PI Network Impedance Matching
Written by Bryce Ringwood   

 

This network is often used to match two resistive impedances Z1 and Z2 as shown below. They crop up in valve transmitter circuits and in antenna tuners. Q is a number representing the quality factor of a tuned circuit. 

Pi Network

Data Entry

As with all these programs, data must be entered in SI units. It is up to you to enter the correct multiplier in all of the fields (p for pico, u for micro and so on.) 

 

Enter Input Resistive Impedance Z1  - Ohms

Enter Output Resistive Impedance Z2 - Ohms

Enter Frequency - Hz   

Enter Circuit Q(2 - 50)


Capacitor C1 - F                                    

Capacitor C2 - F                                    

Inductance - L - H                                  

                  
 

 

The example provided uses the same parameters that the impedance calculation uses in the article on impedance.

Note that ${\displaystyle {{Z_2} \over{ Z_1}}({Q^2}+1) \gt 1}$

Formulae

The following formulae were used:

$\displaystyle{{X_{C1}}={{Z_1} \over Q}}$
$\displaystyle{ {X_{C2}}={ {Z_2} \over{ {\sqrt{ {{ {Z_2}\over {Z_1}}  {(Q^2 + 1) -1}}    } }}} }$
$\displaystyle{  {X_L}={{Q{Z_1}}\over{Q^2 + 1}} \left (1+{{Z_2}\over{QX_{C2}}}\right ) }$
$\displaystyle{ {C_1} = {1\over{2\pi f{X_{C1}}}}     }$
$\displaystyle{  {C_2} = { 1\over{2\pi f{X_{C2}}}  }     }$
$\displaystyle{ L = {{X_L}\over {2\pi f}}    }$

One can imagine the pain of trying to calculate these formulae on even a non-programmable calculator! The HP 65 didn't think quietly. While doing the calculations, the display would flash through all the intermediate results. The calculation would take maybe a second or two. I haven't attempted to program this on newer models of calculator.

 
 

Reference

Hewlett Packard EE Pack for HP65 EE1-09A Hewlett Packard, 1974
 
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