡
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where Pd* is the LED power at which maximum φν occurs. (Note that ke is a negative value.)

From (3), the corresponding LED current at which maximum φν occurs can be obtained as:




  
    
      
        
          I
          d
          *
        
        =
        
          -
          
            
              [
              
                1
                +
                
                  
                    k
                    e
                  
                  ⁡
                  
                    (
                    
                      
                        T
                        a
                      
                      -
                      
                        T
                        o
                      
                    
                    )
                  
                
              
              ]
            
            
              2
              ⁢
              
                k
                e
              
              ⁢
              
                
                  k
                  h
                
                ⁡
                
                  (
                  
                    
                      R
                      jc
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                V
                d
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




Several significant observations can be made from (10) and (11).

    
    
        1. Equations (10) and (11) relate the optimal Pd and Id, respectively, to the thermal design of the LED system (i.e. thermal resistance of the heatsink Rhs and Rjc.
        2. The maximum luminous flux will occur approximately at a lamp power Pd* specified in (10). This Pd* will shift to a lower value if (Rjc+NRhs) is increased. This leads to the possibility that the Pd* may occur at a power level that is less than the rated power Pd(rated) of the LED.
        3. Based on the above comment, one should expect that the Pd* could be shifted to higher power level if a larger heatsink with lower Rhs is used.
        4. For many applications such as head lamps of vehicles and compact lamps for replacement of incandescent lamps, the size of the heatsink is highly restricted and the ambient temperature is high. In these cases, there is a high possibility that Pd* (at which maximum luminous flux is produced) will occur at a power level less than the rated power.
    
    


In practice, Rjc of the LED increases with lamp power. Therefore, a vigorous equation can be obtained from (8) as:




  
    
      
        
          
            d
            
              φ
              v
            
          
          
            d
            
              P
              d
            
          
        
        =
        
          
            NE
            o
          
          ⁢
          
            {
            
              
                [
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                
                ]
              
              +
              
                2
                ⁢
                
                  k
                  e
                
                ⁢
                
                  
                    k
                    h
                  
                  ⁡
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                
                ⁢
                
                  P
                  d
                
              
              +
              
                
                  (
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  )
                
                ⁢
                
                  
                    d
                    
                      R
                      jc
                    
                  
                  
                    d
                    
                      P
                      d
                    
                  
                
              
            
            }
          
        
      
    
    
      
        (
        12
        )
      
    
  




The function of Rjc is highly complex and it depends on several factors such as thermal resistance of the heatsink, ambient temperature, the LED size and mounting structure and even the orientation of the heatsink. Equation (7b) in fact provides the physical meaning of effects of the temperature-dependent Rjc. Since Rjc increases with lamp power Pd, the two negative terms (with ke which is negative) in (7b) will accelerate the reduction of the luminous flux as Pd increases. This effect should be noticeable when Pd exceeds the Pd*, resulting in a slightly asymmetric parabolic luminous flux function.
In order to verify the theory two types of LEDs are used: 3W cool white LEDs and 5W cool white LEDs from Luxeon K2 Star series. They are mounted on several heatsinks with thermal resistances of 6.3° C./W, 3.9° C./W and 2.2° C./W so that experiments can be performed to evaluate their luminous output under different lamp power operations.
Since the junction-to-case thermal resistance Rjc is a complex and nonlinear function of the lamp heat dissipation Pheat (which is equal to khPd) and the thermal design of the mounting structure, the theoretical prediction is based on a simplified linear function as follows:

Rjc=Rjco(1+kjcPd)  (13)

where Rjco is the rated junction-to-case thermal resistance at 25° C. and kjc is a positive coefficient. A typical linear approximation of Rjc is shown in FIG. 6.

If equation (13) is used in (7b), a more accurate luminous flux equation can be derived as:

φν=NEo{[1+ke(Ta−To)]Pd+[kekh(Rjco+NRhs)]Pd2+[kekhkjcRjco]Pd3}  (7c)

A: Tests on 3W LEDs

(i) On a Heatsink with Thermal Resistance of 6.3° C./W

A group of eight identical Luxeon K2 Cool-white 3W LEDs are mounted on a standard heatsink with a thermal resistance of 6.3° C./W. The efficacy of the LEDs is measured at rated power in an integrating sphere. The parameters required for the equation (7) are:
ke=−0.005, kh=0.85, Ta=28° C., T0=25° C., E0=41 Lumen/Watt, N=8, Rhs=6.3° C./W, Rjco=10° C./W and kjc=0.1° C./W2.
Now two equations can be derived from (7). If the Rjc is assumed to be constant as a first approximation (i.e. Rjc=Rjco)=

φν=323.08×Pd−84.2×Pd2  (14)

If Rjc is assumed to obey (13),

φν=323.08×Pd−84.2×Pd2−1.39Pd3  (15)

The luminous flux is measured in an integrating sphere. The measured total luminous flux for eight LEDs is used for comparison with calculated values. The measured and calculated total luminous flux values are plotted, not against the total power sum of eight LEDs but against one LED power because the eight LEDs are identical and are connected in series. Using the power of one LED in the x-axis allows one to check easily if the optimal operating point is at the rated LED power or not. The measured results and calculated results from (14) and (15) are plotted in FIG. 7. Several points should be noted:

    
    
        1. The theoretical curves generally have the same trends as the measured curve. This confirms the validity of the general theory.
        2. The maximum lumen/Watt point occurs at about Pd=1.9W, which is less than the rated power of 3 W. This result shows that the general theory can predict accurately the Pd* operating point which may not be the rated power. Equation (10) indicates that a large NRhs term will shift Pd* to the low power level of the LED.
        3. The two negative terms in this example can also be seen in (15). The asymmetry after the peak luminous output point is more noticeable in the theoretical curve obtained from (15) than from (14). Comparison of (14) and (15) shows that the effect of the variation of Rjc, which is reflected in the extra third term in (15) is the reason for the obvious asymmetry of φν.
        4. In summary, the simplified model (7b), which is the basis for (14), has the form of φν=α1Pd−α2Pd2, while the more vigorous model (7c), which is the basis for (15), has the form of φν=α1Pd−α2Pd2−α3Pd2. Therefore, the model in (7c) is more accurate than the model (7b) particularly when Pd has exceeded Pd*. However, since both simplified and vigorous models are accurate enough for the power less than Pd*, which is also the recommended useable power range of LEDs, both equations can be used in the design optimization procedure to be explained.
    
    


Based on (6), the efficacy function can also be obtained.

E=40.39−10.52Pd assuming Rjc is constant  (16)

E=40.39−10.52Pd−0.17Pd2 assuming Rjc obeys (13)  (17)

The measured efficacy values and the calculated values from (16) and (17) are displayed in FIG. 6. It is noted that the calculated values are consistent with measurements. The results obtained from (17) are more accurate than those from (16) when Pd is large.
(ii) On a Heatsink with Thermal Resistance of 4.5° C./W
Eight identical 3W LEDs are mounted on a larger heatsink with thermal resistance of 4.5° C./W. The measured and calculated total luminous output as a function of single LED power Pd are shown in FIG. 9. It is noted that the calculated values are generally consistent with measurements, except at very low power where the light output is low and the relative measurement error is large. The Pd* is 2.4W at an efficacy of 21 lumens/Watt in this case. The use of a larger heatsink with a smaller thermal resistance means that the NRhs term in the denominator of (10) is smaller than that in the previous case (with Rhs6.3° C./W). Therefore, Pd* has increased from 1.9W to 2.4W as expected from (10) and the efficacy from 21 lumens/Watt to 23 lumens/Watt.
The corresponding measured and calculated efficacy are shown in FIG. 10 and it can be seen that they are in good agreement.
(iii) On a Heatsink with Thermal Resistance of 2.2° C./W
Another eight 3W LEDs are mounted on an even larger heatsink with thermal resistance of 2.2° C./W for evaluation. The measured and calculated luminous output as a function of LED power Pd are shown in FIG. 11 and the corresponding results of the efficacy are included in FIG. 12.
The theoretical Pd* is now about 3.5W, which is higher than the rated power of 3 W. This again confirms the prediction by the theory (10) that Pd* will shift to the higher power level with a decreasing term of NRhs (i.e a larger heatsink with a lower Rhs). Therefore, the theory can be used to design the optimal heatsink for a particular operating power. On the other hand, it can also be used to predict the optimal operating power for a given heatsink.
B: Tests on 5W LEDs
In order to ensure that the theory can be applied to other LEDs, 5W LEDs are used for evaluation. They are mounted on two heatsinks with thermal resistance of 6.8° C./W and 10° C./W respectively
(i) On a Heatsink with Thermal Resistance of 10° C./W
Two 5W LEDs are mounted on a heatsink with thermal resistance of 10° C./W. For the theoretical calculation, the parameters used in (10) are ke=−0.00355, kh=0.85, Ta=28° C., T0=25, E0=38 Lum/W, N=2, Rhs=10° C./W, Rjc=13° C./W and kjc=0.13° C./W2. Fitting these parameters into (7) and assuming that Rjc will rise linearly with temperature, the luminous flux equation and the efficacy equation are expressed as (18) and (19), respectively, and they are plotted with practical measurements in FIG. 13 and FIG. 14, respectively. Despite only two 5W LEDs being used, the theoretical predictions based on the averaged values are in general agreement with the measurements.

