_{1}

^{*}

In a given linear, multistage, cascaded amplifier [ 1] comprising passive coupling circuits and active two-ports alternatively, the problem is where in the amplifier the stabilizing circuit elements should be placed to eliminate instability, and of what type and value. Our investigations are based on a new recursive formula for the determinant of tridiagonal matrices. Relation of our results to the Stern stability factor has been obtained. A verification in numerical examples has also been provided.

Stability Theory is currently being revived in many disciplines. This is because of the novel results in chaotic systems and investigations into its robustness. In electrical engineering as well, these problems are of distinguished actuality in areas such as in automatics, control theory, and power electronics. However, in this study, we concentrate on electrical circuits.

In circuit theory, stability related problems are in close connection with appearance and application of feedback. These investigations were based on the observation that exotic phenomena are in close connection with singularities of the model. Characteristic of the early period is the graphical investigation of the loop gain. A later, well distinguished period can be described as a numeric investigation of stability that was often in connection with invariance properties. During the 1970s and 1980s, clarifications and multidimensional system investigations were performed. The above-mentioned renaissance began in the 1980s when Professor Chua and his colleagues, based on results from physics, started studying exotic phenomena in nonlinear circuits. Most important results of this period are the simplest circuit exhibiting chaotic phenomena and systematic investigation of chaotic behavior. More recent investigations are related to the utilization of chaotic circuits for information transfer. Besides, up to the most recent time, you can often find some novel results for describing chaotic circuit operation.

In Section 2, we overview some publications from the history above outlined that relate more closely to our work. Section 3 contains our result for the determinant of tridiagonal matrices. Section 4 describes stabilization of one stage amplifiers, with a numerical example. In Section 5, two stages are considered. In Section 6, factors of the determinant are related to the Stern stability factor. Generalization for more than two stages is provided in Section 7. In Section 8, numerical examples are found.

References in this Section are organized as linear, time invariant, then time variant, then nonlinear circuits. All references are provided in time order. Relation of the references to the proposed research is that all of them are founding papers of this topic.

An overview figure for the proposed research is in

1) The Section begins with results for linear, time-invariant, active circuits.

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As stability and passivity are in so close connection, we give the reference that provides the most refined concept of passivity for linear multiports [

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[_{1} ports are terminated by a k_{1} port, the next k_{2} port by a k_{2} port, and so on, with k_{1} + k_{2} + ∙∙∙ = n. This k-stability is connected to the stability concept known till date in such a manner that k_{1} = 1, k_{2} = 1, ∙∙∙ He gives for absolute stability a necessary and sufficient condition. He points out that k-stability in general is weaker than traditional stability.

For time invariant, lumped element circuits, stability is often investigated by the Hurwitz test. [

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2) As two examples, we continue with investigations of linear, time-variant circuits in time and complex frequency domain.

[

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3) This overview is finished with some stability investigation results for nonlinear circuits.

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Besides general stability conditions, several researchers publish conditions for specific nonlinear circuits [

In our early study [

This topic has other Hungarian-related results as well: Information transfer using chaotic circuits has been investigated by [

Let A _ _ be a square matrix having nonzero entries only in the main and neighboring diagonals (tridiagonal matrix). The submatrix comprising the rows with serial number p 1 , p 2 , ⋯ , p u and columns q 1 , q 2 , ⋯ , q v is denoted as follows [

A q 1 , q 2 , ⋯ , q v p 1 , p 2 , ⋯ , p u (1)

For determinant of A _ _ , we introduce the notation | A _ _ | .

Theorem 1. The determinant of the n × n matrix A _ _ can be expressed as

| A _ _ | = | A 1 , 2 , ⋯ , p 1 , 2 , ⋯ , p | ∗ | A p + 1 , p + 2 , ⋯ , n p + 1 , p + 2 , ⋯ , n | − a p , p + 1 a p + 1 , p | A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | ∗ | A p + 2 , p + 3 , ⋯ , n p + 2 , p + 3 , ⋯ , n | (2)

where a_{i}_{,k} is the entry of A _ _ in the ith row and kth column, and p is a positive integer with 1 ≤ p ≤ n .

Meaning of (2) is illustrated in an example. Let n = 8 and p = 3. Submatrices in (2) are denoted by thick lines in

Proof: Full induction is applied where for n = 2 and p = 1 the following holds:

| a c b d | = a d − b c (3)

| A 1 , 2 , ⋯ , p 1 , 2 , ⋯ , p | = a (4)

| A p + 1 , p + 2 , ⋯ , n p + 1 , p + 2 , ⋯ , n | = d (5)

a p + 1 , p = b (6)

a p , p + 1 = c (7)

and we define

| A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | = | A p + 2 , p + 3 , ⋯ , n p + 2 , p + 3 , ⋯ , n | = 1 (8)

when the first row or column index is greater than the last one. If n = 2 and p = 2, then (2) holds as well.

