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Network Analysis Laplace Transform N - Port Networks Summary

1. Review of Network Theory

S. S. Dan and S. R. Zinka

Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus

January 22, 2016

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Outline

1 Network Analysis

2 Laplace Transform

3 N - Port Networks

4 Summary

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Outline

1 Network Analysis

2 Laplace Transform

3 N - Port Networks

4 Summary

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

What is Network Analysis?

You tell me ...

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

What is Network Analysis?

You tell me ...

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

A Few Important Theorems & Concepts

1 Millman’s Theorem

2 Superposition Theorem

3 Thevenin’s Theorem

4 Norton’s Theorem

5 Thevenin-Norton Equivalencies

6 Maximum Power Transfer Theorem

7 Y⇔ ∆ Conversions

8 Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2 +

-

+

-

R1

R2

R34 Ω

2 Ω

1 Ω

V328 V 7 V V1

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2 +

-

+

-

R1

R2

R34 Ω

2 Ω

1 Ω

V328 V 7 V V1

Vcommon

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2 +

-

+

-

R1

R2

R34 Ω

2 Ω

1 Ω

V328 V 7 V V1

Vcommon

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2 +

-

+

-

R1

R2

R34 Ω

2 Ω

1 Ω

V3

+

-

+

-

+

-

+

-

+

-

+

-

V1 V3

R1

R2

R3

28 V 7 V

8 V

1 V

8 V

20 V

28 V 7 V V1

Vcommon

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

1. Millman’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

V1 V2 +

-

+

-

R1

R2

R34 Ω

2 Ω

1 Ω

V3

+

-

+

-

+

-

+

-

+

-

+

-

V1 V3

R1

R2

R3

28 V 7 V

8 V

1 V

8 V

20 V

+

-

+

-

+

-

+

- +

-

R1

R2

R3

28 V 7 V

8 V

20 V 1 V

IR3

IR2

IR1

5 A4 A

1 A

V1 V3

28 V 7 V V1

Vcommon

Vcommon =

V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

R1

4 Ω

28 V R2 2 Ω

R3

1 Ω

B1 7 V

R1

R2

R3

4 Ω

2 Ω

1 Ω

B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

R1

4 Ω

28 V R2 2 Ω

R3

1 Ω

B1 7 V

R1

R2

R3

4 Ω

2 Ω

1 Ω

B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem ... Cont’d

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem ... Cont’d

R1

28 V R2

R3

+

-

+ - + -

+

-

24 V

4 V

4 V

6 A 4 A

2 A

B1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem ... Cont’d

R1

28 V R2

R3

+

-

+ - + -

+

-

24 V

4 V

4 V

6 A 4 A

2 A

B1

+

-

+

-

+-+-

R1

R2

R3

4 V

4 V

3 V

7 V

1 A

2 A

3 A

B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem ... Cont’d

R1

28 V R2

R3

+

-

+ - + -

+

-

24 V

4 V

4 V

6 A 4 A

2 A

B1

+

-

+

-

+-+-

R1

R2

R3

4 V

4 V

3 V

7 V

1 A

2 A

3 A

B2

+

-

+

-

+

-

+ - + -

R1

R2

R3

28 V 7 V

20 V

8 V

1 V

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2. Superposition Theorem ... Cont’d

R1

28 V R2

R3

+

-

+ - + -

+

-

24 V

4 V

4 V

6 A 4 A

2 A

B1

+

-

+

-

+-+-

R1

R2

R3

4 V

4 V

3 V

7 V

1 A

2 A

3 A

B2

+

-

+

-

+

-

+ - + -

R1

R2

R3

28 V 7 V

20 V

8 V

1 V

B1 B2

+

-

+

-

R1

R2

R3

28 V 7 V

5 A

4 A

1 A

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem

28 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B228 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B228 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