φν=75.2Pd−7.57d2−0.296Pd3  (18)

E=37.6−3.78Pd−0.149Pd2  (19)

(ii) On a Heatsink with Thermal Resistance of 6.8° C./W

The previous experiments are repeated by mounting the two 5W LEDs on a larger heatsink with a thermal resistance of 6.8° C./W. FIG. 15 and FIG. 16 show the comparisons of the measured and calculated luminous flux and efficacy, respectively. In general, calculated and measured results are in good agreement. Comparisons of the peak luminous flux in FIG. 13 and FIG. 14 confirm once again that using a larger heatsink (with lower thermal resistance) can shift the optimal operating point to a higher lamp power level. For the heatsink with Rhs=10° C./W, the optimal point occurs at about 3.8W. For the heatsink with Rhs=6.8° C./W, this optimal point has shifted to about 6W.
An important conclusion can be drawn from these results. The peak luminous flux (i.e. maximum φν) occurs at a LED power Pd* that depends on the thermal design (i.e. the heatsink thermal resistance). In general, the larger the heatsink (the lower the heatsink thermal resistance or the better the cooling effects), the higher the peak luminous flux can be achieved. Since operating the LEDs at a power higher than their rated power will shorten the lifetime of LEDs drastically, the theory can be used to project the maximum luminous flux for a given thermal design. It can also be used to predict the optimal thermal design for maximum luminous flux output if the LEDs are designed to operate at rated power.
Pd* can be controlled by using different heatsinks with different thermal resistance. For a larger heatsink, Rhs becomes small and therefore Pd* will be shifted to the higher power level as shown in FIG. 17, where the values of Pd* are labeled as A, B, C and D as the size of heatsink (or cooling effect) increases. By assuming Rhs=0, a theoretical limit can be plotted as shown in FIG. 17. It is important to note that the operating LED power must not exceed the rated LED power (Prated) otherwise the lifetime of the LED will be shortened. Therefore, the intersection points of these curves with the rated power limit indicate how the light output can be maximized.
It should be noted that a reduction of Rhs corresponds to an increase in the cooling effect. One way to achieve increased cooling is to increase the size of the heatsink. In FIG. 17, it can be seen that two curves with maximum points marked by C and D have relatively high luminous flux at the rated power. The curve with maximum point D has a smaller Rhs and thus a larger heatsink than that with maximum point C. The increase of luminous flux at the rated power from using curve C to curve D is small, but the increase in the size of the heatsink is much larger in proportion.
Three important points are highlighted here:

    1. The maximum φν is the point of inflexion of the luminous flux function (7b) or (7c). As Pd increases from zero, the positive slope of the curve






  (
  
    i
    .
    e
    .
    
      
    
    ⁢
    
      
        d
        
          φ
          v
        
      
      
        d
        
          P
          d
        
      
    
  
  )




is gradually decreasing to zero when the peak of the curve is reached. A large positive slope means that a relatively small increase of Pd can result in a relatively large increase of φν. So the initial linear portion of the curve results in good efficacy. As Pd is moved to the region at and around Pd*, the slope is zero or relatively small. Therefore, a relatively large increase in Pd will give a relatively small increase in φν.

    2. The LED power Pd must not exceed the rated LED power Pd(rated). Otherwise, the lifetime of the LED will be shortened. Therefore, the intersection points of these curves with the rated power limit should indicate how the light output can be maximized. The intersection points of these curves and the rated power line are denoted as “a”, “b”, “c” and “d” as shown in FIG. 17.
    3. The values of φν at “c” and “d” are higher than that at “b”. But the curve with peak φν at D requires a much larger heatsink than that with peak φν at C. The difference of φν at “c” and “d” may not be significant enough to justify an increase in cost and size of the heatsink.