We assume that the theorem is valid for n × n tridiagonal matrices when 1 ≤ p ≤ n and prove that it holds for (n + 1) × (n + 1) if 1 ≤ p ≤ n + 1 .

In the pth column, three entries are different from zero: a_{p}_{-1,p}, a_{p}_{,p}, a_{p}_{+1,p}. Thus

| A 1 , 2 , ⋯ , n + 1 1 , 2 , ⋯ , n + 1 | = − a p − 1 , p | A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 2 , p , ⋯ , n + 1 | + a p , p | A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 | − a p + 1 , p | A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p , p + 2 , ⋯ , n + 1 | (9)

On the right side, all three determinants are determinants of n x n tridiagonal matrices; for that the assumption above holds:

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 2 , p , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 2 1 , 2 , ⋯ , p − 2 | ∗ | A p − 1 , p + 1 , ⋯ , n + 1 p , ⋯ , n + 1 | − a p − 2 , p − 1 a p , p − 2 | A 1 , 2 , ⋯ , p − 3 1 , 2 , ⋯ , p − 3 | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | (10)

From the tridiagonal property, it follows that

a p , p − 2 = 0 (11)

Therefore, (10) can be rewritten as

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 2 , p , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 2 1 , 2 , ⋯ , p − 2 | ∗ | A p − 1 , p + 1 , ⋯ , n + 1 p , ⋯ , n + 1 | (12)

Now we apply the assumption for the second determinant on the right side of (9):

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | − a p − 1 , p + 1 a p + 1 , p − 1 | A 1 , 2 , ⋯ , p − 2 1 , 2 , ⋯ , p − 2 | ∗ | A p + 2 , ⋯ , n + 1 p + 2 , ⋯ , n + 1 | (13)

Because of the tridiagonal property, all entries with a difference of more than one between row and column indices is zero:

a p − 1 , p + 1 = a p + 1 , p − 1 = 0 (14)

Thus (13) has a form of

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | (15)

Finally, the third determinant on the right side of (9) is expanded:

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p , p + 2 , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p , p + 2 , ⋯ , n + 1 | − a p , p − 1 a p − 1 , p + 1 | A 1 , 2 , ⋯ , p − 2 1 , 2 , ⋯ , p − 2 | ∗ | A p + 2 , ⋯ , n + 1 p + 2 , ⋯ , n + 1 | (16)

As

a p − 1 , p + 1 = 0 (17)

(16) is simplified as

| A 1 , ⋯ , p − 1 , p + 1 , ⋯ , n + 1 1 , ⋯ , p , p + 2 , ⋯ , n + 1 | = | A 1 , 2 , ⋯ , p − 1 1 , 2 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p , p + 2 , ⋯ , n + 1 | (18)

Because of (9, 12, 15, 18):

| A 1 , 2 , ⋯ , n + 1 1 , 2 , ⋯ , n + 1 | = − a p − 1 , p | A 1 , ⋯ , p − 2 1 , ⋯ , p − 2 | ∗ | A p − 1 , p + 1 , ⋯ , n + 1 p , ⋯ , n + 1 | + a p , p | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | − a p + 1 , p | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p , p + 2 , ⋯ , n + 1 | (19)

On the right side of the last equation, we reformulate the second determinant, applying (2):

| A p − 1 , p + 1 , ⋯ , n + 1 p , ⋯ , n + 1 | = | A p − 1 p | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | + a p + 1 , p − 1 a p , p + 1 ∗ 1 ∗ | A p + 2 , ⋯ , n + 1 p + 2 , ⋯ , n + 1 | (20)

Because

| A p − 1 p | = a p , p − 1 (21)

and

a p + 1 , p − 1 = 0 (22)

(20) can be rewritten as

| A p − 1 , p + 1 , ⋯ , n + 1 p , ⋯ , n + 1 | = a p , p − 1 ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | (23)

The last step is getting to know that the second and third term on the right side of (19) can be joined as

a p , p | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | − a p + 1 , p | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | | A p + 1 , ⋯ , n + 1 p , p + 2 , ⋯ , n + 1 | = | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | | A p , ⋯ , n + 1 p , ⋯ , n + 1 | (24)

Based on (19, 23, 24):

| A 1 , 2 , ⋯ , n + 1 1 , 2 , ⋯ , n + 1 | = − a p − 1 , p | A 1 , ⋯ , p − 2 1 , ⋯ , p − 2 | a p , p − 1 ∗ | A p + 1 , ⋯ , n + 1 p + 1 , ⋯ , n + 1 | + | A 1 , ⋯ , p − 1 1 , ⋯ , p − 1 | ∗ | A p , ⋯ , n + 1 p , ⋯ , n + 1 | (25)

and that is what we wanted to see. Pathologic cases (n < p) are trivial.

