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RF and Microwave CircuitRF and Microwave CircuitDes gnDes gn

Course code: ELE 552

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A relatively easy way to implement lowpass

filters in microstri or stri line is to use

alternative sections of very high and verylow characteristics impedance.

Such filters are referred as stepped-

impedance or hi-Z, low-Z filters.

Occupy less space than open stub filters

Suitable for applications where sharp cut-

off is not needed

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LPF Contd..LPF Contd..

Regardless of the construction of this two port

, - ,

consisting of three impedances as shown in figure.

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For example, consider a length lof a

transm ss on ne a symmetr c two-port

network ) whose ABCD parameters are given

0cos sinl jZ lA B

0 s n cos

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By using conversion, Z parameters are

o ta ne as o ows

cotA

Z Z Z l

12 21 0

1

csc

C

Z Z jZ lC

By simple calculation we obtain

1 1 1 2 0 tanl

Z Z jZ

If is small and expressed in radiansl

1 1 1 2 0

lZ Z j Z

0

12 2 1 0

cscZ

Z Z jZ lj l

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Thus an electrically small transmission line

can be modeled as follows

Now, consider the case where thecharacteristic impedance of the

.

denote this characteristic impedance as Zh

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numerator (Zh) is relatively large, and thedenominator (l)is small, the impedance of

shunt device is very large.

So that we can approximate it as an opencircuit and now we have a further

simplification of our model as follows

hjX jZ l j L

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h h

jZ l j L Z l L

hZ lL

Now, consider the case where the

transmission line has a relatively low value,denoted as Z

land the lis small, the series

impedance value is very small considered asshort circuit.

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LPF Contd..LPF Contd..

So we can simplify the model as follows

ljl C

1lZ l

ll C Z

l

lZ

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So inductance and capacitance can be

and low impedances.

applying frequency and impedance scaling

the lengths of the inductor and capacitorsection can be calculated as below

Inductor 0h

lZ

C Z

Capacitor0Z

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Formulas forFormulas for MicrostripMicrostrip

Effective dielectric constant1 1 1

r r

2 2 1 21

eff

dW

dielectric constant

for2

8 A

A

e

/ 2W d

12 0.611 ln(2 1) ln( 1) 0.39r

dB B B

orr r

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01 1 0.11

0.23r rZ

A

377

r r

B

02 rZ

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cut-off frequency of 2.5 GHz. It isnecessary to have more than 20 dB

insertion loss at 4 GHz. The filter

impedance is 50 ohm. The highest andowes prac ca mpe ances are o m

and 20 ohm respectively. The substrate

0.158mm and relative permittivity of 4.2

and loss tan ent of 0.02