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UNIT-4

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SUBJECT-NAS

(EEC-206)

CONCEPT OF NETWORK FUNCTION

1. Driving Point Functions

1.1 Driving point Impedance Functions

1.2 Driving point Admittance Functions

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2. Driving Point Transfer Functions

2.1 Voltage Transfer Functions

2.2 Current Transfer Functions

2.3 Transfer Impedance Functions

2.4 Transfer Admittance Functions

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Driving Point Impedance Function

Ratio of Laplace Transform of Voltage and current at either Port 1-1’ or 2-2’

Z11=V1(s)/I1(s) 1^st Port

or

Z22=V2(s)/I2(s) 2^nd Port

Driving Point Admittance Function

Ratio of Laplace Transform of Current and Voltage at either Port 1-1’ or 2-2’

Y11=I1(s)/V1(s) 1^st Port

or

Y22=I2(s)/V2(s) 2^nd Port

R,L, C in Impedance Function

Time Domain Element S (Freq) Domain Element

Resistance R R (ohm)

Inductance L LS (ohm)

Capacitance C 1/CS (ohm)

R,L C in Admittance Function

Time Domain Element S (Freq) Domain Element

Resistance R 1/R (mho)

Inductance L 1/LS (mho)

Capacitance C CS (mho)

Hurwitz Polynomials

Network Function Z(s)=P(s)/Q(s)

Then Q(s) must be Hurwitz Polynomial

Properties of Hurwitz Polynomial

1. All Coefficient must be positive
2. No missing term in Polynomial
3. If any term missing, Then only odd term or even term in polynomial
4. All factors of continued fraction must be positives

LC CIRCUIT PROPERTIES

1. All Poles and Zeros are lying on jw axis
2. Poles and Zeros are lying alternate
3. Slope of LC circuit Network Function is positive.

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Zero

Pole

Real Axis

Imaginary Axis

FOSTER’S 1^ST FORM

Valid only for Impedance Function

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Z(s)=K0/S + 2KiS/( S+Wi ) + KS

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Poles at Origin Poles at Imaginary Poles at infinity

2

1

2

Source: ECEmaterial.com

Example of Foster 1^st form

FOSTER’S 2^nd FORM

Valid only for Admittance Function

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Y(s)=K0/S + 2KiS/( S+Wi ) + KS

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Poles at Origin Poles at Imaginary Poles at infinity

2

2

1

Source: slideshare.net

Example of Foster 2^nd form

CAUER 1^st FORM & CAUER 2^nd FORM

1. 1^st Form deals with Pole at infinity

2. Do continued fraction & series element is inductor

3. 2^nd form deals with poles at zero

4. Do continued fraction & series element is capacitor

1^st Form Circuit Example

Note: 2^nd Form circuit is having just reverse means L replace with C and vice versa

RC, RL Network Function

Z & Y Function Properties

1. Poles & Zeros are lying on Real Axis of S Plane
2. Poles & Zeros are lying Alternate on Real Axis
3. Lowest Critical Freq. is Pole, Highest Critical Freq. is Zero

RC

RL

Pole

Zero

RC,RL Network Function

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Z & Y Function Properties

1. Poles & Zeros are lying on Real Axis of S Plane
2. Poles & Zeros are lying Alternate on Real Axis
3. Lowest Critical Freq. is Zero, Highest Critical Freq. is Pole

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RC

RL

Source: slideplayer.com

CAUER FORM 1^st & 2^nd OF RL , RC Immittance Function

Cauer 1^st Form

Source: ecematerial.com

Cauer 2^nd Form

Source: ecematerial.com

Low Pass Filter (Passive Filter)

RC Circuit and Frequency Response of Low Pass Filter

Source: electronicshub.org

High Pass Filter (Passive Filter)

RC Circuit and Frequency Response of High Pass Filter

Higher Order Low Pass Filter

Source: electronicstutorials.org

Note: Higher Order Means Higher Slope, Approaches to ideal Filter

Higher Order High Pass Filter