All about Miller's theorem and its applications
The beginning of the electrical impedance connecting the input and output ports of the amplifier complicates the analysis procedure. One method frequently used to reduce circuit complexity in some applications is using Miller's theorem. This theory is very useful in designing equivalent circuits.
Miller's theory is an important tool, commonly used in the design and analysis of various types of amplifiers as voltage-shift feedback. The network theory is proven to be a multiple of Miller's theory, so it is applicable in the design and analysis of amplifiers through current chain feedback. This article provides an overview of miller theory and how it works with examples of problems.
What is Miller's theory?
Miller's theorem states that: In an amplifier circuit if the impedance is connected between the input and output nodes, including the reference node “N”, then the connected impedance can be changed by two impedances. One impedance can be connected between the input and reference node while another is connected between the o/p node and the reference node.
Miller's theories explain important circuit phenomena regarding impedance modulation that include Miller effect, negative impedance, bootstrapping, virtual ground and help in designing various circuits such as negative impedance transformers, feedback amplifiers, resistive and time-dependent transformers. This theory is very useful for circuit analysis especially for feedback circuits and transistor based amplifiers at extreme frequencies.
The main relationship between Miller theory and Miller effect are; In Miller's theory, this is a simplification of the effect and this effect can be thought of as a special case of Miller's theory.
Miller's Theorem Statement
In general, Miller's theorem is mainly used to modify any circuit from one configuration to another. In any linear network that has a common terminal and two terminals the ratio of its voltage with respect to the common terminal is given
V2 = A * V1
In any network, if we want to change the network to an equivalent circuit, the two terminals have to connect to each other with the help of impedance (Z). So this equivalent circuit includes a similar linear network with two impedances where each impedance inside the network terminal is transmitted to the common terminal. So the values of these two impedances are
Z1 = Z / 1-A
Z2 = AZ / A-1
Derivation of Miller's Theorem
We know that Miller's theorem is used to change the configuration of a circuit into another as follows.
In the following circuit, if impedance “Z” is connected between two nodes like 1 and 2, then this node can be changed by two impedances like Z1 and Z2. Here a connection can be made between two impedances like this; The “Z1” impedance is connected between the first node and the ground terminal while the “Z2” impedance is connected between the second node and the ground terminal.
Miller's theorem states that the effect of resistance Z on an input circuit is the ratio of the input voltage V to the current I that flows from the input to the output.
Millers Theory Guide
According to Miller's theory, the effect of impedance “Z” on an input circuit is the ratio of the input voltage and the current “I” supplied from the input to the output.
proof of the theory
Therefore, 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.
Therefore, the above is a formula for Miller's theorem.
Solve Miller's Theorem Problems
To get hie = 1kΩ & hfe = 50, calculate the net voltage gain for the following circuit.
Example of Miller's Theory Circuit
Example of Miller's Theory Circuit
Once Miller's theorem is applied to the resistance between the inputs and the outputs in the above circuit
When i/p RM = 100k / (1-K) = RI
Output, RN = 100k / (1-K-1) = 100k
Internal voltage gain (K) = -hferRL'/hi
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
Net voltage gain = KRI’/(RS + RI’) = - 192 x * 0.337/2 K + 0.337 K = -27.68.
Miller's Dual Theory
Miller's theorem is also available in a dual version based on Kirchhoff's laws as KCL. Typically, Miller's dual theory can be implemented by an arrangement that includes two voltage sources providing a "Z" grounded resistance using floating impedances. Here, the voltage sources and their impedances can form two current sources like main and auxiliary.
In Miller's theory, a secondary voltage is usually generated by a voltage amplifier depending on the type of amplifier and the gain therefore, the input impedance of the circuit may be nearly unlimited, increasing, decreasing, negative or zero.
In the circuit, if there is a branch with a resistance "Z", it connects a node, and two currents I1 and I2 will meet, we can change this branch through two conducting the indicated currents. The impedance is equal to (1+ a) Z & (1+ a) Z/a, where a = I2/I1.
In fact, changing the two-port network through its equivalent is shown in the circuit below.
branch in a circle
It provides the circuit on the left in the following figure and then, applying the source absorption theorem, the circuit on the right.
Change the two-port network
The dual version of Miller's theorem is a very effective tool, used to design and analyze circuits depending on the change of impedance through additional current. Therefore, applications of double mill theory mainly include; Exotic circuits including negative impedances such as load enclosures, Howland current source, capacitive neutral and their Deboo integrator.
Application of source absorption theory
Miller's theory Advantages
The advantages of Miller's theory include the following.
This theory helps reduce the complexity of circuits such as circuits with feedback by changing them to simple circuits.
Circuit capacitance can be protected by the Miller effect.
Miller's theory Applications
Applications of Miller's theory include the following.
This theory is used to analyze high frequency amplifier circuits.
This is implemented in an amplifier setup called a Millers amplifier which is used as an auxiliary voltage source to change the actual impedance into a virtual impedance.
This theory is used for application in the design process of equivalent circuits.
Miller's theorem is used for all tripartite devices.
It is a very powerful tool, which is used to design and understand different circuits depending on the change of impedance through additional voltage
1). What does the Miller effect do?
The Miller effect enhances circuit capacitance by locating the impedance between the input and output nodes in the circuit. Here the Miller capacity is only an additional capacity.
2). What is a miller and bootstrap sweep generator?
The most commonly used integrating circuit is Miller sweep within many devices. It is a widely used sawtooth generator.
In a boot sweep generator circuit, the output is given to the input as feedback to improve or reduce the circuit's input impedance. So this primer is mainly used to achieve a stable charging current. The polarity of the sweep voltage in a Miller circuit is negative while it is positive in a smoothing sweep circuit.
3). What technique is used in Miller's theory?
The technology used in this theory is the equivalent 2-portnetwork technology.
4). What is the constant k in Miller's theorem?
In Miller's theory, the constant "K" is the internal voltage gain of the circuit (K. = V2/V1)
5). What is the miller amplitude in IGBT?
In IGBT, the Miller capacitance is nothing but the result of the interference of gate-end metallization and the N-region. In the equivalent circuit of IGBT & MOSFET, the Miller capacitance lies between the gate, drain, and collector terminals.
6). What causes the Miller effect?
In an amplifier design, the Miller effect can cause a sharp drop in the amplifier's gain when the frequency increases beyond the known gain. So, the coupling impedance in these amplifiers is a parasitic capacitance.
Thus, this is all about taking an overview of Miller's theory, derivation, proof and their applications. In general, this theory is mainly used to change the circuit from one configuration to another. Here is a question for you, what are the limits of Miller's theory?
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