Figure 1: Coupling gap sources in the Port Definition dialog by associating more than one source with a single port.
[[Image:image106.png|thimb|300px|A series-parallel RLC combination that can be modeled as a lumped circuit in EM.Picasso.]][[Image:PMOM64.png|thumb|400px|EM.Picasso's Lumped Element dialog.]]=== Modeling Lumped Elements In Planar MoM in EM.Picasso ===
Lumped elements are components, devices, or circuits whose overall dimensions are very small compared to the wavelength. As a result, they are considered to be dimensionless compared to the dimensions of a mesh cell. In fact, a lumped element is equivalent to an infinitesimally narrow gap that is placed in the path of current flow, across which the device's governing equations are enforced. Using Kirkhoff's laws, these device equations normally establish a relationship between the currents and voltages across the device or circuit. Crossing the bridge to Maxwell's domain, the device equations must now be cast into a from o boundary conditions that relate the electric and magnetic currents and fields. [[EM.Cube]]'s [[Planar Module]] allows you to define passive circuit elements: '''Resistors'''(R), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them as shown in the figure below:
[[File:image106.png]]
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Figure 1: A series-parallel RLC combination that can be modeled as a lumped circuit in [[Planar Module]].
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Lumped elements are conceptualized in a similar way as gap or probe sources. They are indeed considered as infinitesimally narrow gaps placed in the path of current flow, across which Ohm's law is enforced. If a lumped element is placed on a PEC or conductive sheet trace, it is treated as a series connection. The boundary condition at the location of the lumped element is:
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:<math> V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{x}\cdot \mathbf{E_{gap}} \, dx = Z_L \int_W \hat{y} \cdot \mathbf{J_s} \, dy </math>
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where Z<sub>L</sub> is the total impedance across the two terminals of the series element. If the lumped element is placed on a slot trace, it is treated as a shunt connection that creates a current discontinuity. In this case, the magnetic current across the gap is continuous, and the boundary condition at the location of the lumped element is:
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:<math> I_{gap} = Y_L V_{in} \quad\quad \int_{\delta} J_Y^{fila} \, dx = Y_L \int_W E_y \, dy </math>
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:<math> \int_{\delta} \hat{x}\cdot\hat{n} \times (\mathbf{H_{gap}^+ - H_{gap}^-}) \, dx = Y_L \int_W \hat{y}\cdot\mathbf{M_s} \, dy </math>
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where Y<sub>L</sub> is the total admittance across the two terminals of the shunt element. If a lumped element is placed on a PEC via that is connected to a metal strip from one side and to a PEC ground plane from the other end, it is indeed as a series connection across a gap discontinuity at the middle plane of the via. If the via is short, it is meshed using a single prismatic element. In that case, the lumped element in effect shunts the metal strip to the ground. The boundary condition at the location of the lumped element across the PEC via is:
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:<math> V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{z}\cdot \mathbf{E_{gap}} \, dz = Z_L \int_S \hat{z} \cdot \mathbf{J_p} \, ds </math>
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[[File:PMOM69.png]]
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Figure 1: Using a shunt lumped element on a PEC via to terminate a metallic strip line.
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=== Defining Lumped Circuits ===
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[[File:PMOM64.png|thumb|400px|Lumped Element dialog]]
To define a lumped RLC circuit in your planar structure, follow these steps:
* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed out.<br />
[[EM.Cube]]'s [[Planar Module]] Picasso allows you to define a voltage source in series with a series-parallel RLC combination and place them across the gap. This is called an active lumped element. If you choose the '''Active with Gap Source''' option of the '''Lumped Circuit Type''' section of the dialog, the right section of the dialog entitled '''Source Properties''' becomes enabled, where you can you can specify the '''Source Amplitude''' in Volts (or in Amperes in the case of PMC traces) and the '''Phase''' in degrees. Also, the box labeled '''Direction''' becomes relevant in this case which contains a gap source. Otherwise, a passive RLC circuit does not have polarity.
If the project workspace contains an array of rectangle strip objects or PEC via objects, the array object will also be listed as an eligible object for lumped element placement. A lumped element will then be placed on each element of the array. All the lumped elements will have identical direction, offset, resistance, inductance and capacitance values. If you define an active lumped element, you can prescribe certain amplitude and/or phase distribution to the gap sources just like in the case of gap and probe sources. The available amplitude distributions include '''Uniform''', '''Binomial'''''', Chebyshev''' and '''Data File'''.