PTL

S Parameter

• S Parameters are used to characterized two port network at High Frequency rather than (Z,Y,H).

• S parameter are defined in term of travelling wave, which are natural variable to be used in a TL environment.

Travelling waves and S parameter• Consider a two port network:

• For Linear two port network, incident and reflected wave can be expressed in matrix form as:

Scattering matrix

Important Points

• Treat each wave travelling toward each port as incident wave.

• A travelling wave (incident or reflected) has both voltage and current component.

• Voltage and current component of travelling wave

• To obtain the scattering parameter we use variables ‘a’ and ‘b’. Which is defined as

• In terms of wave variables, we can write

• Power Flow is

• These wave variables are w.r.t input plane, So to find the wave variable in other plane, they are multiplied by the phase factors.

‘a’ and ‘b’ are called power variables

Definition of S parameter• S11:

– When the output is matched i.e a2=0.

• S22:– When the input is matched i.e a1=0.

• S21:– When the output is matched i.e a2=0.

• S12:– When the input is matched i.e a1=0.

• Similarly for three port network, we can have

• Two port network with power variables:

Definition

• S11 is the input reflection coefficient (i.e. ) when the output is matched.

• S22 is the output reflection coefficient (i.e. ) when the input is matched.

• S21 is the forward transfer (or ‘transmission’) coefficient with the output matched.

• S12 is the reverse transfer (or ‘transmission’) coefficient with the input matched.

Shifting of Reference Plane• Consider a plane AA is shifted to A’A’ by l1 distance.• a1’ and b1’ represents inward and outward travelling waves• Similarly• Consider a plane BB is shifted to B’B’ by l2 distance.• a2’ and b2’ represents inward and outward travelling waves

Example• Calculate the scattering parameters of the two port

network, normalized to Zo, also assumed that the two-port networks are connected at both the input and output to transmission lines of characteristic impedance Zo.

SOl

• Since input and output are connected to Zo, So no reflection S11=S22=0.

• Phase change due to the propagation delay through the 0.6λ, length of transmission line is 1.2π radians.

• The scattering matrix

Example• Calculate the scattering parameters of the two port network,

normalized to Zo, also assumed that the two-port networks are connected at both the input and output to transmission lines of characteristic impedance Zo.

SOl• We can redraw the same circuit as

(a) Embedded into two transmission lines of characteristic impedance Zo.

(b) Evaluating the scattering parameters of the central part

Sol• So, first we calculate the scattering parameters of just the section

containing the resistor Zo, redrawn as a line of zero length.• S11:

• Zin, is given by the resistance, Zo, in parallel with the load Impedance Zo, i.e. Zin = Zo/2.

• S11 =

Sol• S21 or Sf : To evaluate Sf, an incident wave ei1 is launched and

total voltage that present at A-A is evaluated.• This total voltage is thus: ei1 + er1

• Now, Looking at A-A from the output side, total voltage at A-A is also equal to ei2 + er2

• Since ei2 = 0, there being a matched load on the output side, which means

• S21=

Sol• Since the network is symmetrical from input to output, So

S11 and S21 are identical to S11 and S12.• The scattering parameters of the zero length section is

• To get the scattering parameters of the full circuit, we can add the appropriate phase factors.

• Phase shift introduced by 3λ/8 line is 135°• Phase shift introduced by λ/4 line is 90°• scattering parameters of the complete two-port network is

Transmission Parameter (ABCD)• Scattering parameters do not always present the simplest way

of dealing with certain problems, we use other parameters like ABCD parameters.

• The transmission parameters are most useful when two-port networks are cascaded.

• Multiplying the matrices of the individual two-port networks (in the correct order) simply gives the transmission matrix for the combination.

Cascade network

ABCD of Lossless TL• Derive the ABCD-parameters of a lossless transmission line of

length l?• The voltages and currents at any point ‘z’ on a transmission

line are given by

• When the output open-circuited

• When the output short-circuited

• Two port network is reciprocal and symmetrical, so A=D and AD - BC=1

• Using these two properties, we get

• So the complete ABCD matrix of a two port network (lossless TL) is

• We can also convert the ABCD parameter to find ‘S’ parameter and vice versa.

