Miller theorem

AJAL.A.J – ASSISTANT PROFESSOR [  ECE DEPARTMENT ]

UNIVERSAL ENGINEERING COLLEGE

Vallivattom P.O., Konathakunnu (Via), Near MathilakamThrissur, Kerala, India. Pin - 680 123

MOB:  0 - 890 730 5642 Mail: ec2reach@gmail.com

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.

Kirchhoff's circuit laws:

Miller theorem

Dual Miller

theoremwith regards to impedance supplied by two current sources connected in parallel.

The two versions are based on the two

Advantages

• The theorems are useful in 'circuit analysis' especially for analyzing

1. circuits with feedback

and

2. certain transistor amplifiers at high frequencies

Miller theorem (for voltages)

DEFINITION

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.

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

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.

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

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.

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

The input current decreases/increases

and the input impedance of the circuit seen from the side of the input source

accordingly increases/decreases

• 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.

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.

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

Subtracting V2 from V1

V2 vs V1 V2 = 0 0 < V2 < V1 V2 = V1 V2 > V1

Impedance normal increased infinite negative with current inversion

Adding V2 to V1

V2 vs Vz V2 = 0 0 < V2 < Vz V2 = Vz V2 > Vz

Impedance normal decreased zero negative with voltage inversion

• 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

Implementation

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

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

• Usually, a voltage amplifier with gain ofserves as such a linear network

The Miller amplifier arrangement has two aspects:

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.

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.

Applications based on subtracting V2 from V1

1.

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

2.

3.

Applications based on adding V2 to V1

1.

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

2.

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.

3.

Dual Miller theorem

(for currents)

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.

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

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.