The following rules are proposed as an optimization.
Rule 1:
The function of the luminous flux φν versus LED power Pd is a parabolic curve with a maximum point. The operating point Pd should be chosen at or below the maximum point Pd*. This means that for a given luminous flux output, the lower LED power should be chosen. Within this recommended power range, either (7) or (14) can provide sufficiently accurate predictions.
Rule 2:
If the thermal design is restricted by limited space for the heatsink so that the Pd* occurs at a power less than or equal to the rated power P(rated), then the LED system should be operated at Pd* for each LED device. [For example, points A and B are optimal operating points for the respective curves as their Pd* values do not exceed P(rated).]
Rule 3:
If the thermal design is flexible, then the LED system should be designed in such a way that (i) the theoretical maximum φν point (or Pd*) occurs at a power higher than P(rated) of the LED and (ii) the intersection point of the theoretical φν-Pd curve and the rated power line should have a value of about 80% to 96% of the theoretical maximum φν value. The rated power should be chosen as the operating power for each LED.
Rule 3 is an important idea. Where the theoretical maximum (Pd* for maximum φν) occurs at a point higher than the rated power, one should still operate the LED system at the rated power. As can be seen from FIGS. 17 and 18 the slope of the curve shows that as Pd* is approached the increase in luminous flux becomes very small and in terms of efficiency the gain in flux is not worth the additional power used. By using Δφν|p11=0.04 ̃0.20 of the maximum φν point in the curve the 4%-20% range for Δφν from the maximum φν point offers a good compromise of the light output and the size and thus cost of the heatsink.
If forced cooling is applied, the φν-Pd curve will change dynamically. This can be visualized as having a dynamically changing thermal resistance Rhs. The optimal operating point should follow the three rules explained previously. It should be kept along the operating lines as highlighted in the bold solid lines in FIG. 19 in order to maximize the luminous flux output.
FIG. 20 shows schematically an apparatus that may be used in embodiments of the invention. The apparatus comprises a microprocessor control unit (MCU) 20 that performs functions to be described below, database 21, user input means 22 which may be a keyboard, touchscreen or any other means that enables a user to display data, and output display means 23 which may be a screen, print out or any other means for data output to be communicated to a user. Database 21 includes details obtained from datasheets of the physical and electrical parameters of all known commercially available LEDs. This database may be provided directly as part of the apparatus or may a database kept elsewhere and accessed remotely. A user will input selected parameters of a desired LED lighting system using input means 22. These parameters will include at least the desired luminous flux output required from the system, and may further include any other parameters that the user wishes to fix, including for example the number N of LEDs and the size of the heatsink if that is fixed.
MCU 20 is programmed to carry out the steps shown in the flowcharts of FIGS. 21 and 22. Beginning with FIG. 21, a user may input a required flux output φν. The MCU may then select a candidate LED from the database and calculated the maximum flux that the LED is capable of achieving from Eq.7 above. If this maximum flux is obtained at a power below the rated power, and if this maximum flux is equal to or greater than the required flux, then such an LED is capable of being used in an LED system meeting the desired flux output and the process can stop. If the answer to either question is negative, ie if the maximum flux would only be obtained at a power greater than the rated power, or if the maximum flux is insufficient, then another LED is chosen and the process repeats.
If no LED can be found by the process of FIG. 21, either after a number of attempts or after all LEDs in the database have been exhausted, the MCU runs the process of FIG. 22 (or alternatively a user may go directly to the process of FIG. 22 if preferred). This process corresponds to the situation where the peak of the flux output occurs at a power above the rated power. In such cases, as discussed above, it is preferable to select an LED with a peak flux output such that the maximum rated power is between 80% and 96% of the power at which the maximum flux is obtained. As can be seen from lines C and D in FIG. 17 the output of these lines where they cross the rated power is higher than the equivalent flux of an LED that has its peak flux output exactly when operated at the rated power (the condition of line B). In FIG. 21 a peak output is calculated by the MCU at a power higher than a rated power and then various LEDs are chosen from the database until one is found where the flux output φν at the rated power is equal to or greater than the required power φreq.
As mentioned above, unlike traditional power electronic circuits, the main factor that needs to be optimized in a lighting system is the luminous efficacy instead of energy efficiency. Good luminous efficacy would automatically imply good efficacy, but not vice versa. The general (steady-state) photo-electro-thermal (PET) theory for LED systems, as described above, has pointed out both theoretically and practically that an LED device will not generate the maximum amount of luminous output at its rated power unless the LED system is optimally designed in an integrated manner. Optimal design of LED systems can only be achieved with the proper choice of LED devices and array structures, LED drivers, operating powers and thermal designs. Various aspects of LED systems such as the thermal management, non-linear behavior of junction-to-case thermal resistance, LED drivers and current sharing techniques for LED strings have been reported. While the interactions of photometric, electric and thermal aspects of the LED systems have been linked together, very limited research on the time-dependency of the luminous performance of LED systems has been reported.
This time factor is in fact highly relevant to the actual luminous output of any LED system. For LED systems designed for continuous operation, such as road lighting systems and LED bulbs, the luminous output will drop from the initial to steady-steady operation. The main reason is due to the gradual increase in the LED junction and heatsink temperature values. Therefore, the time factor is critical in understanding how much luminous reduction an LED system will have so that proper LED systems can be designed. For LED systems with momentarily or discontinuous operation, such as traffic lights and signal indicators of vehicles, the design criteria would be different from those for continuous operation.
The PET theory describes the steady-state performance of an LED system. The following is a photo-electro-thermo-temporal (PETT) theory that incorporates the time domain into the PET theory so that the dynamic behavior of the LED systems can be studied. The PETT theory can also be termed a dynamic PET theory. This PETT theory or dynamic PET theory will converge to the steady-state theory under steady-state conditions. Since the luminous efficacy is the main design factor for LED systems, this dynamic theory provides a useful tool not only for studying the behavior of LED systems, but more importantly, for optimizing LED system designs for both continuous and discontinuous operations.
As a special application of this PETT theory, the dynamic variation of the junction temperature can be predicted accurately. This theory therefore provides a valuable numerical design tool for LED systems.
In the following description of the PETT theory, the symbols and labels for various quantities and terms in formulae refer primarily to those quantities and terms under the described PETT theory. However, it will be appreciated that some of the symbols and labels refer to equivalent quantities and terms under the PET theory described above. Furthermore, where a formula under the PETT theory has an equivalent formula under the PET theory described above, the reference numeral of the equivalent formula under the PET theory appears in square brackets “[ ]” and in italics after the reference numeral of the formula under the PETT theory, so that the equivalent formulae can be easily compared.
FIG. 23 shows a simplified thermal equivalent circuit of N LEDs mounted on a heatsink. The heat source Pheat is the amount of heat generated by the LED. The internal junction temperature of the LED device is Tj. The thermal resistor and thermal capacitor of the power device are labeled as Rjc and Cjc respectively. Ths is heatsink surface temperature, Rhs and Chs are heatsink thermal resistor and thermal capacitor respectively. In practice, the LED packages are mounted on the heatsink with some form of electrical insulation or with the use of thermal paste to ensure good thermal contact. The thermal resistance of this thermally conductive layer is usually much smaller than Rjc and Rhs and is thus ignored in the following analysis.
The amount of heat generated by an LED can be expressed as:

Pheat=khPd  (T1)

where kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat and Pd is the input power of each LED.

In FIG. 23, some heat (P1) flows from the equivalent heat source through the equivalent thermal resistor, and another portion of the heat (PCj) flows into the equivalent thermal capacitor of the LED package. Based on the circuit analysis technique, it can be shown that:




  
    
      
        
          P
          
            C
            j
          
        
        =
        
          
            
              Nk
              h
            
            ⁢
            
              P
              d
            
          
          -
          
            P
            1
          
        
      
    
    
      
        (
        T2
        )
      
    
  
  
    
      
        
          P
          
            C
            j
          
        
        =
        
          
            NC
            jc
          
          ⁢
          
            
              d
              
                (
                
                  
                    T
                    j
                  
                  -
                  
                    T
                    a
                  
                
                )
              
            
            
              d
              t
            
          
        
      
    
    
      
        (
        T3
        )
      
    
  
  
    
      
        
          P
          1
        
        =
        
          
            
              T
              j
            
            -
            
              T
              hs
            
          
          
            
              R
              jc
            
            /
            N
          
        
      
    
    
      
        (
        T4
        )
      
    
  




Based on (T2), (T3) and (T4),




  
    
      
        
          
            NC
            jc
          
          ⁢
          
            
              d
              
                (
                
                  
                    T
                    j
                  
                  -
                  
                    T
                    a
                  
                
                )
              
            
            
              d
              t
            
          
        
        =
        
          
            
              Nk
              h
            
            ⁢
            
              P
              d
            
          
          -
          
            
              
                T
                j
              
              -
              
                T
                hs
              
            
            
              
                R
                jc
              
              /
              N
            
          
        
      
    
    
      
        (
        T5
        )
      
    
  




Rearranging (T5) gives




  
    
      
        
          
            
              d
              
                T
                j
              
            
            
              d
              t
            
          
          +
          
            
              1
              
                
                  C
                  jc
                
                ⁢
                
                  R
                  jc
                
              
            
            ⁢
            
              T
              j
            
          
        
        =
        
          
            
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            
              C
              jc
            
          
          +
          
            
              T
              hs
            
            
              
                C
                jc
              
              ⁢
              
                R
                jc
              
            
          
        
      
    
    
      
        (
        
          T
          ⁢
          
            
          
          ⁢
          6
        
        )
      
    
  




In reality, both of the junction temperature and the heatsink temperature will change with time under normal operation. In practice, the thermal time constant of the LED package (τjc=RjcCjc which is typically tens of seconds) is much smaller than that of the heatsink (τhs=RhsChs which is typically hundreds of seconds). The heatsink temperature Ths will change much slowly than LED junction temperature Tj. In order to avoid confusion, the terms ‘fast transient’ and ‘slow transient’ refer to the time frames in the order of the τjc and τhs, respectively. Under fast transient situation, Ths can be considered as a constant. Based on this argument, the dynamic relationship between Tj and Ths under the fast transient condition can be obtained from (T6) as:



  
    [
    
      cf
      .
      
        
      
      ⁢
      
        (
        
          5
          ⁢
          a
        
        )
      
    
    ]
  




  
    
      
        
          
            T
            j
          
          =
          
            
              
                R
                jc
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                (
                
                  1
                  -
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                      
                    
                  
                
                )
              
            
            +
            
              T
              hs
            
          
        
      
      
        
          (
          T7
          )
        
      
    
  



For the heatsink, heat flows into the thermal capacitor Chs and the thermal resistor Rhs of the heatsink. The heat flow component P1 can be expressed as:

P1=PChs+P2  (T8)

where PChs is the heat flowing into the thermal capacitor of heatsink, and P2 is the heat through the thermal resistor of the heatsink. They can be formulated as:





  
    
      
        
          P
          
            C
            hs
          
        
        =
        
          
            C
            hs
          
          ⁢
          
            
              d
              
                (
                
                  
                    T
                    hs
                  
                  -
                  
                    T
                    a
                  
                
                )
              
            
            
              d
              t
            
          
        
      
    
    
      
        (
        T9
        )
      
    
  
  
    
      
        
          P
          2
        
        =
        
          
            
              T
              hs
            
            -
            
              T
              a
            
          
          
            R
            hs
          
        
      
    
    
      
        (
        T10
        )
      
    
  




