millers theorem

1. Miller theorem
AJAL.A.J – ASSISTANT PROFESSOR [ ECE DEPARTMENT ]
UNIVERSAL ENGINEERING COLLEGE
Vallivattom P.O., Konathakunnu (Via), Near Mathilakam
Thrissur, Kerala, India. Pin - 680 123
MOB: 0 - 890 730 5642 Mail: ec2reach@gmail.com

2. Miller theorem
• The Miller theorem refers to the process of
creating equivalent circuits.
• It asserts that a floating impedance element,
supplied by two voltage sources connected in
series, may be split into two grounded
elements with corresponding impedances.

3. Kirchhoff's circuit laws:
Miller
theorem
Dual
Miller
theorem
with regards to impedance supplied by
two current sources connected in parallel.
The two versions are based on the two

4. Advantages
• The theorems are useful in 'circuit analysis'
especially for analyzing
1. circuits with feedback
and
2. certain transistor amplifiers at high frequencies

5. Miller theorem (for voltages)
DEFINITION

6. Explanation
• Miller theorem implies that an impedance
element is supplied by two arbitrary (not
necessarily dependent) voltage sources that
are connected in series through the common
ground.

7. In practice,
one of them acts as a main (independent)
voltage source with voltage V1
and the other –
as an additional (linearly dependent)
voltage source with voltage

8. The idea of Miller theorem (modifying circuit
impedances seen from the sides of the input
and output sources) is revealed below by
comparing the two situations
1. with connecting an additional voltage source V2.
and
2.without connecting an additional voltage source V2.

9. Condition -1
• If V2 was zero (there was not a second voltage
source or the right end of the element with
impedance Z was just grounded), the input
current flowing through the element would be
determined, according to Ohm's law, only by
V1
and the input impedance of the circuit
would be

10. Why the name ,Miller theorem for
voltages.
• This version of the Miller theorem is based on
Kirchhoff's voltage law; for that reason, it is
named also Miller theorem for voltages.

11. What happens when a second
voltage source is included ?
• As a second voltage source is included, the
input current depends on both the voltages.
• According to its polarity, V2 is subtracted from
or added to V1; so, the input current
decreases/increases

12. The input current decreases/increases

13. and the input impedance of the circuit
seen from the side of the input source
accordingly increases/decreases

14. • So, Miller theorem expresses the fact that
connecting a second voltage source with
proportional voltage
in series with the input voltage source changes the
effective voltage, the current and respectively, the
circuit impedance seen from the side of the input
source.
• Depending on the polarity, V2 acts as a
supplemental voltage source helping or opposing
the main voltage source to pass the current
through the impedance.

15. Other advantages :
• Besides by presenting the combination of the
two voltage sources as a new composed
voltage source, the theorem may be explained
by combining the actual element and the
second voltage source into a new virtual
element with dynamically modified
impedance.

16. From this viewpoint,
• V2 is an additional voltage that artificially
increases/decreases the voltage drop Vz across
the impedance Z thus decreasing/increasing
the current. The proportion between the
voltages determines the value of the obtained
impedance

17. Subtracting V2 from V1
V2 vs V1 V2 = 0 0 < V2 < V1 V2 = V1 V2 > V1
Impedance normal increased infinite
negative with
current inversion

18. Adding V2 to V1
V2 vs Vz V2 = 0 0 < V2 < Vz V2 = Vz V2 > Vz
Impedance normal decreased zero
negative with voltage
inversion

19. • The circuit impedance, seen from the side of the
output source, may be defined similarly, if the
voltages V1 and V2 are swapped and the
coefficient K is replaced by 1/K

20. Implementation
A typical implementation of Miller theorem based
on a single-ended voltage amplifier

21. Miller theorem may be observed in:
• Most frequently, the Miller theorem may be
observed in, and implemented by, an
arrangement consisting of an element with
impedance Z connected between the two
terminals of a grounded general linear
network

22. • Usually, a voltage amplifier with gain of
serves as such a linear network

23. The Miller amplifier arrangement has
two aspects:

24. Application - 1
• The introduction of an impedance that connects
amplifier input and output ports adds a great
deal of complexity in the analysis process.
But ,
Miller theorem helps reduce the complexity in
some circuits particularly with feedbackby
converting them to simpler equivalent circuits.

25. Application - 2
• Miller theorem is not only an effective tool
for creating equivalent circuits; it is also a
powerful tool for designing and
understanding circuits based on modifying
impedance by additional voltage.

26. Applications based on subtracting V2
from V1

27. 1.

28. The op-amp non-inverting amplifier is a typical circuit
with series negative feedback based on the Miller
theorem, where the op-amp differential input
impedance is apparently increased up to infinite
NOTE

29. 2.

30. 3.

31. Applications based on adding V2 to V1

32. 1.

33. The op-amp inverting amplifier is a typical circuit,
with parallel negative feedback, based on the Miller
theorem, where the op-amp differential input
impedance is apparently decreased up to zero
NOTE

34. 2.

35. NOTE
• In all these op-amp inverting circuits with
parallel negative feedback, the input current is
increased to its maximum. It is determined
only by the input voltage and the input
impedance according to Ohm's law; it does
not depend on the impedance Z.

36. 3.

37. Dual Miller theorem
(for currents)

39. Explanation
• Dual Miller theorem actually expresses the fact
that connecting a second current source
producing proportional current
in parallel with the main input source and the
impedance element changes the current flowing
through it, the voltage and accordingly, the circuit
impedance seen from the side of the input source.

40. • Depending on the direction, I2 acts as a
supplemental current source helping or
opposing the main current source I1 to create
voltage

41. Applications
• As the main Miller theorem, besides helping
circuit analysis process, the dual version is a
powerful tool for designing and understanding
circuits based on modifying impedance by
additional current.