# Design of Optimal Spiral Inductors in Expert

*Goran Stojanovic, University of Novi
Sad, Serbia*

**Introduction**

One of the key components in many RF ICs applications is the
inductor. It is very important that the inductor has optimal design, meaning
optimal geometry with the best possible characteristics [1]. Our computer-aided
optimization technique using geometric programming (GP) has been used to find
the optimum design for spiral inductors with different layout [2-4]. The goal
is selection of the best geometry of inductors to find optimal values of inductor
parameters (the number of turns and layout dimensions) and then drawing layouts
of optimal inductors. Silvaco ** Expert** is a powerful
tool for drawing spiral inductors with different mask geometry (square, octagonal
or circular). In this paper we presents a simple way to draw layouts of optimal
inductors in

**, without using parameterized cells.**

*Expert*

**Layout of Square Spiral Inductors**

The purpose of * Expert* Wiring tool
is to draw a wire in the active layer. In case of inductors we use Metal2 layer
to draw all spiral conductor segments, Metal1 to draw underpass, and layer Via
to link them. The current layer name is displayed in the Wire panel and all
wires are built in this layer until altered. To run a wire, select

**Tools > Wiring**… menu command. This launches the Wire panel, which can be seen in Figure 1. In this panel we enter input data for Wire width and Wire spacing, which are output results of our geometric programming optimization algorithm. When the wiring parameters are adapted, then drawing of inductor layout can start.

Figure 1. Adjustment of wire width and spacing. |

To start drawing inductor select **Edit > Create object
> Wire** (or second icon )

When drawing the angle of 90 should be adjusted, what is obtained
by selecting **Edit > Angle Mod > **90 deg (or by selecting
the third icon from the picture,

the first icon designates that we have selected any angle,
the second one means we have selected 45 angle and the third one designates
that we have selected 90 angle). In this way we acctually adjust the angle
used to start our wire shaped structure. If we select **Edit > Angle
Mod >** **All **and select** Edit > Numeric
input** (icon)

we can choose arbitrary angle by which our structure will move, and at the same time we can change the angle and width of the metal structure while we are drawing. In the course of drawing once we achive the desired length by the click of the mouse we designate the end of drawing because each and every part of the inductor has to be drawn with given precision in order to get suitable distance between the segments, in our cases 2 µm or 4 µm spacing.

It is clear that there are two metal layers: metal2 and metal1, as well as the layer for via. All these layers differ in color (Figure 2). The metal2 is green, metal1 is blue and via is yellow. In the metal2 the spiral itself is realized, while metal1 serves for creation of the underpass conductor for linking of the spiral’s center to the second external port.

Figure 2. The view of Layer bar. |

It also should be taken into account that in the course of drawing the parts of wire should be prolonged in order to prevent the cut interruption of the structure.

Figure 3. The square spiral inductor
in (zooming).Expert |

Lengths of the first four segments of the structure from the Figure 3 are:

(1) | |

(2) | |

(3) | |

(4) |

where* d _{IN}*,

*s*, w are layout parameters. The iner diameter is

*d*, the width of conductor segment is

_{IN}*w*, the spacing between adjacent conductor segment is

*s*. All of these parameters are output result of GP optimization algorithm. In such a case the first part of the equation (4) represents the previous length of the segment, while the second part of the equation is a step of enlargement of the part of the inductor. We must take care that we always have the two equal segments, except at the start, and after enlargement of segments. This means that the segment

*l*is equal to

_{4}*l*, (

_{5}*l*=

_{4}*l*), while segments

_{5}*l*and

_{6}*l*, (

_{7}*l*=

_{6}*l*) are enlarged for

_{7}*w+s*.

**Layout of Octagonal Spiral Inductors**

Let us fit the wire’s width to suit the width of our structure, (for example w=10.4 µm). The angle that the wire can move should be adjusted to be 45. In the course of drawing when we achieve a desired length of the segment by click of the mouse we pass to drawing of the next segment. Here also should be taken care of prolonging of the segments in order to get corresponding distance between segments of the inductor.