From (T4), (T8), (T9) and (T10), the heatsink temperature is therefore:




  
    
      
        
          
            
              T
              j
            
            -
            
              T
              hs
            
          
          
            
              R
              jc
            
            /
            N
          
        
        =
        
          
            
              C
              hs
            
            ⁢
            
              
                d
                
                  (
                  
                    
                      T
                      hs
                    
                    -
                    
                      T
                      a
                    
                  
                  )
                
              
              
                d
                t
              
            
          
          +
          
            
              
                T
                hs
              
              -
              
                T
                a
              
            
            
              R
              hs
            
          
        
      
    
    
      
        (
        
          T
          ⁢
          
            
          
          ⁢
          11
        
        )
      
    
  




which can be rewritten as:





  
    
      
        
          
            d
            
              T
              hs
            
          
          
            d
            t
          
        
        =
        
          
            
              N
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  jc
                
              
            
            ⁢
            
              T
              j
            
          
          -
          
            
              
                
                  NR
                  hs
                
                +
                
                  R
                  jc
                
              
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  hs
                
                ⁢
                
                  R
                  jc
                
              
            
            ⁢
            
              T
              hs
            
          
          +
          
            
              T
              a
            
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
      
    
    
      
        (
        T12
        )
      
    
  




Now, the relationship of Tj and Ths obtained in (T7) can be used in (T12):




  
    
      
        
          
            d
            
              T
              hs
            
          
          
            d
            t
          
        
        =
        
          
            
              N
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  jc
                
              
            
            ⁢
            
              (
              
                
                  
                    -
                    
                      R
                      jc
                    
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    P
                    d
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                      
                    
                  
                
                +
                
                  
                    R
                    jc
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    P
                    d
                  
                
                +
                
                  T
                  hs
                
              
              )
            
          
          -
          
            
              
                
                  NR
                  hs
                
                +
                
                  R
                  jc
                
              
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  hs
                
                ⁢
                
                  R
                  jc
                
              
            
            ⁢
            
              T
              hs
            
          
          +
          
            
              T
              a
            
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
      
    
    
      
        (
        T13
        )
      
    
  




Solving equation (T13), the heatsink temperature Ths can be obtained as:




  
    
      
        
          
            T
            hs
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                
                  -
                  
                    R
                    jc
                  
                
                ⁢
                
                  C
                  jc
                
                ⁢
                
                  NR
                  hs
                
                ⁢
                
                  k
                  h
                
                ⁢
                
                  P
                  d
                
              
              
                
                  
                    C
                    jc
                  
                  ⁢
                  
                    R
                    jc
                  
                
                -
                
                  
                    C
                    hs
                  
                  ⁢
                  
                    R
                    hs
                  
                
              
            
            ⁢
            
              e
              
                -
                
                  t
                  
                    
                      C
                      jc
                    
                    ⁢
                    
                      R
                      jc
                    
                  
                
              
            
          
          +
          
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          +
          
            A
            ⁢
            
              
            
            ⁢
            
              e
              
                -
                
                  t
                  
                    
                      C
                      hs
                    
                    ⁢
                    
                      R
                      hs
                    
                  
                
              
            
          
          +
          
            T
            a
          
        
      
    
    
      
        (
        T14
        )
      
    
  




where A represents a constant, which can be determined from the physical boundary condition of the heatsink. The boundary condition of a heatsink is that at t=0 the heatsink temperature is equal to ambient temperature. That is,

Ths(t=0)=Ta  (T15)

Putting (T15) into (T14), the coefficient A can be obtained as:




  A
  =
  
    -
    
      (
      
        
          
            
              -
              
                R
                jc
              
            
            ⁢
            
              C
              jc
            
            ⁢
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          
            
              
                C
                jc
              
              ⁢
              
                R
                jc
              
            
            -
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
        +
        
          
            NR
            hs
          
          ⁢
          
            k
            h
          
          ⁢
          
            P
            d
          
        
      
      )
    
  




Putting A into equation (T13), the heatsink temperature can be obtained as:




  
    
      
        
          
        
        ⁢
        
          [
          
            cf
            .
            
              
            
            ⁢
            
              (
              4
              )
            
          
          ]
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            T
            hs
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                
                  -
                  
                    R
                    jc
                  
                
                ⁢
                
                  C
                  jc
                
                ⁢
                
                  NR
                  hs
                
                ⁢
                
                  k
                  h
                
                ⁢
                
                  P
                  d
                
              
              
                
                  
                    C
                    jc
                  
                  ⁢
                  
                    R
                    jc
                  
                
                -
                
                  
                    C
                    hs
                  
                  ⁢
                  
                    R
                    hs
                  
                
              
            
            ⁢
            
              e
              
                -
                
                  t
                  
                    
                      C
                      jc
                    
                    ⁢
                    
                      R
                      jc
                    
                  
                
              
            
          
          -
          
            
              k
              h
            
            ⁢
            
              
                P
                d
              
              ⁡
              
                (
                
                  
                    
                      
                        -
                        
                          NR
                          hs
                        
                      
                      ⁢
                      
                        R
                        jc
                      
                      ⁢
                      
                        C
                        jc
                      
                    
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          R
                          jc
                        
                      
                      -
                      
                        
                          C
                          hs
                        
                        ⁢
                        
                          R
                          hs
                        
                      
                    
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
            
            ⁢
            
              e
              
                -
                
                  t
                  
                    
                      C
                      hs
                    
                    ⁢
                    
                      R
                      hs
                    
                  
                
              
            
          
          +
          
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          +
          
            T
            a
          
        
      
    
    
      
        (
        T16
        )
      
    
  




By putting the result of (T16) into (T7), the complete dynamic relationship of Tj and Ths is:




  
    
      
        [
        
          cf
          .
          
            
          
          ⁢
          
            (
            
              5
              ⁢
              b
            
            )
          
        
        ]
      
    
    
      
        
      
    
  
  
    
      
        
          T
          j
        
        =
        
          {
          
            
              
                -
                
                  R
                  jc
                
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          NR
                          hs
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        jc
                      
                      ⁢
                      
                        R
                        jc
                      
                    
                  
                
              
            
            -
            
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          -
                          
                            NR
                            hs
                          
                        
                        ⁢
                        
                          R
                          jc
                        
                        ⁢
                        
                          C
                          jc
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        hs
                      
                      ⁢
                      
                        R
                        hs
                      
                    
                  
                
              
            
            +
            
              
                (
                
                  
                    R
                    jc
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
              ⁢
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            +
            
              T
              a
            
          
          }
        
      
    
    
      
        (
        T17
        )
      
    
  




The luminous efficacy (E) has the following relationship with the junction temperature Tj of the LED.

E=Eo└1+ke(Tj−To)┘  (T18)

Now, Tj obtained in (T17) can be used in (T18):




  
    
      
        [
        
          cf
          .
          
            
          
          ⁢
          
            (
            6
            )
          
        
        ]
      
    
    
      
        
      
    
  
  
    
      
        E
        =
        
          
            E
            o
          
          ⁡
          
            [
            
              1
              +
              
                
                  k
                  e
                
                ⁡
                
                  (
                  
                    
                      T
                      a
                    
                    -
                    
                      T
                      o
                    
                  
                  )
                
              
              +
              
                
                  k
                  e
                
                ⁢
                
                  
                    k
                    h
                  
                  ⁡
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                
                ⁢
                
                  P
                  d
                
              
              -
              
                
                  k
                  e
                
                ⁢
                
                  R
                  jc
                
                ⁢
                
                  k
                  h
                
                ⁢
                
                  
                    P
                    d
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            NR
                            hs
                          
                          ⁢
                          
                            C
                            jc
                          
                        
                        
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                          -
                          
                            
                              C
                              hs
                            
                            ⁢
                            
                              R
                              hs
                            
                          
                        
                      
                      +
                      1
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    -
                    
                      t
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          R
                          jc
                        
                      
                    
                  
                
              
              -
              
                
                  k
                  e
                
                ⁢
                
                  k
                  h
                
                ⁢
                
                  
                    P
                    d
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            -
                            
                              R
                              jc
                            
                          
                          ⁢
                          
                            C
                            jc
                          
                          ⁢
                          
                            NR
                            hs
                          
                        
                        
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                          -
                          
                            
                              C
                              hs
                            
                            ⁢
                            
                              R
                              hs
                            
                          
                        
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    -
                    
                      t
                      
                        
                          C
                          hs
                        
                        ⁢
                        
                          R
                          hs
                        
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        T19
        )
      
    
  




So the total luminous flux φν is




  
    
      
        [
        
          cf
          .
          