RThevenin

R2 (Load)EThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem ... Cont’d

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem ... Cont’d

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem ... Cont’d

R1 R3

28 V 7 VLoad resistorremoved

4 Ω 1 Ω

B1 B2

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem ... Cont’d

R1 R3

28 V 7 VLoad resistorremoved

4 Ω 1 Ω

B1 B2

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B2

R1 R3

28 V 7 V

4 Ω 1 Ω

+

-

+

-

+ -16.8 V

+ -4.2 V

4.2 A 4.2 A

11.2 V+

-B1 B2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

3. Thevenin’s Theorem ... Cont’d

R1 R3

28 V 7 VLoad resistorremoved

4 Ω 1 Ω

B1 B2

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B2

R1 R3

28 V 7 V

4 Ω 1 Ω

+

-

+

-

+ -16.8 V

+ -4.2 V

4.2 A 4.2 A

11.2 V+

-B1 B2

R1 R3

4 Ω 1 Ω

0.8 Ω

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

4. Norton’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

4. Norton’s Theorem

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B228 V 7 V2 Ω R2

R1 R3

4 Ω 1 Ω

B1 B2

INorton RNortonR2

2 Ω(Load)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

4. Norton’s Theorem ... Cont’d

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

4. Norton’s Theorem ... Cont’d

R1 R3

28 V 7 V

4 Ω 1 Ω

B1 B2

R1

R2 (Load)

R3

28 V 7 V

4 Ω

2 Ω

1 Ω

B1 B2

R1 R3

4 Ω 1 Ω

0.8 Ω

R1 R3

28 V 7 V

4 Ω 1 Ω7 A+

-

+

-

7 A

14 A

Ishort = IR1 + IR2

B1 B2

Load resistorremoved

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

5. Thevenin-Norton Equivalencies

INorton RNortonR2

2 Ω(Load)

14 A

0.8 ΩEThevenin

RThevenin

11.2 V

0.8 Ω

R22 Ω

(Load)

EThevenin = INortonRNorton = INortonRThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

5. Thevenin-Norton Equivalencies

INorton RNortonR2

2 Ω(Load)

14 A

0.8 ΩEThevenin

RThevenin

11.2 V

0.8 Ω

R22 Ω

(Load)

EThevenin = INortonRNorton = INortonRThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

5. Thevenin-Norton Equivalencies

INorton RNortonR2

2 Ω(Load)

14 A

0.8 ΩEThevenin

RThevenin

11.2 V

0.8 Ω

R22 Ω

(Load)

EThevenin = INortonRNorton = INortonRThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

6. Maximum Power Transfer Theorem

EThevenin

RThevenin

11.2 V

0.8 Ω

RLoad 0.8 Ω

RLoad = RThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

6. Maximum Power Transfer Theorem

EThevenin

RThevenin

11.2 V

0.8 Ω

RLoad 0.8 Ω

RLoad = RThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

6. Maximum Power Transfer Theorem

EThevenin

RThevenin

11.2 V

0.8 Ω

RLoad 0.8 Ω

RLoad = RThevenin

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

7. Y⇔ ∆ Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

Delta (Δ) network Wye (Y) network

RA =RABRAC

RAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

7. Y⇔ ∆ Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

Delta (Δ) network Wye (Y) network

RA =RABRAC

RAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

7. Y⇔ ∆ Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

Delta (Δ) network Wye (Y) network

RA =RABRAC

RAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

7. Y⇔ ∆ Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

Delta (Δ) network Wye (Y) network

RA =RABRAC

RAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

7. Y⇔ ∆ Conversions ... Cont’d

10 V

R112 Ω

R218 Ω

R3

6 ΩR418 Ω 12 Ω

R510 V

RA

RB RC

R4 R518 Ω 12 Ω

2 Ω 3 Ω

6 Ω

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

8. Miller’s Theorem

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

8. Miller’s Theorem

+

V2= KV1

1 2Z II

+

V1

Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

8. Miller’s Theorem

+

V2= KV1

1 2Z II

+

V2= KV1Z1 Z2

1 2

+

V1

+

V1

Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

8. Miller’s Theorem

+

V2= KV1

1 2Z II

+

V2= KV1Z1 Z2

1 2I2= II1= I

+

V1

+

V1

Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

8. Miller’s Theorem

+

V2= KV1

1 2Z II

+

V2= KV1Z1 Z2

1 2I2= II1= I

+

V1

+

V1

Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Outline

1 Network Analysis

2 Laplace Transform

3 N - Port Networks

4 Summary

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Laplace Transform – Definition

The Laplace transform is a frequency domain approach for continuous timesignals irrespective of whether the system is stable or unstable. The Laplacetransform of a function f (t), defined for all real numbers t ≥ 0, is the functionF(s), which is a unilateral transform defined by:

F(s) =∫ ∞

0−f (t) e−st dt. (1)

The parameter s is the complex number frequency s = σ + jω, with real num-bers σ and ω.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Laplace Transform – Definition

The Laplace transform is a frequency domain approach for continuous timesignals irrespective of whether the system is stable or unstable. The Laplacetransform of a function f (t), defined for all real numbers t ≥ 0, is the functionF(s), which is a unilateral transform defined by:

F(s) =∫ ∞

0−f (t) e−st dt. (1)

The parameter s is the complex number frequency s = σ + jω, with real num-bers σ and ω.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Laplace Transform – Properties

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Laplace Transform – Properties

Time domain s domain

Linearity af (t) + bg(t) aF(s) + bG(s)

Frequency-domain derivative tnf (t) (−1)nF(n)(s)

Time-domain derivative f (n)(t) snF(s)−∑nk=1 sn−kf (k−1)(0)

Frequency-domain integration 1t f (t)

∫ ∞s F(σ) dσ

Time-domain integration∫ t

0 f (τ) dτ = (u ∗ f )(t) 1s F(s)

Frequency shifting eatf (t) F(s− a)

Time shifting f (t− a)u(t− a) e−asF(s)

Time scaling f (at) 1a F( s

a

)Convolution (f ∗ g)(t) =

∫ t0 f (τ)g(t− τ)dτ F(s) ·G(s)

Complex conjugation f ∗(t) F∗(s∗)

Cross-correlation f (t) ? g(t) F∗(−s∗) ·G(s)

Periodic function f (t) 11−e−Ts

∫ T0 e−stf (t) dt

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Table of Selected Laplace Transforms

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Table of Selected Laplace Transforms

Time domain s domain

Unit impulse δ(t− τ) e−τs

Unit step u(t− τ) 1s e−τs

Ramp t · u(t) 1s2

nth power ( for integer n) tn · u(t) n!sn+1

Exponential decay e−αt · u(t) 1s+α

Sine sin(ω0t) · u(t) ω0s2+ω2

0

Cosine cos(ω0t) · u(t) ss2+ω2

0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Steady State Analysis of Electric Networks

R

Vi C Vo++

RVi

C

Vo++

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Steady State Analysis of Electric Networks

R

Vi C Vo++

RVi

C

Vo++

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks ... Cont’d

+

−

+

−V0

+

−

+

−VoutR

t = 0

Vout (s) = V0

[1

s + (1/RC)

]

Vout (t) = V0e−(1/RC)tu (t)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks ... Cont’d

+

−

+

−V0

+

−

+

−VoutR

t = 0

Vout (s) = V0

[1

s + (1/RC)

]

Vout (t) = V0e−(1/RC)tu (t)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks ... Cont’d

+

−

+

−V0

+

−

+

−VoutR

t = 01/sC

+

−

+

−VoutR

V0 /s −+

Vout (s) = V0

[1

s + (1/RC)

]

Vout (t) = V0e−(1/RC)tu (t)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks ... Cont’d

+

−

+

−V0

+

−

+

−VoutR

t = 01/sC

+

−

+

−VoutR

V0 /s −+

Vout (s) = V0

[1

s + (1/RC)

]

Vout (t) = V0e−(1/RC)tu (t)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Transient Analysis of Electric Networks ... Cont’d

+

−

+

−V0

+

−

+

−VoutR

t = 01/sC

+

−

+

−VoutR

V0 /s −+

Vout (s) = V0

[1

s + (1/RC)

]

Vout (t) = V0e−(1/RC)tu (t)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Outline

1 Network Analysis

2 Laplace Transform

3 N - Port Networks

4 Summary

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2 Port Network

+ +

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters

+ +

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters

+ +

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters

+ +

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters

+ +

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters

+ +

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters

+ +

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

g Parameters

+ +

I1 I2

V1 V2

[I1

V2

]=

[g11 g12

g21 g22

] [V1

I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

g Parameters

+ +

I1 I2

V1 V2

[I1

V2

]=

[g11 g12

g21 g22

] [V1

I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

ABCD Parameters

+ +

I1 - I2

V1 V2

[V1

I1

]=

[A BC D

] [V2

−I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

ABCD Parameters

+ +

I1 - I2

V1 V2

[V1

I1

]=

[A BC D

] [V2

−I2

]

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

By the way ... why 2-port network representations are very important inunderstanding this course ?

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

By the way ... why 2-port network representations are very important inunderstanding this course ?

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2 −+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2 −+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2 −+g11

g12I2 g21V1

g22

+

−

+

−

I1 I2

V1 V2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z, Y, h, g Parameters

* h12, h21, g12, g21 are dimensionless quantities

* h22, y11, y22, g11 are admittance values. So, actual resistance values are 1/h22, etc.

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2 −+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2 −+g11

g12I2 g21V1

g22

+

−

+

−

I1 I2

V1 V2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters (Impedance Parameters)

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters (Impedance Parameters)

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters (Impedance Parameters)

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters (Impedance Parameters)

−+ −+

z11

z12I2 z21I1

z22

+

−

+

−

I1 I2

V1 V2

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters – An Example

Z2

Z1 Z3

Z11 = Z1 + Z2

Z21 = Z2

Z22 = Z3 + Z2

Z12 = Z2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters (Admittance Parameters)

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters (Admittance Parameters)

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters (Admittance Parameters)

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters (Admittance Parameters)

y11

y12V2 y21V1

y22

+

−

+

−

I1 I2

V1 V2

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters – An Example

Y1 Y3

Y2

Y11 = Y1 + Y2

Y21 = −Y2

Y22 = Y3 + Y2

Y21 = −Y2

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters (Hybrid Parameters)

−+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

h11 = V1I1

∣∣∣V2=0

h12 = V1V2

∣∣∣I1=0

h21 = I2I1

∣∣∣V2=0

h22 = I2V2

∣∣∣I1=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters (Hybrid Parameters)

−+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

h11 = V1I1

∣∣∣V2=0

h12 = V1V2

∣∣∣I1=0

h21 = I2I1

∣∣∣V2=0

h22 = I2V2

∣∣∣I1=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters (Hybrid Parameters)

−+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

h11 = V1I1

∣∣∣V2=0

h12 = V1V2

∣∣∣I1=0

h21 = I2I1

∣∣∣V2=0

h22 = I2V2

∣∣∣I1=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h Parameters (Hybrid Parameters)

−+

h11

h12V2

h21I1

h22

+

−

+

−

I1 I2

V1 V2

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

h11 = V1I1

∣∣∣V2=0

h12 = V1V2

∣∣∣I1=0

h21 = I2I1

∣∣∣V2=0

h22 = I2V2

∣∣∣I1=0

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Why do we need so many types of parameters to represent the same two portnetwork?

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Series-Series Connection

CombinedPort 1

CombinedPort 2

[Z]1

[Z]2

[Z] = [Z]1 + [Z]2 (2)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Series-Series Connection

CombinedPort 1

CombinedPort 2

[Z]1

[Z]2

[Z] = [Z]1 + [Z]2 (2)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Shunt-Shunt Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[Y]1

[Y]2

[Y] = [Y]1 + [Y]2 (3)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Series-Shunt Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[h]1

[h]2

[h] = [h]1 + [h]2 (4)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Shunt-Series Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[g]1

[g]2

[g] = [g]1 + [g]2 (5)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Cascade Connection

CombinedPort 1

CombinedPort 2

[a] = [a]1 [a]2 (6)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters of N-port Networks

V1

I1

+

V2 I

2

+

VN-1

IN-1

+

VN

IN

+

......

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z Parameters of N-port Networks

V1

I1

+

V2 I

2

+

VN-1

IN-1

+

VN

IN

+

......

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters of N-port Networks

V1

I1

+

V2 I

2

+

VN-1

IN-1

+

VN

IN

+

......

I1

I2

...IN

=

Y11 Y12 · · · Y1N

Y21

......

...YN1 · · · · · · YNN

V1

V2

...VN

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Y Parameters of N-port Networks

V1

I1

+

V2 I

2

+

VN-1

IN-1

+

VN

IN

+

......

I1

I2

...IN

=

Y11 Y12 · · · Y1N

Y21

......

...YN1 · · · · · · YNN

V1

V2

...VN

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z to Y Conversion

From the definition of Z parameters,

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

⇒

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

−1 V1

V2

...VN

=

I1

I2

...IN

Since we know that [I] = [Y] [V],

Y11 · · · Y1N

......

YN1 · · · YNN

=

Z11 · · · Z1N

......

ZN1 · · · ZNN

−1

. (7)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z to Y ConversionFrom the definition of Z parameters,

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

⇒

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

−1 V1

V2

...VN

=

I1

I2

...IN

Since we know that [I] = [Y] [V],

Y11 · · · Y1N

......

YN1 · · · YNN

=

Z11 · · · Z1N

......

ZN1 · · · ZNN

−1

. (7)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z to Y ConversionFrom the definition of Z parameters,

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

⇒

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

−1 V1

V2

...VN

=

I1

I2

...IN

Since we know that [I] = [Y] [V],

Y11 · · · Y1N

......

YN1 · · · YNN

=

Z11 · · · Z1N

......

ZN1 · · · ZNN

−1

. (7)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Z to Y ConversionFrom the definition of Z parameters,

V1

V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1

I2

...IN

⇒

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

−1 V1

V2

...VN

=

I1

I2

...IN

Since we know that [I] = [Y] [V],

Y11 · · · Y1N

......

YN1 · · · YNN

=

Z11 · · · Z1N

......

ZN1 · · · ZNN

−1

. (7)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

h to Z Conversion ***

From the definition of h parameters,[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

⇒[

I1

V2

]=

[h11 h12

h21 h22

]−1 [V1

I2

]

⇒[

I1

V2

]=

1∆h

[h22 −h12

−h21 h11

] [V1

I2

], where ∆h = h11h22 − h12h21

⇒ I1 =h22∆h

V1 −h12∆h

I2

⇒ V1 =∆hh22︸︷︷︸Z11

I1 +h12h22︸︷︷︸Z12

I2.

Similarly, it can be proved that Z21 = − h21h22

and Z22 = 1h22

.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Conversion Table (2 Port Networks) ***

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Conversion Table (2 Port Networks) ***S Z Y ABCD

S11 S11(Z11 − Z0)(Z22 + Z0)− Z12 Z21

Z(Y0 −Y11)(Y0 +Y22)+Y12Y21

YA + B/ Z0 −C Z0 − DA + B/ Z0 +C Z0 + D

S12 S122Z12 Z0

Z−2Y12Y0

Y2(AD − BC)

