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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
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