ABCD of Lossy TL

ABCD to S

Microstrip Transmission Line• A microstrip transmission line may be seen as a logical

transformation in stages from the familiar coaxial line as is seen in Figure

• Microstrip :– Strip is separated from the ground plane by dielectric substrate– The complete transmission line structure is no longer homogeneous

Coaxial line • Consider a Coaxial line• Charge on the center conductor, q per unit length• Leading to the electric flux density, Dr• Electric field strength Er• Potential difference between the conductors due to the charge

leads to the capacitance per unit length

• Where, a and b are inner and outer radii.

……………….(1)

Coaxial line • In circuit and component design, the characteristic impedance, Zo,

and the propagation coefficient, Y, are required. For lossless line

• Let Cd and Co be capacitances of the transmission line configuration with and without the dielectric filling, for homogeneously filled transmission line

• The inductance per unit length for any transmission line is independent of the dielectric properties of the line.

• Now the characteristic impedance, Zo,

(After substitution from eq(1))

Example

Sol

Stripline• A thin conducting strip of width W is centered between two wide

conducting ground planes, and the region between the ground planes is filled with a dielectric material.

• Striplines are usually constructed by etching the center conductor on a grounded dielectric substrate of thickness b/2 and then covering with another grounded substrate.

Stripline• Supports a TEM wave propagation.

Synthesis/Analysis?

• Synthesis: Electrical Parameters (Zo, Ɛr,h) Physical

parameters(W/h, b etc)

• Analysis:Physical parameters(W,h etc) Electrical

Parameters (Zo, Ɛr)

Analysis of Stripline (t=0)

• Given Parameters:– Width of the centre strip (w)– Permittivity of the substrate(Ɛr)– Height of the substrate(b)– Assume thickness of the centre strip to zero(t=0)

• Determine:– Characteristic Impedance (Zo)

Analysis of Stripline (t=0)

• Characteristic Impedance[17:Cohn SB]:

• (Where K(k) is the complete elliptic integral)

• An accurate and simple expression of the ratio K/K’ is given by [21: Hilberg W]

Analysis of Stripline (t=0)

• HOWE’S APPROXIMATE FORMULAS

COLLIN’S APPROXIMATION FORMULAS

• COLLIN’S APPROXIMATION FORMULAS

COLLIN’S Conductor Loss

Example

• Find the characteristic impedance of a stripline having following dimensions:

w= 6 mm b=20 mm Ɛr=2.32 t=0Compare the two formulae.

Sol• Step1: First calculate ‘k’, k=0.439199

• Step2: Select the correct expression for K/K’• Step3: Calculate K/K’ and K/K’=0.72835 K’/K=1.372 • Step4: Calculate Zo Zo=84.89 ohms (can be compared with TXLine)

Analysis of Stripline (t≠0)

• Characteristic Impedance (t≠0)[19:Wheeler HA]:

Stripline: Plot of Zo

strip thickness(t) vs Zo

• Given Parameters:– Characteristic Impedance (Zo)– Permittivity of the substrate(Ɛr)– Height of the substrate(b)Assume thickness of the centre strip to zero(t=0)

• Determine:– Width of the centre strip to thickness ratio(w/b)

Stripline Synthesis

Stripline Synthesis Formula

• For t=0

Stripline Synthesis Formula(t=0)

• HOWE’S APPROXIMATE FORMULAS

Repeat Same Example againFind the characteristic impedance of a stripline having following dimensions: w= 6 mm b=20 mm Ɛr=2.32 t=0

Ans: Zo=84.49 Ohms

Now do the synthesis of the same and find width(w).Given: Zo =84.49 Ohmb=20 mmƐr=2.32t=0Determine:W?

Sol• Step1

– Calculate x– x=1.371

• Step2– Calculate k– K=sqrt(0.196)=0.442

• Step3– Calculate w/b=0.3027

• Step4– W=6.055mm

Synthesis of Stripline (t≠0)

• Width(w) of a stripline (t≠0)[19:Wheeler HA]:

In case of a stripline

Dielectric Loss is given by:

Attenuation Loss is given by: