Goals

• Design a RF Butterworth low-pass filter with passive elements.

• Populate the filter on printed circuit board (PCB) with surface mounted components and SMA connectors.

• Calculate ABCD and S parameters theoretically .

• Analyze S-parameters with network analyzer.

S: Refers to Scattering.

Definition of S-Parameter

S-parameters are power wave descriptors that permits us to define the input – output relations of a network in terms of incident and reflected power wave. The system must be in the linear mode of operation to perform S-parameter calculation.

S-Parameter Benefits

• Relate to familiar measurements (gain, loss, reflection coefficient, VSWR ..)

• Can cascade S-parameters of multiple devices to predict system performance. Change S-parameters to (ABCD or Chain Scattering Parameters) for computation.

• Relatively easy to obtain at high frequencies- measure voltage traveling waves with a vector network

analyzer- don't need shorts/opens which can cause active devices to

oscillate or self-destruct

• can compute ABCD, H, Y, or Z parameters from S-parameters if desired.

• can easily import and use S-parameter files in our electronic-simulation tools. (CAD programs, Flow-graph analysis)

Incident TransmittedS21

S11Reflected S22

Reflected

Transmitted Incident

b1

a1b2

a2S12

DUT

b1 = S11a1 + S12 a2

b2 = S21 a1 + S22 a2

Port 1 Port 2

a1: Incident power wave at Port 1b1: Total Reflected power wave at Port 1

(Reflected from Port 1, and Transmitted from Port 2)a2: Incident power wave at Port 2b2: Total Reflected power wave at Port 2

(Reflected from Port 2, and Transmitted from Port 1)

Sxy: x-destination port, y- indicates the measurement port

S11: Input Reflection coefficient at Port 1S21: Forward transmission coefficient (Gain or Loss)S22: Output reflection coefficient at Port 2S12: Reverse transmission coefficient (Isolation)

Measuring S-Parameters

Measured at Port 1

Measured at Port 2

These equations give the S-Parameters in terms of incident, and reflected power waves(Incident are ‘a’s, and Reflected are ‘b’s).

This model is from Agilent Corporation

Measuring S-Parameters

S 11 = ReflectedIncident

=b1a 1 a2 = 0

S 21 =Transmitted

Incident=

b2

a 1 a2 = 0

S 22 = ReflectedIncident

=b2a 2 a1 = 0

S 12 =Transmitted

Incident=

b1

a 2 a1 = 0

Incident TransmittedS 21

S 11Reflected

b 1

a 1

b 2

Z 0Load

a2 = 0DUTForward

IncidentTransmitted S 12

S 22Reflected

b 2

a2b

a1 = 0DUTZ 0

Load Reverse

1

NA matches the Load to line impedance

NA matches the IP or the source Impedance to line impedance

This slide is from Agilent Corporation

Measurement of Filter Parameters Using Network Analyzer

Butterworth Low-Pass Filter Design with DOSBOX Design Tool

Types of “Ladder filters”

Maximally Flat or Butterworth

Butterworth Low-Pass Filter LC Ladder Circuitwith DOSBOX Design Tool

•Ladder Filter: A network composed of alternating series and shunt reactive elements.

•The network is “doubly terminated” with the same source and load resistance.

Butterworth Low Pass-Filter Bode Plot with DOSBOX Design Tool

Theoretical Design of Butterworth Low-Pass Filter

•Maximally flat low-pass filter with impedance of 50 Ω.•Cutoff frequency (fc) of 1.8 MHz for AM band (535-1605 kHz). •Insertion loss(IL) at 2 fc (3.6 MHz) ≥ 20dB.

Design Criteria

Design•Butterworth low-pass filter has maximally flat frequency response in the pass band.•IL=10 log (1+ (ω/ωc) 2N) , ωc=2π fc, N=order of the filter.•IL≥20, ω/ωc=2 gives N≥3.31.•Selected order of the filter N=4.•From the table, coefficients for Butterworth low-pass filter:• g0 =1, g1 =0.7654, g2 =1.8478, g3 =1.8478, g4 =0.7654, g5 =1

Butterworth FilterOrder g1 g2 g3 g4 g5

1 22 1.414 1.4143 1 2 14 0.765 1.848 1.848 0.7655 0.618 1.618 2 1.618 0.618

Impedance and Frequency Normalized Butterworth Low-Pass Filter Table

Theoretical Design of Butterworth Low-Pass Filter

c

kk

LRLϖ⋅

=′ 0

c

kk R

CCϖ⋅

=′

0

cc fπϖ 2= Ω= 500R

nFR

Cc

35.17654.0

01 =

⋅=

ϖ

HRLc

µϖ

17.88478.101 =

⋅=

nFR

Cc

26.38478.1

02 =

⋅=

ϖ

HRLc

µϖ

38.37654.002 =

⋅=

Using these equations the scaled element values obtained.

