This document discusses power gain in microwave engineering and antennas. It introduces four definitions of power gain: available gain, delivered gain, maximum gain, and transducer gain. Available gain is defined as the ratio of power available at the load to power available at the source accounting for input impedance mismatch. Delivered gain is defined as the ratio of power delivered to the load to power delivered to the input accounting for output impedance mismatch. Maximum gain assumes optimal power transfer with no impedance mismatch. Transducer gain accounts for mismatch at both the input and output. An example calculation is shown to illustrate the differences between these power gain definitions using the scattering parameters of an RF amplifier.

1. Microwave Engineering
and Antennas
Power Gain – Part II
Domine Leenaerts, Professor
Department of Electrical Engineering
Center for Wireless Technologies Eindhoven
Note: In these slide we have used the peak power in the
definition of Pav,s and related power definitions.
In the book and quizzes we will use the time-average
(rms) value with an additional factor ½.

2. Power Gain – Part II
Objective of this lecture
• Introduce the 4 definitions of power gain
• Provide an example to show the differences

3. Four definitions of Power Gain
• Available (power) Gain 𝐺𝐺𝑎𝑎𝑎𝑎 =
𝑃𝑃𝑎𝑎𝑎𝑎,𝑜𝑜
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
mismatch conditions at input
• Delivered (power) Gain 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑖𝑖
mismatch conditions at output
• Maximum (power) Gain 𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 =
𝑃𝑃𝑎𝑎𝑎𝑎,𝑜𝑜
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑖𝑖
optimal power transfer
• Transducer (power) Gain 𝐺𝐺𝑇𝑇 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
mismatch at input and output
Z
V
Z
Z
V
Z
o
s
o
s
i l
V
l
V
i
system S
source load
input impedance
output impedance

4. Available and Delivered (Power) Gain
Z
V
Z
Z
V
Z
o
s
o
s
i l
V
l
V
i
system S
source load
input impedance
output impedance
𝐺𝐺𝑎𝑎𝑎𝑎 =
𝑃𝑃𝑎𝑎𝑎𝑎,𝑜𝑜
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
𝑅𝑅𝑖𝑖
𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑠𝑠
2
𝐴𝐴𝑣𝑣
2
𝑅𝑅𝑠𝑠
𝑅𝑅𝑜𝑜
𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑖𝑖
= 𝐴𝐴𝑣𝑣
2 𝑅𝑅𝑖𝑖𝑅𝑅𝑙𝑙
𝑅𝑅𝑙𝑙 + 𝑅𝑅𝑜𝑜
2
𝐴𝐴𝑣𝑣 =
𝑉𝑉
𝑜𝑜
𝑉𝑉𝑖𝑖
=
𝑉𝑉𝑙𝑙
𝑉𝑉
𝑠𝑠
Zo=0 Ω; Zi = ∞ Ω:

5. Maximum and Transducer (Power) Gain
Z
V
Z
Z
V
Z
o
s
o
s
i l
V
l
V
i
system S
source load
input impedance
output impedance
𝐴𝐴𝑣𝑣 =
𝑉𝑉
𝑜𝑜
𝑉𝑉𝑖𝑖
=
𝑉𝑉𝑙𝑙
𝑉𝑉
𝑠𝑠
𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 =
𝑃𝑃𝑎𝑎𝑎𝑎,𝑜𝑜
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑖𝑖
= 𝐴𝐴𝑣𝑣
2 𝑅𝑅𝑖𝑖
4𝑅𝑅𝑜𝑜
𝐺𝐺𝑇𝑇 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
𝑅𝑅𝑖𝑖
𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑠𝑠
2
𝐴𝐴𝑣𝑣
2 4𝑅𝑅𝑙𝑙𝑅𝑅𝑠𝑠
𝑅𝑅𝑙𝑙 + 𝑅𝑅𝑜𝑜
2
𝐺𝐺𝑇𝑇
𝑅𝑅𝑙𝑙=𝑅𝑅𝑜𝑜
𝐺𝐺𝑎𝑎𝑎𝑎
𝐺𝐺𝑇𝑇
𝑅𝑅𝑖𝑖=𝑅𝑅𝑠𝑠
𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑
𝐺𝐺𝑇𝑇
𝑅𝑅𝑙𝑙=𝑅𝑅𝑜𝑜, 𝑅𝑅𝑠𝑠=𝑅𝑅𝑖𝑖
𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚

6. Power Gain and scatter parameters: reflection
𝛤𝛤
𝑖𝑖𝑖𝑖 =
𝑏𝑏1
𝑎𝑎1
= 𝑆𝑆11 +
𝑆𝑆12𝑆𝑆21
1
𝛤𝛤
𝐿𝐿
− 𝑆𝑆22
𝑏𝑏1
𝑏𝑏2
=
𝑠𝑠11 𝑠𝑠12
𝑠𝑠21 𝑠𝑠22
𝑎𝑎1
𝑎𝑎2
𝛤𝛤
𝑜𝑜𝑜𝑜𝑜𝑜 =
𝑏𝑏2
𝑎𝑎2
= 𝑆𝑆22 +
𝑆𝑆12𝑆𝑆21
1
𝛤𝛤
𝑆𝑆
− 𝑆𝑆11
System S

7. Available, and delivered Power Gain
𝐺𝐺𝑎𝑎𝑎𝑎 𝛤𝛤
𝑆𝑆, 𝑆𝑆 =
𝑃𝑃𝑎𝑎𝑎𝑎,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
1 − 𝛤𝛤
𝑆𝑆
2
1 − 𝑆𝑆11𝛤𝛤
𝑆𝑆
2
𝑆𝑆21
2
1
1 − 𝛤𝛤
𝑜𝑜𝑢𝑢𝑢𝑢
2
𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤
𝐿𝐿, 𝑆𝑆 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑠𝑠
=
1 − 𝛤𝛤
𝐿𝐿
2
1 − 𝑆𝑆22𝛤𝛤
𝐿𝐿
2
𝑆𝑆21
2
1
1 − 𝛤𝛤
𝑖𝑖𝑖𝑖
2
𝐺𝐺𝑎𝑎𝑎𝑎 𝛤𝛤
𝑆𝑆, 𝑆𝑆 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤
𝐿𝐿, 𝑆𝑆

8. Relation between all power gains
𝑀𝑀𝑆𝑆 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑖𝑖𝑖𝑖
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
1 − Γ𝑖𝑖𝑖𝑖
2
1 − Γ𝑆𝑆
2
1 − Γ𝑖𝑖𝑖𝑖Γ𝑆𝑆
2
𝑀𝑀𝐿𝐿 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑙𝑙
=
1 − Γ𝑂𝑂𝑂𝑂𝑂𝑂
2
1 − Γ𝐿𝐿
2
1 − Γ𝑜𝑜𝑢𝑢𝑢𝑢Γ𝐿𝐿
2
𝐺𝐺𝑇𝑇 Γ𝑆𝑆, Γ𝐿𝐿, 𝑆𝑆 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑠𝑠
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑠𝑠
= 𝑀𝑀𝑆𝑆𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 Γ𝐿𝐿, 𝑆𝑆
𝐺𝐺𝑇𝑇 Γ𝑆𝑆, Γ𝐿𝐿, 𝑆𝑆 ≤ 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 Γ𝐿𝐿, 𝑆𝑆
𝐺𝐺𝑇𝑇 Γ𝑆𝑆, Γ𝐿𝐿, 𝑆𝑆 =
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
=
𝑃𝑃𝑎𝑎𝑎𝑎,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑠𝑠
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑,𝑙𝑙
𝑃𝑃𝑎𝑎𝑎𝑎,𝑙𝑙
= 𝐺𝐺𝑎𝑎𝑎𝑎 Γ𝑆𝑆, 𝑆𝑆 𝑀𝑀𝐿𝐿
𝐺𝐺𝑇𝑇 Γ𝑆𝑆, Γ𝐿𝐿, 𝑆𝑆 ≤ 𝐺𝐺𝑎𝑎𝑎𝑎 Γ𝑆𝑆, 𝑆𝑆
𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑
𝐺𝐺𝑎𝑎𝑎𝑎
𝐺𝐺𝑇𝑇

9. Example: an RF amplifier
𝑠𝑠11 = 0.61, 165𝑜𝑜
𝑠𝑠12 = 0.05, 42𝑜𝑜
𝑠𝑠21 = 3.72, 59𝑜𝑜
𝑠𝑠22 = 0.45, −48𝑜𝑜
𝑍𝑍0 = 50Ω
𝑍𝑍𝑠𝑠 = 40Ω
𝑉𝑉
𝑠𝑠 = 5 ∗ sin(𝜔𝜔𝜔𝜔) 𝑍𝑍𝐿𝐿 = 73Ω

