## What is Miller Theorem : Derivation & Its Applications

- Frank
- May 25, 2022
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The beginning of electrical impedance that connects the input & output ports of an amplifier includes a complexity within the analysis procedure. One method is used frequently to decrease the circuit complexity in some applications using **Miller Theorem**. This theorem is very helpful in designing equivalent circuits.

The theory of Miller’s is a significant tool, used commonly in the design & analysis of different types of amplifiers like voltage-shunt feedback. A network theorem is proved to be double to Miller’s theorem, so it is applicable in the design & analysis of amplifiers through current-series feedback. This article discusses an overview of the miller theorem and its working with example problems.

### What is Miller Theorem?

Miller’s theorem states that; in an amplifier circuit if the impedance is connected in between the input & output nodes, including a reference node ‘N’, then connected impedance can be changed through two impedances. One impedance can be connected in between the input & reference node whereas another one is connected in between the o/p & the reference node.

Miller theorems explain significant circuit phenomena regarding impedance modifying which includes Miller effect, negative impedance, bootstrapping, virtual ground & assists in designing different circuits like negative impedance converters, feedback amplifiers, resistive & time-dependent converters. This theorem is very useful for the analysis of circuits particularly for feedback circuits & transistor-based amplifiers at maximum frequencies.

The main relationship between the Miller theorem & the Miller effect is; in the miller theorem, it is considered as a simplification of the effect & this effect may be considered from a special case of the miller theorem.

### Miller Theorem Statement

Generally, miller’s theorem is mainly used to modify any circuit from one configuration to another. In any linear network which have a common terminal & two terminals whose voltage ratio, is given with respect to the common terminal is

**V2 = A * V1**

In any network, if we want to change the network into an equivalent circuit then the two terminals need to interconnect with the help of impedance (Z). So this equivalent circuit includes a similar linear network with two impedances where each impedance within a network terminal moves to the common terminal. So the values of these two impedances are

**Z1 = Z/1-A**

**Z2 = AZ/A-1**

#### Miller Theorem Derivation

We know that miller’s theorem is used to change one circuit configuration to other like the following.

In the following circuit, if the ‘Z’ impedance is connected in between two nodes like 1 & 2 then this node can be changed through two impedances like Z1 & Z2. Here the connection of two impedances can be done like this; impedance ‘Z1’ is connected in between the first node & ground terminal whereas impedance ‘Z2’ is connected in between the second node & ground terminal.

Miller’s theorem states that the effect of resistance Z on the input circuit is a ratio of the input voltage ‘V’ to the current ‘I’ which flows from the input to the output.

#### Millers Theorem Proof

According to miller’s theorem, the impedance effect ‘Z’ on the input circuit is a ratio of the input voltage & the current ‘I’ which supplies from the input to the output.

So, **Z = V1/I**

**I = Vi-V0/Z**

**I = Vi (1-(V0/Vi)/Z)**

**I = Vi (1-Av/Z)**

**Z1 = Z/1-K**

**Z2 = V0/I**

**I = V0-Vi/Z**

**I = V0(1-Vi/V0)/Z)**

**I = V0(1-1/Av)/Z)**

**Z2 = Z/1-1/K**

So, the above shown is miller’s theorem formula.

### Miller’s Theorem Solved Problems

For given hie = 1kΩ & hfe = 50, .calculate net voltage gain for the following circuit.

Once the miller’s theorem is applied to resistance in between input & output in the above circuit

At i/p RM = 100k/(1-K) = RI

Output, RN=100k / (1-K-1) = 100k

Gain of internal voltage (K) = -hfeRL’/hie

K = – 50*Rc||(100k/1k) = – 50*4*100/104 = – 192

RI = 100k/(1+192) = 0.51kΩ

RI’ = RI||hie = 0.51k||1k = 0.51*1/1.51 = 0.337kΩ

Gain of net voltage = K.RI’/(RS+RI’) = – 192 x* 0.337/2k + 0.337k = -27.68.

### Dual Miller’s Theorem

The miller theorem is also available in a dual version based on Kirchoff’s Laws like KCL. Usually, the Dual Miller theorem can be implemented through an arrangement that includes two voltage sources that provide grounded impedance ‘Z’ using floating impedances. Here, the voltage sources and their impedances can form two current sources like main & auxiliary.

In the Miller theorem, usually, the secondary voltage is generated through a voltage amplifier based on the kind of amplifier & the gain so, the input impedance of the circuit may be almost infinite, increased, decreased, and negative or zero.

In a circuit, if a branch with impedance ‘Z’ is there then it connects a node, and two flows of currents I1 & I2 will meet, we can change this branch through two performing the referred currents. The impedances correspondingly equivalent to (1+ a) Z & (1+ a) Z / a, wherever a = I2/I1.

Actually, changing the two-port network through its equivalent is shown in the circuit below.

It provides the circuit on the left within the next figure and after that, applying the theorem of source absorption, the circuit on the right.

The dual version of the Miller theorem is a very efficient tool, used to design & analyze the circuits depending on changing impedance through extra current. So, dual miller theorem applications mainly include; exotic circuits including negative impedance like load cancellers, Howland current source, capacitance neutralizers & its derivative Deboo integrator.

### Advantages

The **Miller Theorem advantages** include the following.

- This theorem helps in reducing the circuit complexity like the circuits with feedback through changing them into simple circuits.
- The capacitance of the circuit can be protected by the miller effect.

### Applications

The **Miller Theorem applications** include the following.

- This theorem is used to analyze high-frequency-based amplifier circuits.
- This is applied in the setting of an amplifier called Millers amplifier which is used as an additional voltage source to change the actual impedance into a virtual impedance.
- This theorem is used to apply in the process of designing equivalent circuits.
- Miller’s theorem is used for all three-terminal devices.
- It is a very powerful tool, used to design and understand different circuits depending on changing impedance through extra voltage

**1). What does the Miller effect do?**

Miller effect enhances the capacitance of a circuit by locating impedance in between input & output nodes in the circuit. Here miller capacitance is nothing but extra capacitance.

**2). What is Miller and bootstrap sweep generator?**

The most frequently used integrator circuit is Miller sweep within several devices. It is an extensively used sawtooth generator.

In the bootstrap sweep generator circuit, the output is given to the input like feedback to enhance or reduce the circuit’s input impedance. So this bootstrapping is mainly used to attain a stable charging current. The sweep voltage’s polarity in the miller sweep circuit is negative whereas, in the bootstrap sweep circuit, it is positive.

**3). What technique is used in Miller Theorem?**

The technique used in this theorem is the equivalent 2-portnetwork technique.

**4). What is the constant k in Miller’s theorem?**

In the miller theorem, the constant ‘K’ is the internal voltage gain of the circuit (K. = V2/V1)

**5). What is Miller capacitance in IGBT?**

In IGBT, the miller capacitance is nothing but the outcome of the overlap of the metallization of the gate terminal & the N- region. In IGBT & MOSFET equivalent circuit, the miller capacitance is situated in between gate, drain & collector terminals.

**6). What causes the Miller effect?**

In amplifier design, the Miller effect can cause a drastic decrease of amplifier gain when frequency increases known gain roll-off. So, the coupling impedance in these amplifiers is a parasitic capacitance.

Thus, this is all about an overview of the miller theorem, derivation, proof, and its applications. Generally, this theorem is mainly used to change the circuit from one configuration to another. Here is a question for you, what are the miller theorem limitations?