A. Kruger Two-Port Theory, Slide 1 55:141: Advanced Circuit Techniques The University of Iowa, 2014

55:141 Advanced Circuit Techniques

Two-Port Theory

Material: Lecture Notes

A. Kruger Two-Port Theory, Slide 2 55:141: Advanced Circuit Techniques The University of Iowa, 2014

What Are Two-Ports?

2-Port parameters are given on data sheets because they are easy to measure.

2-Port theory unifies feedback amplifier theory.

2121111 vyvyi += 2221212 vyvyi +=

Basic idea: replace a complex linear circuit with a simple equivalent model

Concept similar to replacing a circuit with its Thevenin/Norton equivalent.

Why is this useful?

A. Kruger Two-Port Theory, Slide 3 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Example (from 2n3903 Datasheet)

ℎ𝑓𝑓𝑓𝑓 ℎ𝑜𝑜𝑓𝑓

ℎ𝑖𝑖𝑓𝑓 ℎ𝑟𝑟𝑓𝑓

A. Kruger Two-Port Theory, Slide 4 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Example (from 2n3903 Datasheet)

ℎ𝐹𝐹𝐹𝐹 ≡dc 𝛽𝛽

A. Kruger Two-Port Theory, Slide 5 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Notational Conventions

A. Kruger Two-Port Theory, Slide 6 55:141: Advanced Circuit Techniques The University of Iowa, 2014

z-Parameters

A. Kruger Two-Port Theory, Slide 7 55:141: Advanced Circuit Techniques The University of Iowa, 2014

z-Parameters

Terminations for z-parameters

Terminations for 𝑧𝑧21 and 𝑧𝑧11

Terminations for 𝑧𝑧12 and 𝑧𝑧22

𝒛𝒛𝟏𝟏𝟏𝟏 =𝒗𝒗𝟏𝟏𝒊𝒊𝟏𝟏�𝒊𝒊𝟐𝟐=𝟎𝟎

𝒛𝒛𝟐𝟐𝟏𝟏 =𝒗𝒗𝟐𝟐𝒊𝒊𝟏𝟏�𝒊𝒊𝟐𝟐=𝟎𝟎

𝒛𝒛𝟏𝟏𝟐𝟐 =𝒗𝒗𝟏𝟏𝒊𝒊𝟐𝟐�𝒊𝒊𝟏𝟏=𝟎𝟎

𝒛𝒛𝟐𝟐𝟐𝟐 =𝒗𝒗𝟐𝟐𝒊𝒊𝟐𝟐�𝒊𝒊𝟏𝟏=𝟎𝟎

A. Kruger Two-Port Theory, Slide 8 55:141: Advanced Circuit Techniques The University of Iowa, 2014

y-Parameters

2121111 vyvyi += 2221212 vyvyi +=

y-Parameters are also called the admittance parameters.

y-Parameter model

A. Kruger Two-Port Theory, Slide 9 55:141: Advanced Circuit Techniques The University of Iowa, 2014

y-Parameters

2121111 vyvyi +=

021

111

=

=vv

iy02

1

221

=

=vv

iy

012

112

=

=vv

iy01

2

222

=

=vv

iy

2221212 vyvyi +=

+

- 2v

+

- 1v

A. Kruger Two-Port Theory, Slide 10 55:141: Advanced Circuit Techniques The University of Iowa, 2014

y-Parameters

2121111 vyvyi +=

K101

021

111 ==

=vviy

021

221

=

=vv

iy

012

112

=

=vv

iy01

2

222

=

=vv

iy

2221212 vyvyi +=

+

- 1v K10

12

vi =−

K101

21 −=y

+

- 2v

K102

1vi =−

K101

12 −=yK51

22 =y

A. Kruger Two-Port Theory, Slide 11 55:141: Advanced Circuit Techniques The University of Iowa, 2014

h-Parameters

2121111 vhihv +=

01

111 2 =

∆

= vivh

01

221 2 =

∆

= viih

02

112 1=

∆

= ivvh

02

222 1=

∆

= ivih

2221212 vhihi +=

One can determine model parameters through analysis

More common–determine parameters experimentally

This circuit could be arbitrarily complex

Model

2121111 vhihv +=

2221212 vhihi +=

A. Kruger Two-Port Theory, Slide 12 55:141: Advanced Circuit Techniques The University of Iowa, 2014

h-Parameter Model for CE NPN

cerebiebe vhihv +=

Special notation for CE BJT

2121111 vhihv += 2221212 vhihi +=

ceoebfec vhihi +=

Linear Small Signal

A. Kruger Two-Port Theory, Slide 13 55:141: Advanced Circuit Techniques The University of Iowa, 2014

h-Parameter and Hybrid-𝝅𝝅 Model Equivalence

µπ rrrh bie +=o

oe rrh 11

++

=µ

β

cerebiebe vhihv +=

ceoebfec vhihi +=

h-Parameter

Hybrid 𝜋𝜋

β

β

π

π

=

=

=

rgIVr

VI

g

m

CQ

T

T

CQm

β=fehµ

π

rrhre ≅

Equivalence

A. Kruger Two-Port Theory, Slide 14 55:141: Advanced Circuit Techniques The University of Iowa, 2014

4 Equivalent 2-Port Networks

Voltage Amplifier

Current Amplifier

Input: Voltage Output: Voltage

Input: Current Output: Current

A. Kruger Two-Port Theory, Slide 15 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Transconductance Amplifier

Transresistance Amplifier

Input: Voltage Output: Current

Input: Current Output: Voltage

4 Equivalent 2-Port Networks

A. Kruger Two-Port Theory, Slide 16 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Ideal Basic Feedback Configurations

