The lowpass \(\pi\) network is equivalent to a pair of backtoback lowpass LPad networks:  \(\hspace{18.9cm}\)


Lowpass \(\pi\) network 
Equivalent circuit 




\(C_1 = \dfrac{1}{\omega\cdot R_{IN}}\sqrt{\dfrac{R_{IN}}{\color{red}{R_X}}1}\hspace{1cm} L_1 = C_1\cdot R_{IN}\cdot\color{red}{R_X} \)
\(C_2 = \dfrac{1}{\omega\cdot R_{L}}\sqrt{\dfrac{R_{L}}{\color{red}{R_X}}1}\hspace{1.7cm} L_2 = C_2\cdot R_{L}\cdot\color{red}{R_X} \)


The equivalent circuit R_{X} represents \(R_L\) for the leftstage and \(R_{IN}\) for the right stage. You solve the left and right stages independently then combine L_{1} and L_{2} into inductor L_{T}. R_{X} is called the transfer impedance \(Z_T\).
The calculator below requires R_{X} to be less than R_{IN}. This makes L_{1} and C_{1} an impedance reducer with L_{2} and C_{2} as an impedance multiplier. With the \(\pi\) calculator's default values, 50 Ω is converted to a \(Z_T\) of 25 Ω and then \(Z_T\) is multiplied to 300 Ω.




The relevance of Qfactor As previously proven for the basic LPad network, these relations are all true:  $$Q \hspace{0.5cm} = \hspace{0.5cm}A_V\hspace{0.5cm} = \hspace{0.5cm}\sqrt{\dfrac{R_L}{R_{IN}}}\hspace{0.5cm} = \hspace{0.5cm}R_L\sqrt{\dfrac{C}{L}}\hspace{0.5cm}=\hspace{0.5cm}\omega_n R_L C$$ This means that Q is not an input variable; it is wholly defined by \(R_{IN}\) and \(R_L\). However, because the \(\pi\) network has a transfer impedance, Qfactor has two identities. For this reason, Q is discarded as an input variable in favour of R_{X}. \(A_V\) is still calculated based on a lossless power transfer:  $$A_V=\sqrt{\dfrac{R_L}{R_{IN}}}$$ 

Benefits of the \(\pi\) network The 1st benefit is that you can manipulate the passbandwidth at the operating frequency. The image below shows the bodeplot response for several values of R_{X}. The upper graph is gain and the lower graphs are \(\angle{Z_{IN}}\) and \(Z_{IN}\):  At 10 MHz, the gains converge at 1.761 dB. In real numbers that's 1.2247 and, when taking into account the 2:1 loss when connecting a 50 Ω source to a network with \(Z_{IN}\) = 50 Ω, the voltage magnification is the same as the lowpass LPad network (2.44949): 
The 2nd benefit; The input impedance phase angle can be tailored to be more closely resistive than the LPad in the operating region around the centre frequency.
The 3rd benefit is that the input impedance and output impedance can be equal. This may not seem much of a benefit for an impedance transformer but, if you consider that several \(\pi\) networks can be cascaded to produce a significant filtering effect, you have a tool that is very useful.


The lowpass \(\pi\) network in cascade Below is a 50 Ω to 50 Ω, 5stage cascaded \(\pi\) network simulated in Microcap 12 (R_{X} = 25 Ω): 
The frequency response, impedance phase angle and impedance magnitude is this: 
The main feature is the very sharp attenuation from about 13 MHz. The impedance is fairly flat from about 8 MHz to 11 MHz with the corresponding impedance phase angle remaining close to 0°.
When cascading \(\pi\) stages, the value calculations are repeated and trivial. 