            
          
          ⁢
          
            (
            
              7
              ⁢
              b
            
            )
          
        
        ]
      
    
    
      
        
      
    
  
  
    
      
        
          
            φ
            v
          
          =
          
            NEP
            d
          
        
        ⁢
        
          

        
        ⁢
        
          
            φ
            v
          
          =
          
            NE
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          [
                          
                            1
                            +
                            
                              
                                k
                                e
                              
                              ⁡
                              
                                (
                                
                                  
                                    T
                                    a
                                  
                                  -
                                  
                                    T
                                    o
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        ⁢
                        
                          P
                          d
                        
                      
                      +
                      
                        
                          k
                          e
                        
                        ⁢
                        
                          
                            k
                            h
                          
                          ⁡
                          
                            (
                            
                              
                                R
                                jc
                              
                              +
                              
                                NR
                                hs
                              
                            
                            )
                          
                        
                        ⁢
                        
                          P
                          d
                          2
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          k
                          e
                        
                        ⁢
                        
                          k
                          h
                        
                        ⁢
                        
                          
                            R
                            jc
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  
                                    NR
                                    hs
                                  
                                  ⁢
                                  
                                    C
                                    jc
                                  
                                
                                
                                  
                                    
                                      C
                                      jc
                                    
                                    ⁢
                                    
                                      R
                                      jc
                                    
                                  
                                  -
                                  
                                    
                                      C
                                      hs
                                    
                                    ⁢
                                    
                                      R
                                      hs
                                    
                                  
                                
                              
                              +
                              1
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              t
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                            
                          
                        
                        ⁢
                        
                          P
                          d
                          2
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        k
                        e
                      
                      ⁢
                      
                        
                          k
                          h
                        
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  -
                                  
                                    R
                                    jc
                                  
                                
                                ⁢
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  NR
                                  hs
                                
                              
                              
                                
                                  
                                    C
                                    jc
                                  
                                  ⁢
                                  
                                    R
                                    jc
                                  
                                
                                -
                                
                                  
                                    C
                                    hs
                                  
                                  ⁢
                                  
                                    R
                                    hs
                                  
                                
                              
                            
                            +
                            
                              NR
                              hs
                            
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          -
                          
                            t
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                      
                      ⁢
                      
                        P
                        d
                        2
                      
                    
                  
                
              
              }
            
          
        
      
    
    
      
        (
        T20
        )
      
    
  




Equations (T16) to (T20) now form the dynamic equations that describe the essential variables of the LED system. These dynamic equations can converge to the steady-state equations as time variable t approaches infinity. As t→∞:
eqn.(T16) becomes

Ths(t)=NRhskhPd+Ta  (T21)[cf.(4)]

eqn.(T17) becomes

Tj=(Rjc+NRhs)khPd+Ta  (T22)[cf.(5b)]

eqn.(T19) becomes

E=Eo└1+ke(Ta−To)+kekh(Rjc+NRhs)Pd┘  (T23)[cf.(6)]

eqn.(T20) becomes

φν=NE{[1+ke(Ta−To)]Pd+kekh(Rjc+NRhs)Pd2}  (T24)[cf.(7b)]

It can be seen that (T21), (T22), (T23) and (T24) are identical to the steady-state equations of the general PET theory described previously (the reference numerals of the equivalent equations of the PET theory are shown in square brackets and italics directly after the reference numerals of these PETT equations).
Because the thermal time constant of the LED device package is typically in the order to tens of seconds (and that of the heatsink is typically tens of minutes), the junction temperature of the LED does not change significantly within tens of micro-seconds. Therefore, it can be assumed that the junction temperature remains constant if the time frame is small compared with the thermal time constant of the LED device package. For example, for an off-line LED system powered by the ac mains, the mains frequency is either 50 Hz or 60 Hz. So the period of the mains half cycle is typically 10 ms or 8.33 ms which is small compared with the time constant of the LED package of tens of seconds. Thus, the steady-state equation (T24) can be rewritten as:




  
    
      
        
          R
          jc
        
        =
        
          
            
              
                
                  φ
                  v
                
                
                  
                    NE
                    o
                  
                  ⁢
                  
                    k
                    e
                  
                  ⁢
                  
                    k
                    h
                  
                
              
              ⁢
              
                P
                d
                
                  -
                  2
                
              
            
            -
            
              
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                
                
                  
                    k
                    e
                  
                  ⁢
                  
                    k
                    h
                  
                
              
              ⁢
              
                P
                d
                
                  -
                  1
                
              
            
            -
            
              
                NR
                hs
              
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              
                P
                d
              
            
          
          >
          0
        
      
    
    
      
        (
        T25
        )
      
    
  




It is important to note that Rjc in (T25) can be determined from the measured LED power Pd and luminous flux φν and other measurable coefficients, heatsink temperature and temperature. With this Rjc obtained, the dynamic junction temperature can be obtained from (T17):




  
    
      
        
          T
          j
        
        =
        
          {
          
            
              
                -
                
                  R
                  jc
                
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          NR
                          hs
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        jc
                      
                      ⁢
                      
                        R
                        jc
                      
                    
                  
                
              
            
            -
            
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          -
                          
                            NR
                            hs
                          
                        
                        ⁢
                        
                          R
                          jc
                        
                        ⁢
                        
                          C
                          jc
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        hs
                      
                      ⁢
                      
                        R
                        hs
                      
                    
                  
                
              
            
            +
            
              
                (
                
                  
                    R
                    jc
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
              ⁢
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            +
            
              T
              a
            
          
          }
        
      
    
    
      
        (
        T17
        )
      
    
  




For the steady-state conditions, eqn.(T17) becomes

Tj=(Rjc+NRhs)khPd+Ta  (T22)

The equations of the dynamic model described above can form the basis of numerical design tools and methods as follows.
Provided the junction thermal resistance is known or approximated, the flowchart of FIG. 24 shows an example of using the above dynamic model in a numerical design tool or a method to determine:

    (i) the luminous flux;
    (ii) the luminous efficacy;
    (iii) the heatsink temperature; or
    (iv) the LED junction temperature.


This approach gives a good estimation of the luminous performance and can thus be used iteratively to fine tune LED system designs.
Further important embodiments of a numerical design tool and a method provide data on the junction temperature in a practical LED system in order to make sure that the junction temperature will not exceed the maximum temperature limits as specified in the LED device data sheets. FIG. 25 shows a flowchart of an embodiment of a prediction method or a numerical design tool based on practical measurements of the luminous flux and LED power. If the peak junction temperature exceeds the maximum limit, then the design should be modified.
Thus, the dynamic model described above can be embodied as numerical tools and methods, or form part of numerical tools and methods. Preferably, these numerical tools and methods are implemented, or partly implemented, on a computer. These computer-implemented numerical tools and methods can be in the form of a software tool, or can involve the use of a software tool. In some embodiments, the computer-implemented numerical tools, or the computer-implemented parts of the methods, are accessible from an internet website. In another embodiment, the computer-implemented numerical tools, or the computer-implemented parts of the methods, are stored on a portable computer-readable medium, such as a CD, DVD, or a flash memory device. These can then be sold separately for implementation on a user's computer.
Tests were carried out as practical confirmation of the accuracy of the numerical tools described above.
(i) Tests on CREE X Lamp XR-E 3W LEDs
Eight CREE 3W LEDs were mounted on heatsinks in two sets of tests with the heatsink thermal resistance equal to 1.8° C./W (Sample A) and 2.9° C./W (Sample B). Instead of using the constant Rjc value in the LED data sheet, the curves formed by the calculated Rjc values based on (T25) are used for predicting relationships of the luminous flux and LED power at different time intervals. Table 1 below shows the measured data and the corresponding calculated Rjc values when the two heatsinks are used.







TABLE 1





Parameter list for the calculation of Rjc of CREE X lamp XR-E LEDs mounted


on heatsinks with thermal resistances of 1.8° C./W and 2.9° C./W.























