Variational Method of Analysis for Microstrip Lines

Purti R. S.

Introduction• Microstrip is a type of electrical transmission line which can

be fabricated using Printed Circuit Board technology, and is used to convey Microwave-frequency signals.

• It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate.

Microstrip

• The dielectric in between

the strip and the ground plane

does not fill the air region

above the strip• Hence most field lines in the

dielectric region concentrate

between the strip and the

ground plane• While some concentrate in the

region above the substrate• Therefore the medium is

inhomogeneous

Quasi – TEM fields

• Fundamental mode of propagation in homogeneous medium – pure TEM

• But in case of microstrip, phase velocity of the TEM fields in the dielectric is c/√εr whereas in air region it would be c .

• Due to the phase mismatch, a microstrip line cannot support a pure TEM mode and fields sustained are known as Quasi TEM fields.

Methods of Analysis• With this quasi TEM mode approximation, the calculation of

propagation parameters or the analysis of the line essentially reduces to the solution of 2D Laplace’s equation, subject to the boundary conditions determined by the geometry of the line.

• Two important methods to solve 2D Laplace’s equation are, – Conformal Transformation Method:

• Exact method• used to find propagation parameters of homogeneous stripline• complicated if used for inhomogeneous medium.

– Variational method:• Approximate method• simpler even for inhomogeneous medium

Basis of Variational Method• Consider a system of perfect conductors

S1, S2, ... , SN with Q1, Q2, ... , QN charges

on the conductors and held at potentials

V1, V2, ... , VN.

• Let the potential function in the space

surrounding the conductors be ϕ. • Like all fundamental forces of nature,

ϕ satisfies and is the solution to Laplace’s

Equation,

∇2ϕ = 0

• Boundary conditions being ϕ = Vi on Si with i = 1, 2, ... , N

• The stored electrostatic energy is given by,

We = (ε/2) ∫∇ϕ .∇ϕ dV

• Here the integration is carried over the entire volume containing the electric field E obtained from the gradient of ϕ.

Basis of Variational Method• Suppose the charges are slightly displaced,

but Vi is kept constant

• There will be a change in potential function

ϕ surrounding the conductors given by

ϕ+δϕ where δϕ is the incremental change

in potential.

• Change in We is given by,

δWe = (ε/2) [∫ (∇ ϕ+δϕ). (∇ ϕ+δϕ) dV - ∫∇ϕ .∇ϕ dV]

δWe = (ε/2) [∫∇ϕ .∇δϕ dV + ∫∇δϕ .∇δϕ dV]• Considering a surface S enclosing the entire volume and then

applying the divergence theorem,

δWe = (ε/2) ∫∇δϕ .∇δϕ dV

Basis of Variational Method

• From this we conclude that if we insert a trial function for the potential distribution, which differs by a small quantity δϕ from its correct value, the resulting value of We will differ from its true value by an amount proportional to (δϕ)2.

• The energy function We is thus a stationary function for the equilibrium conditions.

• We is also seen to be a +ve quantity.

• Hence, the true value of We is a minimum since any change from the equilibrium increases the energy function We

Variational Expression for Upper Bound on Capacitance

• Consider a two conductor transmission line

as shown. Conductor S2 is at +ve potential V0 and

conductor S1 is at zero potential.

• We per unit length along the line is given by,

We = (ε/2)∫∫|∇t ϕ|2dxdy

= (1/2) CV02

• Where C is the capacitance per unit length

of the line & V0 is the potential difference between the two conductors.

• V0 = - = ∇t ϕ . dl

•

• This is the variational expression for the capacitance of C.

Variational Expression for Upper Bound on Capacitance

• The variational method involves the determination of capacitance by employing a trial function containing several variable parameters.

• Let these parameters be αi with i = 1, 2, 3, ... , N.

• So that C = C(α1 , α2 , α3 , ... , αN)

• The condition that the integral is stationary leads then to the simultaneous equations

= 0

• Solution of N homogeneous equations yields the values of αi and hence the minimum value of C

Variational Expression for Lower Bound on Capacitance

• Poisson’s Equation:

Usually used in electrostatics to find the electric

potential ϕ for a charge distribution which is

described by its given density function.

∇2 ϕ = - ρf /ε

• ∇ is the divergence operator

• ρf is the free charge density

• ε is the permittivity• Consider a two conductor TEM Transmission line,

with its cross sectional view as shown.

• Conductor S2 is at +ve potential V0 and

conductor S1 is at zero potential.

Thus in our case, the Poisson’s equation becomes

∇2ϕ = - ρ (x0, y0) / ε

• where ρ (x0, y0) is the unknown charge distribution

on S2

Variational Expression for Lower Bound on Capacitance

• Aim: To find the unknown charge distribution

on S2 using Green’s function technique.

• Green’s function:

It is a type of function used to solve inhomogeneous

differential equations which have initial or boundary

conditions.• To find the Green’s function, we consider potential

due to point charge at (x0,y0)

• Greens function should follow Poisson’s distribution• Hence we have,

∇2ϕ = - δ(x-x0)δ(y-y0) / ε

• δ(x-x0) and δ(y-y0) are Dirac’s Delta functions

• The solution of this equation is the charge distribution

on S2 ϕ(x,y) = G( x,y | x0,y0 ) ρ (x0,y0) dl0

• On the surface of S2 the potential is nothing but V0

V0 = G( x,y | x0,y0 ) ρ (x0,y0) dl0

• This equation is used to obtain a variational expression for the Capacitance in which ϕ becomes a trial function. Multiply both sides by ρ(x,y) and integrate over S2

V0 ρ(x,y) dl = G( x,y | x0,y0 ) ρ(x,y) ρ (x0,y0) dldl0

• As can be seen , the LHS gives total charge per unit length of conductor S2 and is equal to CV0 where C is the capacitance per unit length.

ρ(x,y) dl = Q = CV0

(1/C) = (1/Q2) G( x,y | x0,y0 ) ρ(x,y) ρ (x0,y0) dldl0

(1/C) = (1/Q2) ϕ(x,y) ρ(x,y) dl

• This becomes the variational expression for lower bound on C

Variational Expression for Lower Bound on Capacitance

Variational Expression for Lower Bound on Capacitance

• It is a stationary expression for C for arbitrary first order changes in the functional form of ρ.

• Further when ρ is changed by a small value δρ there is a second order change in C. Squared change is always +ve.

• Thus , for any trial function, ρ (x0,y0), the calculated value of (1/C) is always larger than the true value, and thus we know that the value obtained is the lower bound on the capacitance.

Thank You