Miller's theorem

Two-pole circuit diagram

The Miller Theorem , named after John Milton Miller , deals with the separation of impedances in electrical networks . If there are two networks ( two-pole ) that are connected via a real impedance Z , this impedance can be virtually broken down so that both networks can be viewed separately. The Miller theorem is the generalization of the Miller effect .

The decomposition into Z ₁ and Z ₂ is correct if both networks see the same impedances after the decomposition as before.

The two tensions are about gain

${\displaystyle M={\frac {U_{2}}{U_{1}}}}$

linked, which then results in the decomposed impedances Z ₁ and Z ₂ :

${\displaystyle Z_{1}=Z{\frac {1}{1-M}}}$

${\displaystyle Z_{2}=Z{\frac {M}{M-1}}}$

results.

It should be noted that for a gain M greater than one, the impedance Z _{1 is} negative when Z is positive. With an inverting amplifier M less than minus −1, the impedance Z _{1 is} significantly reduced compared to Z.

Derivation

The voltage U _Z across the impedance Z results from the difference between the terminal voltages , the substitution by M giving the conversion listed.

${\displaystyle I={\frac {U_{Z}}{Z}}={\frac {U_{1}-U_{2}}{Z}}={\frac {U_{1}}{Z}}\left(1-{\frac {U_{2}}{U_{1}}}\right)={\frac {U_{1}}{Z}}(1-M)}$

At the same time applies to the "seen" impedance Z ₁ :

${\displaystyle I=I_{1}={\frac {U_{1}}{Z_{1}}}}$

By equating it follows:

${\displaystyle {\frac {U_{1}}{Z_{1}}}={\frac {U_{1}}{Z}}(1-M)}$

and by equivalence conversion we finally get:

${\displaystyle Z_{1}={\frac {Z}{1-M}}}$.

The same applies to the "seen" impedance Z ₂ :

${\displaystyle I={\frac {U_{Z}}{Z}}={\frac {U_{1}-U_{2}}{Z}}={\frac {U_{2}}{Z}}\left({\frac {U_{1}}{U_{2}}}-1\right)={\frac {U_{2}}{Z}}\left({\frac {1}{M}}-1\right)={\frac {U_{2}}{Z}}{\frac {1}{M}}\left(1-M\right)}$

${\displaystyle I=-1\cdot I_{2}=-1\cdot {\frac {U_{2}}{Z_{2}}}}$

${\displaystyle -1\cdot {\frac {U_{2}}{Z_{2}}}={\frac {U_{2}}{Z}}{\frac {\left(1-M\right)}{M}}}$

${\displaystyle Z_{2}=Z{\frac {M}{M-1}}}$

literature

• Richard C. Li: RF Circuit Design . In: Information and Communication Technology . 2nd Edition. Wiley, 2012, ISBN 978-1-118-12849-7 .