Prolongation of segments of the wire is being done on the basis of the following formula:

(5) | |

(6) | |

(7) |

In all that *l _{1}* and

*l*form the overall circle of the octagon, i.e. the first four segments of length

_{2}*l*and the following four ones are of length

_{1}*l*. The equation 7 designates the step between the two lengths, so that

_{2}*l*, and by that the four following segments have length

_{3}*l*=

_{3}*l*+ Step, and so on, as can be seen in Figure 4. Layouts of all optimally designed spiral inductors are depicted in Figure 5.

_{2}
Figure 4. Drawing process of an octagonal inductor
in .Expert |

Figure 5. Layout of optimal spiral inductors in
Layout Editor.Expert |

**Layout of Circular Spiral Inductors**

The layout of the circular inductor is drawn slightly different
from drawing of the octagonal and square inductors. The principle we use is
to draw first the circumference of corresponding radius and to cut off that
half that we do not need. This way we enlarge diameter to each semi-circle.
The equations used to draw the structure are:

(8) | |

(9) | |

(10) |

We select **Edit > Set Origin**, i.e. we place
the coordinate system there where we draw the structure. Select **Edit
> Create Object >** Circle (icon)

and draw first the circle with greater radius. Then, select
**View > Modify object **and **Edit > Numeric input**
and the window will open data for X and Y axis, X=Y=0 and data for radius, value
for r2 is entered. After that select **Edit > Region Mode > Hole**
(third icon)

first designates the normal mod of drawing, second icon designates
the merge mod, third icon designates hole mod, while the fourth icon also designates
hole selected mod). Then, select **Edit > Create Object > Circle
**and draw the circle of smaller radius. For values of coordinates X
and Y we write down X=Y=0, while for value of radius r1. Here we actually deducted
from the greater circle the smaller one and we obtained as the difference the
surface between these two circles needed for drawing of the circular inductor.
By selecting **Edit > Create Object > Box **we access to
removing of the negative part (lower half) of our circle in order to get the
upper part, which we need, for drawing of the structure.

The equation (10) represents a step between radiuses of different
segments needed for drawing of the circular structure. When drawing this structure,
user needs to pay attention as to which part of the circle is to be removed,
especially when linking segments. The segment that is drawn should be moved
slightly vertically or horizontally by selecting **Edit > Move**.
For fine displacement select** Edit > Move** (icon)

and **Edit > Numeric input **on the basis of
which we can input arbitrary values for displacement of the structure along
X and Y axis, so that we move the segment to the needed place with suitable
accuracy. Layouts of designed optimal circular inductors are depicted in Figure
5.

**Conclusion**

In this paper we present design of different layouts (square,
octagonal or circular) using * Expert* tool. All inductors
are optimized to have maximum quality fator (Q-factor) with constrain for minimal
spacing between adjacent conductor segments and for fixed inductance at operating
frequency (for our examples inductances are 4nH or 10nH at operating frequency
2.5GHz).

**can create all angle polygons for RF and microwave circuit elements, including different geometry of spiral inductors in many process technologies.**

*Expert*

Reference:

- M. Hershenson, S. S. Mohan, S. P. Boyd, and T. H. Lee: “Optimization of inductor circuits via geometric programming”, in Proc. Design Automation Conf., session 54.3, June 1999, pp. 994-998.
- Goran Stojanovic, Lj. D. Zivanov, M. Damnjanovic: “Improved method for optimisation of inductors using geometric programming“, Electronics, Banjaluka, vol. 7, no. 1, september 2003. godine, pp. 31-34.
- G. M. Stojanovic, Lj. D. Zivanov: “Comparison of optimal design of different spiral inductors”, 24th International conference on microelectronics (MIEL 2004), 17-19. May 2004, Ni, Serbia and Montenegro.
- Goran Stojanovic, Ljiljana Zivanov and Mirjana Damnjanovic: “Optimal design of octagonal inductors“, Buletinul Stiintific al Universitatii “Politehnica” din Timisoara, Romania, Seria automatica si calculatoare, periodica politechnica, Transactions on automatic control and computer science, Vol. 49 (63), pp. 219-222, 2004, ISSN 1224-600X.