Electromagnetics:
Microwave Engineering
The T-Junction Power Divider
Slide 1
Lecture Outline
• Lossless Divider
• Resistive Divider
2
1
Lossless Divider
𝐸 −plane waveguide
𝑍1
+
𝑉0 𝑗𝐵
−
𝑍0
𝑍2
𝑌𝑖𝑛
𝐻 −plane waveguide
Microstrip Junction Divider
Slide 3
Lossless Divider
In order for the divider to be matched to
the input line with characteristic
impedance 𝑍0 , we must have
𝑍1
+
𝑉0 𝑗𝐵
−
𝑍0
𝑍2
𝑌𝑖𝑛
𝑌𝑖𝑛 = 𝑗𝐵 +
1
1
1
+
=
𝑍1 𝑍2 𝑍0
Fringing Fields and
higher order modes
For the lossless case, 𝐵 = 0, and we have
1
1
1
+
=
𝑍1 𝑍2 𝑍0
Slide 4
2
Lossless Divider - Example
𝑍0
A lossless T-junction power divider has a source impedance of 50 Ω. Find the output
characteristic impedances so that the output powers are at a 2:1 ratio. Compute the
reflection coefficients seen looking into the ports.
Solution:
𝑉02
𝑃𝑖𝑛 =
2𝑍0
The output powers are
𝑍1
𝑉02
1
+
𝑃1𝑜𝑢𝑡 =
= 𝑃𝑖𝑛
𝑉0
2𝑍1
3
−
2
𝑉0
2
𝑍2
𝑃2𝑜𝑢𝑡 =
= 𝑃𝑖𝑛
2𝑍2
3
𝑌𝑖𝑛
And the resistances are
𝑍1 = 3𝑍0 = 150 Ω
𝑍2 = 3/2 𝑍0 = 75 Ω
Slide 5
Lossless Divider - Example
A lossless T-junction power divider has a source impedance of 50 Ω. Find the output
characteristic impedances so that the output powers are at a 2:1 ratio. Compute the
reflection coefficients seen looking into the ports.
Solution:
𝑍𝑖𝑛 = 75||150 = 50Ω
Looking into the Z1 = 150 Ω output line, the impedance is 30 Ω
𝑍1
Looking into the Z2 = 75 Ω output line, the impedance is 37.5 Ω
+
𝑉0
−
𝑍0
𝑍2
𝑌𝑖𝑛
The reflection coefficients looking into the ports are
Γ1 =
30 − 150
= −0.667,
30 + 150
Γ2 =
37.5 − 75
= −0.333
37.5 + 75
Slide 6
3
Resistive Divider
Equal-Split Three-Port
Resistive Power Divider
Port 2
𝑍0 /3
Port 1
𝑍0
+
𝑉1
−
𝑍𝑖𝑛
+
𝑉 𝑍
−
𝑍0 4
= 𝑍0
3
3
𝑍0
𝑍0 /3
𝑃1
𝑍 = 𝑍0 +
𝑍0 /3
𝑍2
4
4
𝑍0 3 𝑍0
𝑍0
3
=
+
= 𝑍0
4
4
3
𝑍
+
𝑍
3 0 3 0
Port 3
𝑍𝑖𝑛
𝑍0
𝑉 = 𝑉1
2𝑍0 /3
2
= 𝑉1
𝑍0 Τ3 + 2𝑍0 Τ3 3
𝑉2 = 𝑉3 = 𝑉
𝑍0
1
= 𝑉1
𝑍0 + 𝑍0 /3 2
Slide 7
Resistive Divider
Equal-Split Three-Port
Resistive Power Divider
𝑍0
𝑍0 /3
𝑍0
𝑆11 = 𝑆22 = 𝑆33 = 0
𝑍0 /3
Port 1
𝑃1
+
𝑉1
−
+
𝑉 𝑍
−
Since 𝑉2 = 𝑉3 = 1/2𝑉1 ,
𝑍0 /3
𝑍2
𝑍𝑖𝑛
Because the network is symmetric from all
three ports the output ports are also
matched.
Port 2
𝑆21 = 𝑆31 = 𝑆23 = 1/2
Port 3
𝑍0
The scattering matrix is written as
𝑆=
1 0 1 1
1 0 1
2
1 1 0
Slide 8
4
Resistive Divider
Equal-Split Three-Port
Resistive Power Divider
𝑍0
𝑍0 /3
𝑍0 /3
Port 1
𝑃1
𝑍0
+
𝑉1
−
+
𝑉 𝑍
−
and the output powers are
𝑍0 /3
𝑍2
𝑍𝑖𝑛
The power delivered to the input of the
divider is
1 𝑉12
𝑃𝑖𝑛 =
2 𝑍0
Port 2
Port 3
1 1Τ2𝑉1
𝑃2 = 𝑃3 =
2
𝑍0
2
1 𝑉12 1
=
= 𝑃
8 𝑍0 4 𝑖𝑛
𝑍0
Slide 9
5