Heatsink
Power(watt)
1.94
2.27
2.65
3.00
3.36
3.73
4.10


with
Luminous
1247
1367
1476
1546
1617
1674
1711


1.8° C./W
flux (lumen)









thermal
Calculated
9.4
17.5
23.4
28.1
29.9
31.1
31.8


resistance
Rjc (° C./W)









Heatsink
Power(watt)
1.91
2.26
2.60
2.96
3.31
3.67
4.02


with
Luminous
1184
1294
1384
1443
1490
1521
1533


2.9° C./W
flux (lumen)









thermal
Calculated
10.4
18.5
23.6
28.0
30.1
31.3
32.1


resistance
Rjc (° C./W)









(a) On a Heatsink with Thermal Resistance of 1.8° C./W:

The curves of the luminous flux are measured and recorded at different time intervals as shown in FIG. 26. The parameters τLED=CjcRjc, τhs=ChsRhs, Rhs and Rjc are listed in Table 2 below. Based on the dynamic theory (PETT), the theoretical curves of the luminous flux obtained from (T20) at the same time intervals are plotted in FIG. 27.







TABLE 2







LED system parameters.















ke
kh
Eo
Ta
To
N
Rhs
τLED = CjcRjc
τhs = ChsRhs





−0.0025
0.87
90
28
25
8
1.8
45
520









(b) On a Heatsink with Thermal Resistance of 2.9° C./W:

The measured luminous flux curves are shown in FIG. 28. For the theoretical curves, the parameters τLED=CjcRjc, τhs=ChsRhs and Rhs are listed in Table 3 below. The theoretical luminous flux curves versus LED power at different time intervals generated by (T20) are plotted in FIG. 29. Very good agreements between the measured and theoretical curves are observed.







TABLE 3







LED system parameters.















ke
kh
Eo
Ta
To
N
Rhs
τLED = CjcRjc
τhs = ChsRhs





−0.0025
0.87
90
28
25
8
2.9
45
510









Based on (T20), the measured and calculated luminous flux versus time curves of the two examples above are plotted in FIG. 30. These results confirm the accuracy of the method and numerical tool for forming and optimizing the performance of LED systems as proposed in the present invention.
(ii) Off-Line LED Bulb
In another example, an off-line LED bulb (with an electronic LED driver and LED devices) mounted on a heatsink with thermal resistance of 4.1° C./W is simulated. In the simulation, Rjc is determined from equation (T25) using the practical luminous flux measurements. The variations of the junction temperature are predicted and recorded at different intervals (from 2 minutes to 27 minutes) after the LED bulb is turned on.
These junction temperature curves are plotted in FIG. 31. It is clear that this numerical tool can predict the gradual increase in junction temperature (due to the gradual temperature increase in the heatsink which has a longer thermal time constant of tens of minutes). From FIG. 31, it is important to note that, under steady-state conditions, the peak junction temperature exceeds 150° C., which is the maximum temperature of many LED devices. Such operation will definitely reduce the lifetime of the LED product. Therefore, this numerical tool allows the product designer to predict this important factor.
Now the heatsink with thermal resistance of 4.1° C./W is replaced by a large heatsink with thermal resistance of 2.2° C./W and the corresponding junction temperature curves are plotted in FIG. 32. It can be seen that the peak junction temperature now falls below the maximum temperature of 150° C. and therefore the product will be more reliable.
The internal junction temperature of LED packages cannot be easily assessed directly. In LED system design, it is important to make sure that the junction temperature of an LED does not exceed its maximum rating in order to prolong its lifetime. As shown above, the present invention offers a method and a numerical tool that can be used to predict the junction temperature in a computer simulation or other computational environment. It can be used as a design tool for LED systems.
Although the invention has been described with reference to specific examples, it will be appreciated by those skilled in the art that the invention can be embodied in many other forms. It will also be appreciated by those skilled in the art that the features of the various examples described can be combined in other combinations.

## Claims

1. A method of forming or optimizing an LED system to achieve a luminous performance over a time period, the LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, the method comprising the steps of:
(a) calculating a predicted luminous performance over the time period with a computer-implemented dynamic model; and
(b) modifying the LED system such that the predicted luminous performance matches the luminous performance, wherein luminous performance is one or more of luminous flux of the LED system, luminous efficacy of the LED system, heatsink temperature and junction temperature over the time period.

2. A method according to claim 1 wherein the computer-implemented dynamic model comprises the following equations for calculating the luminous flux of the LED system, the luminous efficacy of the LED system, the heatsink temperature and the junction temperature over the time period:



  
    
      
        T
        hs
      
      ⁡
      
        (
        t
        )
      
    
    =
    
      
        
          
            
              -
              
                R
                jc
              
            
            ⁢
            
              C
              jc
            
            ⁢
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          
            
              
                C
                jc
              
              ⁢
              
                R
                jc
              
            
            -
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  jc
                
                ⁢
                
                  R
                  jc
                
              
            
          
        
      
      -
      
        
          k
          h
        
        ⁢
        
          
            P
            d
          
          ⁡
          
            (
            
              
                
                  
                    -
                    
                      NR
                      hs
                    
                  
                  ⁢
                  
                    R
                    jc
                  
                  ⁢
                  
                    C
                    jc
                  
                
                
                  
                    
                      C
                      jc
                    
                    ⁢
                    
                      R
                      jc
                    
                  
                  -
                  
                    
                      C
                      hs
                    
                    ⁢
                    
                      R
                      hs
                    
                  
                
              
              +
              
                NR
                hs
              
            
            )
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  hs
                
              
            
          
        
      
      +
      
        
          NR
          hs
        
        ⁢
        
          k
          h
        
        ⁢
        
          P
          d
        
      
      +
      
        T
        a
      
    
  




  
    
      
    
    ⁢
    
      
        T
        j
      
      =
      
        {
        
          
            
              
                
                  
                    -
                    
                      R
                      jc
                    
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                      
                    
                  
                
                -
              
            
          
          
            
              
                
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                NR
                                hs
                              
                            
                            ⁢
                            
                              R
                              jc
                            
                            ⁢
                            
                              C
                              jc
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
                +
              
            
          
          
            
              
                
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    P
                    d
                  
                
                +
                
                  T
                  a
                
              
            
          
        
        }
      
    
  




  
    
      
    
    ⁢
    
      E
      =
      
        
          E
          o
        
        ⁡
        
          [
          
            
              
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      R
                      jc
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        P
                        d
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
      
    
  




  
    
      
    
    ⁢
    
      
        φ
        v
      
      =
      
        NE
        ⁢
        
          {
          
            
              
                
                  
                    
                      [
                      
                        1
                        +
                        
                          
                            k
                            e
                          
                          ⁡
                          
                            (
                            
                              
                                T
                                a
                              
                              -
                              
                                T
                                o
                              
                            
                            )
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        R
                        jc
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    
                      k
                      h
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                  ⁢
                  
                    P
                    d
                    2
                  
                
              
            
          
          }
        
      
    
    ⁢
    
      

    
  


wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
To is the rated temperature;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat;
Pd is the input power of each LED;
E is the luminous efficacy;
Eo is the rated luminous efficacy at the rated temperature
ke is the relative reduction of efficacy with increase in temperature;
φν is the total luminous flux; and
t is time.

3. A method according to claim 1 wherein the computer-implemented dynamic model is in the form of a software tool on a computer accessible through an internet website.

4. A method according to claim 1 wherein the computer-implemented dynamic model is in the form of a software tool stored on a portable computer-readable medium.

5. A method of optimizing an LED system, the LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction, the method comprising the steps of:
(a) calculating junction temperature over a time period with a computer-implemented dynamic mode; and
(b) modifying the LED system if a calculated peak junction temperature exceeds a maximum rated junction temperature so as to optimize the LED system;
wherein the computer-implemented dynamic model utilizes the following equation for calculating the junction temperature:





  
    T
    j
  
  =
  
    {
    
      
        
          
            
              
                -
                
                  R
                  jc
                
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          NR
                          hs
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        jc
                      
                      ⁢
                      
                        R
                        jc
                      
                    
                  
                
              
            
            -
          
        
      
      
        
          
            
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          -
                          
                            NR
                            hs
                          
                        
                        ⁢
                        
                          R
                          jc
                        
                        ⁢
                        
                          C
                          jc
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        hs
                      
                      ⁢
                      
                        R
                        hs
                      
                    
                  
                
              
            
            +
            
              
                (
                
                  
                    R
                    jc
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
              ⁢
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            +
            
              T
              a
            
          
        
      
    
    }
  



wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat
Pd is the input power of each LED; and
t is time.