A + B/ Z0 +C Z0 + D

S21 S212Z21 Z0

Z−2Y21Y0

Y2

A + B/ Z0 +C Z0 + D

S22 S22(Z11 + Z0)(Z22 − Z0)− Z12 Z21

Z(Y0 +Y11)(Y0 −Y22)+Y12Y21

Y−A + B/ Z0 −C Z0 + DA + B/ Z0 +C Z0 + D

Z11 Z0(1 + S11)(1 − S22)+ S12 S21(1 − S11)(1 − S22)− S12 S21

Z11Y22|Y |

AC

Z12 Z02S12

(1 − S11)(1 − S22)− S12 S21Z12

−Y12|Y |

AD − BCC

Z21 Z02S21

(1 − S11)(1 − S22)− S12 S21Z21

−Y21|Y |

1C

Z22 Z0(1 − S11)(1 + S22)+ S12 S21(1 − S11)(1 − S22)− S12 S21

Z22Y11|Y |

DC

Y11 Y0(1 − S11)(1 + S22)+ S12 S21(1 + S11)(1 + S22)− S12 S21

Z22|Z |

Y11DB

Y12 Y0−2S12

(1 + S11)(1 + S22)− S12 S21

−Z12|Z |

Y12BC − AD

B

Y21 Y0−2S21

(1 + S11)(1 + S22)− S12 S21

−Z21|Z |

Y21−1B

Y22 Y0(1 + S11)(1 − S22)+ S12 S21(1 + S11)(1 + S22)− S12 S21

Z11|Z |

Y22AB

A(1 + S11)(1 − S22)+ S12 S21

2S21

Z11Z21

−Y22Y21

A

B Z0(1 + S11)(1 + S22)− S12 S21

2S21

|Z |Z21

−1Y21

B

C1

Z0

(1 − S11)(1 − S22)− S12 S212S21

1Z21

−|Y |Y21

C

D(1 − S11)(1 + S22)+ S12 S21

2S21

Z22Z21

−Y11Y21

D

|Z | = Z11 Z22 − Z12 Z21; |Y | = Y11Y22 −Y12Y21; Y = (Y11 +Y0)(Y22 +Y0)−Y12Y21; Z = (Z11 + Z0)(Z22 + Z0)− Z12 Z21; Y0 = 1/ Z0.

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

An Application of 2 Port Parameters – Y⇔ ∆Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

[Y11 Y12

Y21 Y22

]=

[Z11 Z12

Z21 Z22

]−1

⇒[

YAB + YAC −YAC

−YAC YBC + YAC

]=

[RA + RB RB

RB RC + RB

]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

An Application of 2 Port Parameters – Y⇔ ∆Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

[Y11 Y12

Y21 Y22

]=

[Z11 Z12

Z21 Z22

]−1

⇒[

YAB + YAC −YAC

−YAC YBC + YAC

]=

[RA + RB RB

RB RC + RB

]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

An Application of 2 Port Parameters – Y⇔ ∆Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

[Y11 Y12

Y21 Y22

]=

[Z11 Z12

Z21 Z22

]−1

⇒[

YAB + YAC −YAC

−YAC YBC + YAC

]=

[RA + RB RB

RB RC + RB

]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

An Application of 2 Port Parameters – Y⇔ ∆Conversions

A

B

C

B

CARAC

RAB RBC

RA RC

RB

[Y11 Y12

Y21 Y22

]=

[Z11 Z12

Z21 Z22

]−1

⇒[

YAB + YAC −YAC

−YAC YBC + YAC

]=

[RA + RB RB

RB RC + RB

]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Self Study

Reciprocal & symmetrical networks

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Outline

1 Network Analysis

2 Laplace Transform

3 N - Port Networks

4 Summary

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Network Analysis – Summary

Millman’s Theorem:

VMillman =V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

Thevenin-Norton Equivalencies:

VThevenin = INortonRNorton = INortonRThevenin

Y⇔ ∆ Conversions:RA =

RABRACRAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

Miller’s Theorem:Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

Network Analysis – Summary

Millman’s Theorem:

VMillman =V1R1

+ V2R2

+ V3R3

1R1

+ 1R2

+ 1R3

Thevenin-Norton Equivalencies:

VThevenin = INortonRNorton = INortonRThevenin

Y⇔ ∆ Conversions:RA =

RABRACRAB + RAC + RBC

RAB =RARB + RBRC + RCRA

RC

Miller’s Theorem:Z1 = Z/ (1− K)

Z2 = Z/(1− 1/K)

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2 Port Networks – Summary

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

[I1

V2

]=

[g11 g12

g21 g22

] [V1

I2

]

[V1

I1

]=

[A BC D

] [V2

−I2

]

[Z] = [Y]−1

[h] = [g]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus

Network Analysis Laplace Transform N - Port Networks Summary

2 Port Networks – Summary

[V1

V2

]=

[Z11 Z12

Z21 Z22

] [I1

I2

]

Zij =ViIj

∣∣∣∣∣Ii=0

[I1

I2

]=

[Y11 Y12

Y21 Y22

] [V1

V2

]

Yij =IiVj

∣∣∣∣∣Vi=0

[V1

I2

]=

[h11 h12

h21 h22

] [I1

V2

]

[I1

V2

]=

[g11 g12

g21 g22

] [V1

I2

]

[V1

I1

]=

[A BC D

] [V2

−I2

]

[Z] = [Y]−1

[h] = [g]−1

1: Review of Network Theory ECE/EEE/INSTR F244, Dept. of EEE, BITS Pilani Hyderabad Campus