Explanation for Equation (25) is that it expresses the same as Equation (2) above and that we wanted to prove.

In this Section, Theorem 1 will be applied for stabilization of a one-port amplifier. The amplifier can be characterized in the following manner:

[ I G 0 0 0 ] = [ Y G + Y 11 ( 1 ) Y 21 ( 1 ) 0 0 Y 21 ( 1 ) Y 22 ( 1 ) + Y 11 ( 2 ) Y 12 ( 2 ) 0 0 Y 21 ( 2 ) Y 22 ( 2 ) + Y 11 ( 3 ) Y 21 ( 3 ) 0 0 Y 21 ( 3 ) Y 22 ( 3 ) + Y L ] [ V G V 12 V 23 V L ] (26)

where the square matrix on the right is called the admittance matrix of the amplifier. Explanation for notations is found in

Transducer power gain is defined as

G T = P L P G max (27)

Power absorbed by the load is

P L = 1 2 | V L | 2 G L (28)

where G L = R e ( Y L ) . Maximum generator power is

P G max = | I G | 2 4 G G (29)

where G G = R e ( Y G ) . From (27)-(29):

G T = 2 G G G L | V L I G | 2 (30)

The transducer power gain (30) is a function of the transfer impedance:

Z T = V L I G = − Y 21 ( 1 ) Y 21 ( 2 ) Y 21 ( 3 ) D (31)

and that is obtained from (26) using Cramer’s rule. D is the determinant of the admittance matrix of the amplifier. The amplifier is unstable if

G T → ∞ (32)

otherwise it is stable. Therefore,

Theorem 2. For stability of the amplifier, it is necessary that the zeroes of the determinant D are placed at the left half of the complex frequency plane. If the admittances in the nominator of (31) do not have right half plane poles, then the given condition is necessary and sufficient.

Proof: Now Theorem 1 will be applied for determining D. The admittance matrix of the amplifier is denoted by Y _ _ :

D = | Y 1 , 2 1 , 2 | | Y 3 , 4 3 , 4 | − y 2 , 3 y 3 , 2 | Y 1 1 | | Y 4 4 | (33)

The determinants at the right side are written based on (26):

| Y 1 , 2 1 , 2 | = ( Y G + Y 11 ( 1 ) ) ( Y 22 ( 1 ) + Y 11 ( 2 ) ) − ( Y 21 ( 1 ) ) 2 (34)

| Y 3 , 4 3 , 4 | = ( Y 22 ( 2 ) + Y 11 ( 3 ) ) ( Y 22 ( 3 ) + Y L ) − ( Y 21 ( 3 ) ) 2 (35)

y 2 , 3 = Y 12 ( 2 ) (36)

y 3 , 2 = Y 21 ( 2 ) (37)

| Y 1 1 | = Y G + Y 11 ( 1 ) (38)

| Y 4 4 | = Y 22 ( 3 ) + Y L (39)

In (34), | Y 1 , 2 1 , 2 | can be rewritten as a product:

| Y 1 , 2 1 , 2 | = ( Y G + Y 11 ( 1 ) ) [ Y 22 ( 1 ) + Y 11 ( 2 ) − ( Y 21 ( 1 ) ) 2 Y G + Y 11 ( 1 ) ] (40)

In the second term on the right, we introduce a new notation. Output admittance of the first block in

Y ( 1 , o ) = Y 22 ( 1 ) − ( Y 21 ( 1 ) ) 2 Y G + Y 11 ( 1 ) (41)

Meaning of the new notation is clarified in

| Y 1 , 2 1 , 2 | = ( Y G + Y 11 ( 1 ) ) ( Y ( 1 , o ) + Y 11 ( 2 ) ) (42)

Based on (35), similarly, | Y 3 , 4 3 , 4 | can be expressed as a product:

| Y 3 , 4 3 , 4 | = ( Y 22 ( 2 ) + Y ( 3 , i ) ) ( Y 22 ( 3 ) + Y L ) (43)

where Y ( 3 , i ) is the input admittance of block 3 in

Y ( 3 , i ) = Y 11 ( 3 ) − ( Y 21 ( 3 ) ) 2 Y 22 ( 3 ) + Y L (44)

Utilizing (42) and (43), D in (33) can be written as follows:

D = ( Y G + Y 11 ( 1 ) ) [ ( Y ( 1 , o ) + Y 11 ( 2 ) ) ( Y 22 ( 2 ) + Y ( 3 , i ) ) − Y 12 ( 2 ) Y 21 ( 2 ) ] ( Y 22 ( 3 ) + Y L ) (45)

