RF Electronic Devices and Systems

Transmission line circuit for mismatched load and generator

linear circuit

Let’s consider the most general case of a circuit, where we have the arbitrary loads Z1, Zand an arbitrary generator.

The input impedance of the terminated transmission line is Zin=Z0ejβl+Γle(jβl)ejβlΓle(jβl ,Γl=Z1Z0Z1+Z0. The voltage on the line V(z)=V0(ejβz+Γlejβz) . To receive the voltage of the generator we must take into consideration z=l, so V0=Z0VgZ0+Z2 ejβl1ΓlΓge2jβl. Γg=Z2Z0Z2+Z0 is the reflection coefficient of the generator. The standing wave ratio here is SWR=1+|ΓL|1|ΓL| . So the power delivered to the load is: P=12|VinIin|=12|Vg|2 Rin(Rin+R2)2+(Zin+2)2  where Zin=Rin+jZin,Z2=R2+jZ2.

1. When the load is matched to the line, Zin=Z0,Гl=0 and P=12|Vg|2  Z0(Z0+R2)2+X22 .

2. When the load is matched to the line, Zin=Z2 the load power is P=12|Vg|2 R24 1(R22+X22).

3. Assuming that the generator impedance is fixed, the condition dPdZin0 leads to the maximum power delivered to the load.

The conjugate matching condition is R(in=Rg,Xin=Xg,or Zin=Zg* . This condition is conjugate matching, that leads to the maximum power delivered to the load for a fixed generator impedance P=12|Vg|2 14Rg. Figure 1 depicts the general case of transmission line circuit for mismatched load and the generator.

Figure 1. The arbitrary transmission line circuit
Figure 1. The arbitrary transmission line circuit

Transmission line with losses

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