10. Example: an RF amplifier
Γ =
𝑍𝑍 − 𝑍𝑍0
𝑍𝑍 + 𝑍𝑍0
, 𝑡𝑡𝑡𝑡𝑡𝑡𝑡 Γ𝑠𝑠 = −0.111, Γ𝐿𝐿 = 0.187
𝛤𝛤
𝑖𝑖𝑖𝑖 = 𝑆𝑆11 +
𝑆𝑆12𝑆𝑆21
1
𝛤𝛤
𝐿𝐿
− 𝑆𝑆22
= −0.594 + 𝑗𝑗𝑗.194 𝛤𝛤
𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑆𝑆22 +
𝑆𝑆12𝑆𝑆21
1
𝛤𝛤
𝑠𝑠
− 𝑆𝑆11
= 0.305 − 𝑗𝑗𝑗.356
𝑍𝑍𝑖𝑖𝑖𝑖 = 11 + 𝑗𝑗𝑗.54 [Ω] 𝑍𝑍𝑜𝑜𝑜𝑜𝑜𝑜 = 63 − 𝑗𝑗𝑗𝑗. 4[Ω]
𝛤𝛤
𝑖𝑖𝑖𝑖 𝛤𝛤
𝑜𝑜𝑜𝑜𝑜𝑜
𝑍𝑍𝑠𝑠 = 40Ω
𝑉𝑉
𝑠𝑠 = 5 ∗ sin(𝜔𝜔𝜔𝜔) 𝑍𝑍𝐿𝐿 = 73Ω

11. Example: an RF amplifier
𝐺𝐺𝑎𝑎𝑎𝑎 𝛤𝛤
𝑆𝑆, 𝑆𝑆 =
1 − 𝛤𝛤
𝑆𝑆
2
1 − 𝑆𝑆11𝛤𝛤
𝑠𝑠
2
𝑆𝑆21
2
1
1 − 𝛤𝛤
𝑜𝑜𝑢𝑢𝑢𝑢
2
= 20.05 𝑜𝑜𝑜𝑜 13.02𝑑𝑑𝑑𝑑
𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤
𝐿𝐿, 𝑆𝑆 =
1 − 𝛤𝛤
𝐿𝐿
2
1 − 𝑆𝑆22𝛤𝛤
𝐿𝐿
2
𝑆𝑆21
2
1
1 − 𝛤𝛤
𝑖𝑖𝑖𝑖
2
= 24.49 𝑜𝑜𝑜𝑜 13.89𝑑𝑑𝑑𝑑
𝐺𝐺𝑇𝑇 𝛤𝛤
𝑆𝑆, 𝛤𝛤
𝐿𝐿, 𝑆𝑆 =
1 − 𝛤𝛤
𝐿𝐿
2
1 − 𝑆𝑆22𝛤𝛤
𝐿𝐿
2
𝑆𝑆21
2
1 − 𝛤𝛤
𝑆𝑆
2
1 − 𝛤𝛤
𝑖𝑖𝑖𝑖𝛤𝛤
𝑠𝑠
2
= 16.89 𝑜𝑜𝑜𝑜 12.27𝑑𝑑𝑑𝑑
𝑍𝑍𝑠𝑠 = 40Ω
𝑉𝑉
𝑠𝑠 = 5 ∗ sin(𝜔𝜔𝜔𝜔) 𝑍𝑍𝐿𝐿 = 73Ω
𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 𝑆𝑆 = 41.5 𝑜𝑜𝑜𝑜 16.18𝑑𝑑𝑑𝑑

12. Example: an RF amplifier
• The available power Pav =
𝑉𝑉𝑠𝑠
2
4𝑅𝑅𝑠𝑠
= 156 mW or Pav = 21.93 dBm.
• The power delivered to the load is the available power multiplied by
the transducer gain. This results in Pdel = PavGT = 2.63W or expressed
in dBm,
𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐺𝐺𝑇𝑇 𝛤𝛤
𝑆𝑆, 𝛤𝛤
𝐿𝐿, 𝑆𝑆 𝑃𝑃𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑 = 21.93 + 12.27 = 34.2𝑑𝑑𝑑𝑑𝑑𝑑
𝑍𝑍𝑠𝑠 = 40Ω
𝑉𝑉
𝑠𝑠 = 5 ∗ sin(𝜔𝜔𝜔𝜔) 𝑍𝑍𝐿𝐿 = 73Ω
Note: The time average Pav = 78 mW or
18.93 dBm. Time average Pdel = 31.2 dBm

13. Summary
• 4 definitions of power gain, depending on the impedance matching
conditions.
• Matching networks at between source and input (Ms) and output and
load (ML) influences the power transfer from source to load
• Transducer (power) gain is equal to or smaller than the available
(power) gain or delivered (power) gain