6. A method according to claim 1 wherein the computer-implemented dynamic model utilizes the following equation for calculating Rjc:




  
    R
    jc
  
  =
  
    
      
        
          
            φ
            v
          
          
            
              NE
              o
            
            ⁢
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            2
          
        
      
      -
      
        
          
            1
            +
            
              
                k
                e
              
              ⁡
              
                (
                
                  
                    T
                    a
                  
                  -
                  
                    T
                    o
                  
                
                )
              
            
          
          
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            1
          
        
      
      -
      
        
          NR
          hs
        
        ⁢
        
          
        
        ⁢
        for
        ⁢
        
          
        
        ⁢
        
          P
          d
        
      
    
    >
    0
  



wherein:
To is the rated temperature;
Eo is the rated luminous efficacy at the rated temperature;
ke is the relative reduction of efficacy with increase in temperature; and
φνis the total luminous flux.

7. A method according to claim 5 wherein the computer-implemented dynamic model is in the form of a software tool on a computer accessible through an internet website.

8. A method according to claim 5 wherein the computer-implemented dynamic model is in the form of a software tool stored on a portable computer-readable medium.

9. A non-transitory computer readable medium containing program instructions for performing the function of a computer-implemented numerical tool comprising:
computer executable code that when executed on a processor performs the following:
calculating a predicated luminous performance over a time period for an LED system comprising a single or a plurality of LEDs on a heatsink with each LED having a junction;
modifying the LED system such that the predicted luminous performance matches a luminous performance;
wherein luminous performance is one or more of luminous flux of the LED system, luminous efficacy of the LED system, heatsink temperature and junction temperature over the time period.

10. A non-transitory computer readable medium according to claim 9 wherein the computer executable code that when executed on the processor further performs the following: calculating one or more of the luminous flux of the LED system, luminous efficacy of the LED system, heatsink temperature and junction temperature over the time period using the following equations:



  
    
      
        T
        hs
      
      ⁡
      
        (
        t
        )
      
    
    =
    
      
        
          
            
              -
              
                R
                jc
              
            
            ⁢
            
              C
              jc
            
            ⁢
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          
            
              
                C
                jc
              
              ⁢
              
                R
                jc
              
            
            -
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  jc
                
                ⁢
                
                  R
                  jc
                
              
            
          
        
      
      -
      
        
          k
          h
        
        ⁢
        
          
            P
            d
          
          ⁡
          
            (
            
              
                
                  
                    -
                    
                      NR
                      hs
                    
                  
                  ⁢
                  
                    R
                    jc
                  
                  ⁢
                  
                    C
                    jc
                  
                
                
                  
                    
                      C
                      jc
                    
                    ⁢
                    
                      R
                      jc
                    
                  
                  -
                  
                    
                      C
                      hs
                    
                    ⁢
                    
                      R
                      hs
                    
                  
                
              
              +
              
                NR
                hs
              
            
            )
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  hs
                
              
            
          
        
      
      +
      
        
          NR
          hs
        
        ⁢
        
          k
          h
        
        ⁢
        
          P
          d
        
      
      +
      
        T
        a
      
    
  




  
    
      
    
    ⁢
    
      
        T
        j
      
      =
      
        {
        
          
            
              
                
                  
                    -
                    
                      R
                      jc
                    
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                      
                    
                  
                
                -
              
            
          
          
            
              
                
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                NR
                                hs
                              
                            
                            ⁢
                            
                              R
                              jc
                            
                            ⁢
                            
                              C
                              jc
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
                +
              
            
          
          
            
              
                
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    P
                    d
                  
                
                +
                
                  T
                  a
                
              
            
          
        
        }
      
    
  




  
    
      
    
    ⁢
    
      E
      =
      
        
          E
          o
        
        ⁡
        
          [
          
            
              
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      R
                      jc
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        P
                        d
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
      
    
  




  
    
      
    
    ⁢
    
      
        φ
        v
      
      =
      
        NE
        ⁢
        
          {
          
            
              
                
                  
                    
                      [
                      
                        1
                        +
                        
                          
                            k
                            e
                          
                          ⁡
                          
                            (
                            
                              
                                T
                                a
                              
                              -
                              
                                T
                                o
                              
                            
                            )
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        R
                        jc
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    
                      k
                      h
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                  ⁢
                  
                    P
                    d
                    2
                  
                
              
            
          
          }
        
      
    
  


wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
To is the rated temperature;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat;
Pd is the input power of each LED;
E is the luminous efficacy;
Eo is the rated luminous efficacy at the rated temperature
ke is the relative reduction of efficacy with increase in temperature;
φν is the total luminous flux; and
t is time.

11. A non-transitory computer readable medium containing program instructions for performing the function of a computer-implemented numerical tool comprising:
computer executable code that when executed on a processor performs the following:
calculating junction temperature over a time period; and
modifying the LED system if a calculated peak junction temperature exceeds a maximum rated junction temperature so as to optimize the LED system;
wherein the computer-implemented numerical tool is arranged to calculate the junction temperature over the time period using the following equation:





  
    T
    j
  
  =
  
    {
    
      
        
          
            
              
                -
                
                  R
                  jc
                
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          NR
                          hs
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        jc
                      
                      ⁢
                      
                        R
                        jc
                      
                    
                  
                
              
            
            -
          
        
      
      
        
          
            
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          -
                          
                            NR
                            hs
                          
                        
                        ⁢
                        
                          R
                          jc
                        
                        ⁢
                        
                          C
                          jc
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        hs
                      
                      ⁢
                      
                        R
                        hs
                      
                    
                  
                
              
            
            +
            
              
                (
                
                  
                    R
                    jc
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
              ⁢
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            +
            
              T
              a
            
          
        
      
    
    }
  



wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat
Pd is the input power of each LED; and
t is time.

12. A non-transitory computer readable medium according to claim 11 wherein the computer executable code that when executed on the processor further performs the following:
calculating Rjc using the following equation:





  
    R
    jc
  
  =
  
    
      
        
          
            φ
            v
          
          
            
              NE
              o
            
            ⁢
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            2
          
        
      
      -
      
        
          
            1
            +
            
              
                k
                e
              
              ⁡
              
                (
                
                  
                    T
                    a
                  
                  -
                  
                    T
                    o
                  
                
                )
              
            
          
          
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            1
          
        
      
      -
      
        
          NR
          hs
        
        ⁢
        
          
        
        ⁢
        for
        ⁢
        
          
        
        ⁢
        
          P
          d
        
      
    
    >
    0
  



wherein:
To is the rated temperature;
Eo is the rated luminous efficacy at the rated temperature;
ke is the relative reduction of efficacy with increase in temperature; and
φν is the total luminous flux.

13. An LED system design optimization system comprising:
a processor:
a non-transitory computer readable storage media operably coupled with the processor, the non-transitory computer readable storage media containing computer executable instructions that when executed perform the following:
calculating a predicated luminous performance of an LED system design over a time period;
wherein the luminous performance is one or more of luminous flux of the LED system, luminous efficacy of the LED system, heatsink temperature and junction temperature over the time period;
wherein the LED system comprises a single or a plurality of LEDs on heatsink with each LED having a junction;
determining an optimization of the LED system design based on the calculation;
further modifying the LED system design based on the predicted luminous performance calculation, such that the modified LED system luminous performance matches an optimized luminous performance; and
manufacturing the LED system design based upon the optimized LED performance calculations.

