-lintegral: computes a line integral

To compute the complex voltage $V$, we enter the section -lintegral. Its menu is:

 ##############################################################################
 # Flags: nomenu, prompt, message,                                            #
 ##############################################################################
 # section: -lintegral                                                        #
 ##############################################################################
 # symbol    = e_1                                                            #
 # quantity  = e                                                              #
 # solution  = 1                                                              #
 #                                                                            #
 # direction = z                                                              #
 # component = z                                                              #
 # startpoint= ( 0.0, 0.0, -1.0e+30 )                                         #
 #    (used) : ( @x0: undefined, @y0: undefined, @z0: undefined )             #
 # length    = auto                                                           #
 #    (@length) : undefined                                                   #
 # beta      = 1.0                                                            #
 # frequency = auto                     -- [auto | Real]                      #
 ##############################################################################
 # @vreal= undefined        @vimag= undefined       @vabs= undefined          #
 ##############################################################################
 # doit, ?, return, end, help                                                 #
 ##############################################################################

In order to compute our voltage, we have to specify what field shall be integrated, what component of the field shall be integrated, along which direction we want to perform the integration, and what the startpoint shall be. We specify this and perform the integration (doit):

 -lintegral
    symbol= e_1
    direction= z
    component= z
    startpoint= ( 0, 0, @zmin)
    length= auto
    doit

Upon entering "?", gd1.pp shows us the changed menu now as:

 ##############################################################################
 # Flags: nomenu, prompt, message,                                            #
 ##############################################################################
 # section: -lintegral                                                        #
 ##############################################################################
 # symbol    = e_1                                                            #
 # quantity  = e                                                              #
 # solution  = 1                                                              #
 #                                                                            #
 # direction = z                                                              #
 # component = z                                                              #
 # startpoint= ( 0.0, 0.0, -360.0e-3 )                                        #
 #    (used) : ( @x0: 0.0, @y0: 0.0, @z0: -360.0e-3 )                         #
 # length    = auto                                                           #
 #    (@length) : 360.0e-3                                                    #
 # beta      = 1.0                                                            #
 # frequency = auto                     -- [auto | Real]                      #
 ##############################################################################
 # @vreal= 13.88403         @vimag= -14.67090       @vabs= 20.19905           #
 ##############################################################################
 # doit, ?, return, end, help                                                 #
 ##############################################################################

We see the startpoint that we entered is the same as the startpoint actually taken, the length of the integration path is 360 mm, and the result of the integration is: Real part is 13.88403 Volts, imaginary part is -14.67090 Volts, and the absolute value of the voltage is 20.19905 Volts. We can write these numbers down on paper, but we can also compute with them within gd1.pp. They are accessible as symbolic variables @length, @vreal, @vimag, @vabs. We will use these variables later.

For our shunt impedance computation, we have to decide what value of the three values we have to take. From a plot of the accelerating field strength as it is shown in figure 3.3, we see that the field strength is even with respect to the plane z=0. The accelerating voltage that would be seen by a particle traversing the whole structure would therefore be twice the real part of the voltage in the half structure. In a full structure, the imaginary part of the complex voltage would vanish. We therefore have to take as $V V^*$ four times the square of the real part @vreal.

Figure 3.3: $E_z$ component of the first mode on the axis x=y=0. Since only the part of the structure below the plane z=0 is modeled, we only have direct information about the field below z=0. Clearly, the $E_z$-component is even with respect to the plane z=0.