g0 =1, g1 =0.7654, g2 =1.8478, g3 =1.8478, g4 =0.7654, g5 =1

Low-Pass filter with normalized elements

Implementation of Butterworth Low-Pass Filter

Practical Schematic with available components

•Using Digi_Key product index the parts were selected.•Surface mounted components (SMC) were selected because of minimal stray contribution. •These values were picked due to size, cost and availability.•The component’s package size is 0805, voltage rating 50 V and tolerance 5%.•If we select a component’s size smaller than 0805, the voltage rating is low and it is very difficult to work with.

1.35nF 3.27nF8.30uH

PCB Fabrication of Practical SchematicSilkscreen Layer

Bottom Copper Layer

•Used Express PCB for schematic layout and fabrication.•Silkscreen layer gives component placement on the double layer PCB board

Bottom copper layer shows the bottom layer of copper trace

PCB Fabrication of Practical SchematicTop Copper Layer

Complete PCB Board Layout

Top copper layer shows the top layer of copper trace

Completed PCB with components

Population of PCB board with SMC

Populated PCB board with SMC

ABDC Parameter Transformation Matrix for Elements

Shunt Admittance ⇔

101

Y

Series Impedance ⇔

10

1 Z

Y

Z

Theoretical Calculations of ABCD-Parameters of Practical Schematic

⋅

⋅

⋅

⋅

⋅

=

1101

101

101

101

101

101

0

2

2

1

1

0

Z

LjCj

LjCj

ZDCBA ϖ

ϖϖ

ϖ

−+−

−−+−405.0)0007.00081.0(

)6204.12489.20()0049.08772.2(j

jj

Calculation of ABCD parameters at cutoff frequency (1.8 MHz).

=

ABCD to S-Parameter Conversion

Theoretical Calculations of S-Parameters of Practical Schematic

=

2221

1211

SSSS

+−−−−−−

)0166.0604.0()000895.0489.0()000438.0489.0()0164.0604.0(

jjjj

Theoretical calculation of S11 at cutoff frequency (1.8 MHz).

Calculated S-parameters at cutoff frequency (1.8 MHz) using S-parameter calculator.

DZCZBA

DZCZBA

S+⋅++

−⋅−+=

00

00

11

jj

0075.0092.40625.0472.2

−−+−

=

j0164.06041.0 −=

Experimental results using Network Analyzer

SC OC

S21 Magnitude Plot using Network Analyzer

S21 Phase Plot using Network Analyzer

Network Analyzer Calibration

• Calibrating a network analyzer is crucial for high-accuracy measurements.

• The accuracy of network analysis is greatly influenced by factors external to the network analyzer

• Calibration removes one or more of the systematic errors using an equation called an error model.

• Each port is calibrated one at a time.• The calibration is done with open end, short end and 50 ohm load.

Network Analyzer Calibration

• Thru calibration is done between two ports, which avoids the leakage between test ports when no DUT is present.

• Measurement of high quality standards (for example, a short, open, load, and thru) allows the analyzer to solve for the error terms in the error model.

Measurement of Filter Parameters Using Network Analyzer

• We used Agilent E5071C four port network analyzer

• Our measurements were taken with two ports.

Measurement of Filter Parameters Using Network Analyzer

Connecting DUT to the Network Analyzer

•Torque wrench is used to tighten the SMA connector.•It ensures proper connection between the network analyzer and the DUT.

Conclusions

• Practically -3 dB attenuation obtained at 1.56 MHz but theoretically it is at 1.8 MHz

• The cutoff frequency change is due to stray effects of the components and the PCB board, soldering effects, availability of components and measurement uncertainty which is not included.

• Filter’s theoretical and software implementation using DOSBOX were comparable.

S-ParametersTheoretical Value Practical Values

Magnitude (dB) Phase (0) Magnitude (dB) Phase (0)

S11 -2.19 -1.56 -3.51 -17.4

S12 -3.11 179.95 -5.18 177

S21 -3.11 179.95 -5.11 178

S22 -2.19 181.87 -2.83 170

Comparison of Theoretical and Practical S-Parameter values at Cutoff Frequency.