The second term on the right side of (45) can be transformed:

( Y ( 1 , o ) + Y 11 ( 2 ) ) [ Y 22 ( 2 ) + Y ( 3 , i ) − Y 12 ( 2 ) Y 21 ( 2 ) Y ( 1 , o ) + Y 11 ( 2 ) ] = ( Y ( 1 , o ) + Y 11 ( 2 ) ) ( Y ( 2 , o ) + Y ( 3 , i ) ) (46)

where Y ( 2 , o ) denotes output admittance of the block 2 in

Y ( 2 , o ) = Y 22 ( 2 ) − Y 12 ( 2 ) Y 21 ( 2 ) Y ( 1 , o ) + Y 11 ( 2 ) (47)

Thus, product decomposition of the determinant is

D = ( Y G + Y 11 ( 1 ) ) ( Y ( 1 , o ) + Y 11 ( 2 ) ) ( Y ( 2 , o ) + Y ( 3 , i ) ) ( Y 22 ( 3 ) + Y L ) (48)

Please recognize that the terms of the product decomposition correspond to the admittances at the connections of blocks as shown in

Theorem 3. Number of terms in the product decomposition of the determinant is equal to the number of interconnections in the amplifier. One of the terms is the admittance of interconnected ports, without any change in the amplifier. The other terms are also admittances at other interconnections, but with other ports short-circuited.

Proof:

Theorem 3 can be applied for finding circuit elements stabilizing the amplifier. For this, you have to recognize that a circuit element connected to the input of block 3 modifies only the third term of the determinant. The circuit element connected to the input of block two modifies the second and the third term, the circuit element at the input of block 1 modifies terms 1, 2, and 3, whereas the circuit element at the output of block three modifies terms 3 and 4. Hurwitz test of the terms specifies which term has to be modified and thus the place where it has to be connected is given.

The last statement is illustrated by an example. A one-stage amplifier is illustrated in

Determinant of the admittance matrix of the amplifier in

Y ( 2 , o ) + Y ( 3 , i ) = 2.499 × 10 − 3 A ( p ) B ( p ) (49)

A ( p ) = 1 + 2.209 × 10 − 9 p + 1.038 × 10 − 19 p 2 + 3.065 × 10 − 30 p 3 + 1.099 × 10 − 41 p 4 − 4.217 × 10 − 67 p 5 (50)

where p denotes complex frequency. A(p) is not a Hurwitz polynomial because the sign of the fifth order term is different from others. For this reason, a stabilizing element to be connected at the input of the block 3 is needed. Let it be a resistor whose element value is determined so that the Rollett stability factor equals to one at the edge of the passband. The stabilized amplifier with its characteristic before and after stabilization are shown in Figures 6-8.

In

amplifier, of course. Det < 1 in both cases, satisfying the stability condition regarding the determinant. Before stabilization, K < 1 at the lower edge of the passband, hurting the stability condition, and K > 1 in the whole passband, after stabilization.

A two-stage cascaded amplifier can be characterized as follows:

[ I G 0 0 0 0 0 ] = [ Y G + Y 11 ( 1 ) Y 21 ( 1 ) 0 0 0 0 Y 21 ( 1 ) Y 22 ( 1 ) + Y 11 ( 2 ) Y 12 ( 2 ) 0 0 0 0 Y 12 ( 2 ) Y 22 ( 2 ) + Y 11 ( 3 ) Y 21 ( 3 ) 0 0 0 0 Y 21 ( 3 ) Y 22 ( 3 ) + Y 11 ( 4 ) Y 12 ( 4 ) 0 0 0 0 Y 12 ( 4 ) Y 22 ( 4 ) + Y 11 ( 5 ) Y 21 ( 5 ) 0 0 0 0 Y 21 ( 5 ) Y 22 ( 5 ) + Y L ] [ V G V 12 V 23 V 34 V 45 V L ] (51)

For determining the power gain, we have to formulate the transfer impedance as it can be seen from (30):

Z T = V L I G = − Y 21 ( 1 ) Y 21 ( 2 ) Y 21 ( 3 ) Y 21 ( 4 ) Y 21 ( 5 ) D (52)

where D denotes now the determinant of the 6 × 6 admittance matrix in (51). In the following, we calculate the product decomposition of D. If Theorem 1 is applied for n = 6 and p = 2, then n = 4 and p = 2, and the result is rearranged as in the previous Section; we arrive at the following expression:

D = ( Y G + Y 11 ( 1 ) ) ( Y ( 1 , o ) + Y 11 ( 2 ) ) ( Y ( 2 , o ) + Y 11 ( 3 ) ) ( Y ( 3 , o ) + Y ( 4 , i ) ) ( Y 22 ( 4 ) + Y ( 5 , i ) ) ( Y 22 ( 5 ) + Y L ) (53)

where the following notations have been introduced:

Y ( 1 , o ) = Y 22 ( 1 ) − ( Y 21 ( 1 ) ) 2 Y G + Y 11 ( 1 ) (54)

the output admittance of block 1 when its input is terminated by Y G ,

Y ( 5 , i ) = Y 11 ( 5 ) − ( Y 21 ( 5 ) ) 2 Y 22 ( 5 ) + Y L (55)

the input admittance of block 5 when its output is terminated by Y L ,

Y ( 2 , o ) = Y 22 ( 2 ) − Y 12 ( 2 ) Y 21 ( 2 ) Y ( 1 , o ) + Y 11 ( 2 ) (56)

the output admittance of block 2 when its input is terminated by Y ( 1 , o ) ,

Y ( 4 , i ) = Y 11 ( 4 ) − Y 12 ( 4 ) Y 21 ( 4 ) Y 22 ( 4 ) + Y ( 5 , i ) (57)

the input admittance of block 4 when its output is terminated by Y ( 5 , i ) ,

Y ( 3 , o ) = Y 22 ( 3 ) − ( Y 21 ( 3 ) ) 2 Y ( 2 , o ) + Y 11 ( 3 ) (58)

the output admittance of block 3 when its input is terminated by Y ( 2 , o ) .

Terms of D in (53) are identical to the admittances in

Thus, as it is written in the previous Section, Hurwitz test of the terms in (53) can be applied for determining the place of the stabilizing circuit elements.

Product decomposition of the determinant is not unique. As a consequence, a specific stabilization problem can be solved at least two ways. Solutions for one- and two-stage amplifiers are summarized in

Port number | Serial number of the non-Hurwitz term | |||
---|---|---|---|---|

1 | 2 | 3 | 4 | |

1 | X | |||

2 | X | X | ||

3 | X | X | X | X |

4 | X |

Port number | Serial number of the non-Hurwitz term | |||
---|---|---|---|---|

1 | 2 | 3 | 4 | |

1 | X | |||

2 | X | X | X | X |

3 | X | X | ||

4 | X |

Port number | Serial number of the non-Hurwitz term | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

1 | X | |||||

2 | X | X | ||||

3 | X | X | X | |||

4 | X | X | X | X | X | X |

5 | X | X | ||||

6 | X |

Port number | Serial number of the non-Hurwitz term | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

1 | X | |||||

2 | X | X | ||||

3 | X | X | X | X | X | X |

4 | X | X | X | |||

5 | X | X | ||||

6 | X |

of the amplifier, we exploit that there are ports where, if a stabilizing circuit element is connected, then not all terms are modified. For example, a circuit element connected to port 1, does not influence term 2, 3, or 4.

The difference between

Instability of the investigated amplifiers are caused by the feedback inside the active elements. Instability can also be traced by investigating the real part of the terms of the determinant. Therefore, it is interesting to investigate the ratio of the real part of the terms with and without feedback. For example, in case of a two-stage amplifier, the third term in (53) is:

y 3 = Y ( 2 , o ) + Y 11 ( 3 ) (59)

and the value of y 3 without the feedback of the active element (block 2) is

y 30 = Y 22 ( 2 ) + Y 11 ( 3 ) (60)

The above-mentioned ratio is

d 3 = R e ( y 3 ) R e ( y 30 ) (61)

Physical interpretation of the terms in d_{3} is given in

Relation between the Stern stability factor and the minimum of d_{3} is

d 3 min = 1 − 1 S 3 (62)

where S_{3} is the Stern stability factor of the circuit in the upper right corner of

We denote by y a term of the determinant that is identical to the admittance at the load side port of an active element. The following idea is valid also for generator side admittance; thus, a full generalization of (62) is made. Denote by y_{0} the value of y when the feedback is omitted. Let us denote by Y_{G} and Y_{L} the terminating admittances according to

Let

d = R e ( y ) R e ( y 0 ) (63)

and let S denote the Stern stability factor of the amplifier in

Theorem 4.

d min = 1 − 1 S (64)

Proof: First we recognize that d depends on G_{G}, G_{L}, and B_{G}, but it does not depend on B_{L} ( Y G = G G + j B G , Y L = G L + j B L ). For this reason, to keep the power gain constant, we keep G_{G}, G_{L} constant and we seek for the minimum as a function of B_{G}. Then we compare the expression for d_{min} to the definition of the Stern stability factor.

Applying the notations of

y = Y 22 − Y 12 Y 21 Y 11 + Y G + Y L (65)

When determining the real part of y, we introduce the following notations:

Y 12 Y 21 = M + j N (66)

Real and imaginary parts of the admittance matrix entries are denoted by G and B, respectively, properly indexed.

g = G 22 − R e M + j N ( G 11 + G G ) + j ( B 11 + B G ) + G L (67)

Let

G 11 + G G = G 1 (68)

G 22 + G L = G 2 (69)

B 11 + B G = B 1 (70)

Real part of y is the following:

g = − M G 1 + N B 1 G 1 2 + B 1 2 + G 2 (71)

and really, g does not depend on B_{L}. We seek g_{min} as a function of B_{G}, therefore, we have to solve the following equation:

∂ g ∂ B G = ∂ g ∂ B 1 = 0 (72)

or in more detailed form,

N ( G 1 2 + B 1 2 ) − 2 B 1 ( M G 1 + N B 1 ) ( G 1 2 + B 1 2 ) 2 = 0 (73)

The solution is

B 1 = G 1 [ − M N ± ( M N ) 2 + 1 ] (74)

Minimum of g is found by substituting (74) into (71):

g min = G 2 − M 2 + N 2 + M 2 G 1 (75)

If the active element does not have inner feedback, then M = N = 0, thus, from (71):

R e ( y 0 ) = G 2 (76)

From (75) and (76):

d min = 1 − M 2 + N 2 + M 2 G 1 G 2 (77)

The definition of the Stern stability factor [

S = 2 G 1 G 2 M 2 + N 2 + M (78)

Therefore, the relation between the Stern stability factor and d_{min} can be expressed as

d min = 1 − 1 S (79)

The Stern stability condition is

S ≥ 1 (80)

Thus from (79) and (80):

d min ≥ 0 (81)

as expected.

Note that the proof contains more to learn. Using the results above, (3) can be proven in a simple way, as follows. For stability, terms of the determinant must not have roots in the closed right half plane. From this and from passivity of y_{0}, (81) follows at the jω axis. From (81) and (79), (80) follows, and finally, from (80) and (78) results in (3).

We also note that for a two-stage amplifier, not considering feedback in active elements does not influence the terms 1, 2, 5, and 6, thus d i = 1 if i = 1, 2, 5, or 6. In these cases, Theorem 2 is useless. On the contrary, in a multistage amplifier, Theorem 2 holds for all terms that are modified when the feedback of the active elements is omitted.

In this Section, the stabilization method introduced in the previous Sections is generalized. Generalization is based on the fact that in the recursive formula for the determinant of tridiagonal matrices, direction of recursion can be changed once.

Theorem 5.

If A _ _ = [ a i k ] is an n × n tridiagonal matrix, and m is a positive integer, 2 ≤ m ≤ n − 1 , then the determinant of A _ _ can be expressed as follows:

| A _ _ | = ∏ k = 1 n a k (82)

a 1 = a 11 (83)

a k = a k k − a k − 1 , k a k k − 1 a k − 1 , k = 2 , 3 , ⋯ , m − 1 (84)

a m = a m , m − a m − 1 , m a m , m − 1 a m − 1 − a m + 1 , m a m , m + 1 a m + 1 (85)

a k = a k k − a k + 1 , k a k k + 1 a k + 1 , k = m + 1 , m + 2 , ⋯ , n − 1 (86)

a n = a n n (87)

Proof: We will see that the Theorem is valid for m = n − 1, then we assume it holds for m and prove that it holds for m − 1 as well.

| A _ _ | = ∏ k = 1 n b k (88)

b 1 = a 11 (89)

b k = a k k − a k − 1 , k a k k − 1 b k − 1 , k = 2 , 3 , ⋯ , n (90)

The n-1th and nth terms in detail are as follows:

b n − 1 = a n − 1 , n − 1 − a n − 2 , n − 1 a n − 1 , n − 2 b n − 2 (91)

b n = a n , n − a n − 1 , n a n , n − 1 b n − 1 (92)

Then the product of these two terms is rewritten as

b n − 1 b n = b n − 1 a n , n − a n − 1 , n a n , n − 1 = ( b n − 1 − a n − 1 , n a n , n − 1 a n , n ) a n , n = ( a n − 1 , n − 1 − a n − 2 , n − 1 a n − 1 , n − 2 b n − 2 − a n − 1 , n a n , n − 1 a n , n ) a n , n (93)

Comparing this with (85), we can see that the Theorem is valid for m = n − 1 as we wanted. Now we assume it is valid for m, and terms m − 1 and m are written as

a m − 1 = a m − 1 , m − 1 − a m − 2 , m − 1 a m − 1 , m − 2 a m − 2 (94)

if m ≥ 2 , and

a m = a m , m − a m − 1 , m a m , m − 1 a m − 1 − a m + 1 , m a m , m + 1 a m + 1 (95)

if m ≤ n − 1 .

Similarly, as in (93), we rewrite the product of these two terms:

a m − 1 a m = a m − 1 a m , m − a m − 1 , m a m , m − 1 − a m − 1 a m + 1 , m a m , m + 1 a m + 1 = ( a m − 1 − a m − 1 , m a m , m − 1 a m , m − a m + 1 , m a m , m + 1 a m + 1 ) ( a m , m − a m + 1 , m a m , m + 1 a m + 1 ) (96)

If in (82)-(87), we leave all terms unchanged except terms m − 1 and m, and at the place of these terms (96) is written, then the theorem is proven.

Now we apply Theorem 5 for the admittance matrix of an n-stage amplifier.

This amplifier can be characterized as follows:

[ I G 0 0 0 0 0 ] = [ Y G + Y 11 ( 1 ) Y 21 ( 1 ) 0 0 0 0 Y 21 ( 1 ) Y 22 ( 1 ) + Y 11 ( 2 ) 0 0 0 0 Y 12 ( 2 ) 0 0 0 0 Y 22 ( m − 1 ) + Y 11 ( m ) 0 0 0 0 Y 21 ( m ) 0 0 0 0 Y 22 ( 2 n + 1 ) + Y L ] [ V G V 12 V 23 V 34 V 45 V L ] (97)

Applying Theorem 5 for the admittance matrix in (97), product decomposition of the determinant is the following:

D = ( Y G + Y 11 ( 1 ) ) ( Y ( 1 , o ) + Y 11 ( 2 ) ) ∗ ⋯ ∗ ( Y ( m , o ) + Y ( m + 1 , i ) ) ∗ ⋯ ∗ ( Y 22 ( 2 n ) + Y ( 2 n + 1 , i ) ) ( Y 22 ( 2 n + 1 ) + Y L ) (98)

where the following notations were introduced: Output admittance of block 1 when the input termination is Y G :

Y ( 1 , o ) = Y 22 ( 1 ) − ( Y 21 ( 1 ) ) 2 Y G + Y 11 ( 1 ) (99)

Output admittance of block m when the input termination is Y ( m − 1 , o ) :

Y ( m , o ) = Y 22 ( m ) − Y 12 ( m ) Y 21 ( m ) Y ( m − 1 , o ) + Y 11 ( m ) (100)

Input admittance of block m + 1 when its output is terminated by Y ( m + 2 , i ) :

Y ( m + 1 , i ) = Y 11 ( m + 1 ) − Y 12 ( m + 1 ) Y 21 ( m + 1 ) Y 22 ( m + 1 ) + Y ( m + 2 , i ) (101)

Input admittance of block 2n + 1 when its output is terminated by Y L :

Y ( 2 n + 1 , i ) = Y 11 ( 2 n + 1 ) − Y 12 ( 2 n + 1 ) Y 21 ( 2 n + 1 ) Y 22 ( 2 n + 1 ) + Y L (102)

Terms of (98) have physical meaning. In case of n-stage amplifier, there are 2n + 2 terms, where 2n + 1 terms are equal to the admittance at a port interconnection, when another interconnection is short-circuited. In case of the remaining term, the short circuit must not be applied. Physical meaning of the terms is summarized in

Equation (98) can be applied for stabilization of n-stage amplifiers as it was described in the previous Sections. Note that the fact that coupling circuits and active elements follow each other alternatively was not deeply exploited. Therefore, our results can be reasonably applied for arbitrary order of components.

It would result in further simplification if the direction of recursion in (82)-(87) could be altered once more. But we can see easily that this is not possible. The reason is that we exploited in (82)-(87) the property of (88)-(90), that is, that terms do not depend on other terms with higher serial number. On the contrary, the mth term in (85) depends on terms m − 1 and m + 1, and all terms of lower serial number depend on the previous ones, whereas those of higher serial number depend on the next terms. Therefore, recursion can be altered only once.

Next, you can find a simple example for illustration that product decomposition of the determinant can be utilized for characterization of stability based on circuit element values.

Product decomposition in (48) is applied here for a one-stage field-effect transistor (FET) amplifier. Circuit schematics is shown in

According to Theorem 3, the determinant is a product of four terms. Roots of first, second, and fourth terms lie in the left half plane. Thus, stability of the amplifier depends on the third term. Regarding (48), the third term can be expressed as follows:

y 3 = G 2 + p C G D − ( g m − p C G D ) ( − p C G D ) p C 1 1 + R G p C 1 + G 1 + p C G S + p C G D + p C 2 1 + R L p C 2 (103)

where p stands for the complex frequency. When C 1 , C 2 → ∞ , the above expression is simplified as

y 3 = G 2 + p C G D − ( g m − p C G D ) ( − p C G D ) 1 R G + G 1 + p C G S + p C G D + 1 R L (104)

Thus, the stability of the amplifier is determined by the real part of the roots of the following equation:

0 = ( G 2 + p C G D + 1 R L ) ( 1 R G + G 1 + p C G S + p C G D ) − ( g m − p C G D ) ( − p C G D ) (105)

Recognize that the right side of (105) is the determinant of the admittance matrix of the FET model completed with resistive elements. If the FET does not have inner feedback, then the second term would appear, and the real part of the roots is always negative. The learning from this example is that our method is very efficient in characterizing stability.