14. An LED system design optimization system according to claim 13, wherein the LED system design optimization system is arranged to calculate one or more of the luminous flux of the LED system, the luminous efficacy of the LED system, the heatsink temperature and the junction temperature over the time period using the following equations:



  
    
      
        T
        hs
      
      ⁡
      
        (
        t
        )
      
    
    =
    
      
        
          
            
              -
              
                R
                jc
              
            
            ⁢
            
              C
              jc
            
            ⁢
            
              NR
              hs
            
            ⁢
            
              k
              h
            
            ⁢
            
              P
              d
            
          
          
            
              
                C
                jc
              
              ⁢
              
                R
                jc
              
            
            -
            
              
                C
                hs
              
              ⁢
              
                R
                hs
              
            
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  jc
                
                ⁢
                
                  R
                  jc
                
              
            
          
        
      
      -
      
        
          k
          h
        
        ⁢
        
          
            P
            d
          
          ⁡
          
            (
            
              
                
                  
                    -
                    
                      NR
                      hs
                    
                  
                  ⁢
                  
                    R
                    jc
                  
                  ⁢
                  
                    C
                    jc
                  
                
                
                  
                    
                      C
                      jc
                    
                    ⁢
                    
                      R
                      jc
                    
                  
                  -
                  
                    
                      C
                      hs
                    
                    ⁢
                    
                      R
                      hs
                    
                  
                
              
              +
              
                NR
                hs
              
            
            )
          
        
        ⁢
        
          e
          
            -
            
              t
              
                
                  C
                  hs
                
                ⁢
                
                  R
                  hs
                
              
            
          
        
      
      +
      
        
          NR
          hs
        
        ⁢
        
          k
          h
        
        ⁢
        
          P
          d
        
      
      +
      
        T
        a
      
    
  




  
    
      
    
    ⁢
    
      
        T
        j
      
      =
      
        {
        
          
            
              
                
                  
                    -
                    
                      R
                      jc
                    
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                      
                    
                  
                
                -
              
            
          
          
            
              
                
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                NR
                                hs
                              
                            
                            ⁢
                            
                              R
                              jc
                            
                            ⁢
                            
                              C
                              jc
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
                +
              
            
          
          
            
              
                
                  
                    (
                    
                      
                        R
                        jc
                      
                      +
                      
                        NR
                        hs
                      
                    
                    )
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    P
                    d
                  
                
                +
                
                  T
                  a
                
              
            
          
        
        }
      
    
  




  
    
      
    
    ⁢
    
      E
      =
      
        
          E
          o
        
        ⁡
        
          [
          
            
              
                
                  1
                  +
                  
                    
                      k
                      e
                    
                    ⁡
                    
                      (
                      
                        
                          T
                          a
                        
                        -
                        
                          T
                          o
                        
                      
                      )
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      R
                      jc
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        P
                        d
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    k
                    h
                  
                  ⁢
                  
                    
                      P
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
      
    
  




  
    
      
    
    ⁢
    
      
        φ
        v
      
      =
      
        NE
        ⁢
        
          {
          
            
              
                
                  
                    
                      [
                      
                        1
                        +
                        
                          
                            k
                            e
                          
                          ⁡
                          
                            (
                            
                              
                                T
                                a
                              
                              -
                              
                                T
                                o
                              
                            
                            )
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      P
                      d
                    
                  
                  +
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      
                        k
                        h
                      
                      ⁡
                      
                        (
                        
                          
                            R
                            jc
                          
                          +
                          
                            NR
                            hs
                          
                        
                        )
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      k
                      e
                    
                    ⁢
                    
                      k
                      h
                    
                    ⁢
                    
                      
                        R
                        jc
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                NR
                                hs
                              
                              ⁢
                              
                                C
                                jc
                              
                            
                            
                              
                                
                                  C
                                  jc
                                
                                ⁢
                                
                                  R
                                  jc
                                
                              
                              -
                              
                                
                                  C
                                  hs
                                
                                ⁢
                                
                                  R
                                  hs
                                
                              
                            
                          
                          +
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          t
                          
                            
                              C
                              jc
                            
                            ⁢
                            
                              R
                              jc
                            
                          
                        
                      
                    
                    ⁢
                    
                      P
                      d
                      2
                    
                  
                  -
                
              
            
            
              
                
                  
                    k
                    e
                  
                  ⁢
                  
                    
                      k
                      h
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              -
                              
                                R
                                jc
                              
                            
                            ⁢
                            
                              C
                              jc
                            
                            ⁢
                            
                              NR
                              hs
                            
                          
                          
                            
                              
                                C
                                jc
                              
                              ⁢
                              
                                R
                                jc
                              
                            
                            -
                            
                              
                                C
                                hs
                              
                              ⁢
                              
                                R
                                hs
                              
                            
                          
                        
                        +
                        
                          NR
                          hs
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        t
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                  
                  ⁢
                  
                    P
                    d
                    2
                  
                
              
            
          
          }
        
      
    
    ⁢
    
      

    
  


wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
To is the rated temperature;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat;
Pd is the input power of each LED;
E is the luminous efficacy;
Eo is the rated luminous efficacy at the rated temperature
ke is the relative reduction of efficacy with increase in temperature;
φν is the total luminous flux; and
t is time.

15. An LED system design optimization system comprising:
a processor:
a non-transitory computer readable storage media operably coupled with the processor, the non-transitory computer readable storage media containing computer executable instructions that when executed perform the following:
calculating junction temperature over a time period: and
modifying the LED system if a calculated peak junction temperature exceeds a maximum rated junction temperature so as to optimize the LED system;
wherein the LED system design optimization system is arranged to calculate the junction temperature over the time period using the following equation:





  
    T
    j
  
  =
  
    {
    
      
        
          
            
              
                -
                
                  R
                  jc
                
              
              ⁢
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          C
                          jc
                        
                        ⁢
                        
                          NR
                          hs
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        jc
                      
                      ⁢
                      
                        R
                        jc
                      
                    
                  
                
              
            
            -
          
        
      
      
        
          
            
              
                k
                h
              
              ⁢
              
                
                  P
                  d
                
                ⁡
                
                  (
                  
                    
                      
                        
                          -
                          
                            NR
                            hs
                          
                        
                        ⁢
                        
                          R
                          jc
                        
                        ⁢
                        
                          C
                          jc
                        
                      
                      
                        
                          
                            C
                            jc
                          
                          ⁢
                          
                            R
                            jc
                          
                        
                        -
                        
                          
                            C
                            hs
                          
                          ⁢
                          
                            R
                            hs
                          
                        
                      
                    
                    +
                    
                      NR
                      hs
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  -
                  
                    t
                    
                      
                        C
                        hs
                      
                      ⁢
                      
                        R
                        hs
                      
                    
                  
                
              
            
            +
            
              
                (
                
                  
                    R
                    jc
                  
                  +
                  
                    NR
                    hs
                  
                
                )
              
              ⁢
              
                k
                h
              
              ⁢
              
                P
                d
              
            
            +
            
              T
              a
            
          
        
      
    
    }
  



wherein:
N is the number of LEDs in the LED system;
Tj is the junction temperature of each LED;
Rjc is the thermal resistance of the junction of each LED;
Cjc is the thermal capacitance of the junction of each LED;
Ths is the heatsink temperature;
Rhs is the thermal resistance of the heatsink;
Chs is the thermal capacitance of the heatsink;
Ta is the ambient temperature;
kh is the heat dissipation coefficient that represents the portion of input power that is dissipated as heat;
Pd is the input power of each LED; and
t is time.

16. An LED system design optimization system of claim 15, wherein the LED system design optimization system is arranged to calculate Rjc using the following equation:




  
    R
    jc
  
  =
  
    
      
        
          
            φ
            v
          
          
            
              NE
              o
            
            ⁢
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            2
          
        
      
      -
      
        
          
            1
            +
            
              
                k
                e
              
              ⁡
              
                (
                
                  
                    T
                    a
                  
                  -
                  
                    T
                    o
                  
                
                )
              
            
          
          
            
              k
              e
            
            ⁢
            
              k
              h
            
          
        
        ⁢
        
          P
          d
          
            -
            1
          
        
      
      -
      
        
          NR
          hs
        
        ⁢
        
          
        
        ⁢
        for
        ⁢
        
          
        
        ⁢
        
          P
          d
        
      
    
    >
    0
  



wherein:
To is the rated temperature;
Eo is the rated luminous efficacy at the rated temperature;
ke is the relative reduction of efficacy with increase in temperature; and
φν is the total luminous flux.