Voltage Amplifier Current Amplifier

Transresistance Amplifier (current in-voltage out)

Transconductance Amplifier (voltage in-current out)

A. Kruger Two-Port Theory, Slide 17 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Ideal Series-Shunt Feedback Voltage Amplifier

2121111 vhihv += 2221212 vhihi +=

Two-port network

• Input signal is transmitted through amplifier and not through β network: h21 = 0 • β network does not load amplifier: neglect h22 and h11.

Implicit Assumptions Made

h12 of the feedback network is the same as β

A. Kruger Two-Port Theory, Slide 18 55:141: Advanced Circuit Techniques The University of Iowa, 2014

y-Parameters Revisited

2121111 vyvyi +=

021

111

=

=vv

iy02

1

221

=

=vv

iy

012

112

=

=vv

iy01

2

222

=

=vv

iy

2221212 vyvyi +=

+

- 2v

+

- 1v

A. Kruger Two-Port Theory, Slide 19 55:141: Advanced Circuit Techniques The University of Iowa, 2014

y-Parameters Revisited

2121111 vyvyi +=

K101

021

111 ==

=vviy

021

221

=

=vv

iy

012

112

=

=vv

iy01

2

222

=

=vv

iy

2221212 vyvyi +=

+

- 1v K10

12

vi =−

K101

21 −=y

+

- 2v

K102

1vi =−

K101

12 −=yK51

22 =y

A. Kruger Two-Port Theory, Slide 20 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Shunt-Shunt Feedback: y-Parameters

• Input signal is transmitted through amplifier and not through β network: h12 = 0

y12 of the feedback network is the same as β

• β network does not load amplifier: neglect y22 and y11

A. Kruger Two-Port Theory, Slide 21 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Ideal Basic Feedback Configurations

Voltage Amplifier Current Amplifier

Transresistance Amplifier (current in-voltage out)

Transconductance Amplifier (voltage in-current out)

h-parameters

y-parameters

A. Kruger Two-Port Theory, Slide 22 55:141: Advanced Circuit Techniques The University of Iowa, 2014

How to Account For Loading?

Step 1: Replace feedback network with y-parameter model

Step 2: Assume no feed-forward through β network: y21 = 0

Step 3: The feedback transfer function is β = y12

Step 4: Turn off feedback y12= 0

Step 5: Compute gain, input- and output resistances: merge all resistances at a port

Step 6: Scale values by (1+ βA) as appropriate to find closed-loop values

Step 7: “Unmerge” resistances

A. Kruger Two-Port Theory, Slide 23 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Discrete Transistor Circuit

Current-in, voltage-out

Transresistance 𝐴𝐴𝑧𝑧

Feedback reduces input impedance

Output voltage and R2 generates a feedback current that reduces current flowing into transistor base

Fg R

1−=β

Shunt-Shunt Feedback

y-parameters

𝐼𝐼𝑖𝑖

A. Kruger Two-Port Theory, Slide 24 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Discrete Transistor Circuit

A. Kruger Two-Port Theory, Slide 25 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Discrete Transistor Circuit

2121111 vyvyi += 2221212 vyvyi +=

A. Kruger Two-Port Theory, Slide 26 55:141: Advanced Circuit Techniques The University of Iowa, 2014

3.5K 3.3K

1/510K 1/510K

V/A

A. Kruger Two-Port Theory, Slide 27 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Rif

A. Kruger Two-Port Theory, Slide 28 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Transform source to a current source

Shunt-shunt feedback

Notice difference between Rif and Rin

A. Kruger Two-Port Theory, Slide 29 55:141: Advanced Circuit Techniques The University of Iowa, 2014

A. Kruger Two-Port Theory, Slide 30 55:141: Advanced Circuit Techniques The University of Iowa, 2014

A. Kruger Two-Port Theory, Slide 31 55:141: Advanced Circuit Techniques The University of Iowa, 2014

A. Kruger Two-Port Theory, Slide 32 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Conversion Between Two-Port Parameters

Δℎ = ℎ11ℎ22 − ℎ12ℎ21

A. Kruger Two-Port Theory, Slide 33 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Terminated Two Ports

z-parameters

h-parameters

h-parameters

A. Kruger Two-Port Theory, Slide 34 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Terminated Two Ports

Δ𝑦𝑦 = 𝑦𝑦11𝑦𝑦22 − 𝑦𝑦12𝑦𝑦21

A. Kruger Two-Port Theory, Slide 35 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Interconnected Two-Ports

Cascade

Series Parallel

Series-Parallel Parallel-Series

A. Kruger Two-Port Theory, Slide 36 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Reciprocal Two-Ports

A. Kruger Two-Port Theory, Slide 37 55:141: Advanced Circuit Techniques The University of Iowa, 2014

Symmetric Two-Ports

A. Kruger Two-Port Theory, Slide 38 55:141: Advanced Circuit Techniques The University of Iowa, 2014