Next, we investigate a two-stage amplifier designed for the 0.5 - 2.5 GHz band. The amplifier does not meet the stability conditions. We analyze the terms of the determinant and we obtain a possible way for stabilization.

Schematics of the amplifier is shown in

Recognize that we have to separate the parts of the amplifier differently than what is shown in

Now (98) applies for seven blocks; thus, the number of terms is 8. The circuit is cut between blocks 4 and 5; thus m = 4. In the product decomposition, obviously terms 1, 2, and 8 do not exhibit instability. Numerators of the other terms A i ( p ) , i = 3 , 4 , ⋯ , 7 are determined in half-symbolic form [

A 3 ( p ) = p + 3.706 × 10 − 11 p 2 + 1.668 × 10 − 21 p 3 (106)

A 4 ( p ) = 1 + 1.703 × 10 − 10 p + 6.687 × 10 − 21 p 2 + 2.023 × 10 − 31 p 3 + 4.528 × 10 − 58 p 4 (107)

A 5 ( p ) = 1 + 6.623 × 10 − 10 p + 9.369 × 10 − 20 p 2 + 5.636 × 10 − 30 p 3 + 2.688 × 10 − 40 p 4 + 6.998 × 10 − 51 p 5 + 1.28 × 10 − 61 p 6 + 4.101 × 10 − 73 p 7 − 1.773 × 10 − 97 p 8 (108)

A 6 ( p ) = 1 + 1.352 × 10 − 10 p + 6.078 × 10 − 20 p 2 + 2.369 × 10 − 31 p 3 (109)

A 7 ( p ) = p + 4.618 × 10 − 12 p 2 (110)

It is obvious that in case of A_{7} there is no stability problem. In case of A_{3}, A_{4}, and A_{7}, the coefficients and all Hurwitz determinants [_{5}. The problem is solved by R_{5} that is connected between blocks 4 and 5. R_{5} modifies only term 5; therefore, no more stabilizing elements are necessary. To have all coefficients of A_{5} positive, R 5 ≤ 0.674 kohm resistor is needed. Circuit schematics and characteristics are given in Figures 19-21.

Our problem was to determine places where stabilizing elements could be connected in a linear multistage cascaded amplifier. We showed that the problem

leads to the product decomposition of the determinant of the admittance matrix of the amplifier. For this reason, we obtained identities for the determinants of tridiagonal matrices, and we applied them for our problem. We showed that Hurwitz test of the terms of the determinant provides a possibility for finding places for the stabilizing elements. We proved that the terms of the determinant are in close connection with the Stern stability factor.

In practice, it is convenient to connect stabilizing elements with ports of blocks of the amplifier. For this reason, we solved our problem based on the admittance matrix description of the amplifier. Other formalisms, for example, impedance matrix, can also be used and that leads to stabilizing elements in series with the ports.

Stability can also be provided using feedback. Using low-loss circuit elements for the feedback, maximum stable gain can be approached. In our method, resulting stabilized gain is smaller than that, but in return, we have a conveniently applicable procedure, and the stabilizing elements can be inserted at the ready amplifier.

The method is not directly applicable for stabilizing distributed circuits. For our method, lumped-element model is necessary.

The product decomposition introduced is not unique. It depends on selecting the value of m in (98). We showed in examples that the method can be efficiently applied in writing circuit-specific stability conditions, and that transfer admittances of the blocks must not have right half plane poles.

With this article, the author remembers his former first master, Dr. A. Baranyi, with whom these investigations were started. We have received the problem from Prof. L. Pap, and we are grateful for that. The author is grateful to Prof. P. Rózsa, his former examiner for the PhD degree, who checked that turning the direction of recursion is really a new result. Prof. T. Berceli, Dr. E. Simonyi, V. Beskid, A. Bartus, and Dr. B. Kovács are also acknowledged for providing the author the best possible research conditions that were necessary for writing this article.

The authors declare no conflicts of interest regarding the publication of this paper.

Ladvánszky, J. (2018) Stabilization of Linear Multistage Amplifiers. Circuits and Systems, 9, 169-195. https://doi.org/10.4236/cs.2018.911017