Synthesisar na Aiki | Kyakkyawan Hurwitz | Funcciyoyi Masu Farko

Tattalin Nau'in Shugaban Sistem

Ma'adaniyar Nau'in Shugaban Sistem

Tattalin nau'in shugaban sistem yana nuna nau'in shugaban sistem da suka dacewa da abubuwan da ake amfani da su don kawo (kamar mafi girma) da kuma abubuwan da ba ake amfani da su don kawo (kamar inductors da capacitors).

A tafara da abubuwan da za a iya gano: wannan shine ma'adaniyar nau'in shugaban sistem? A cikin tsarin frequency, ma'adaniyar nau'in shugaban sistem suna nufin zabi mai karfi da ke divide phasor ta shi ne da phasor ta input.

A harshen mutane, ma'adaniyar nau'in shugaban sistem suna nufin zabi mai karfi da ke divide output phasor da input phasor a lokacin da phasors ke wuce a cikin tsarin frequency. Tsarin ma'adaniyar nau'in shugaban sistem ya zama haka:

Daga baya, a kan tsafta da na ma'adaniyar nau'in shugaban sistem, za a iya bayyana al'amuran da suka fi siffar da dalilin da take daidaita a kan ma'adaniyar nau'in shugaban sistem. Ana da uku masu al'amuran da suka fi siffar da dalilin da take daidaita a kan ma'adaniyar nau'in shugaban sistem, kuma ana ambaci su a nan:

1. Zabubbukan numerator of F(s) ba zan iya kudeta zabubbukan denominator da zaka bi da wahid. Kafin samun lafiya (m – n) ya zama kadan ko kadan.

2. F(s) ba zan iya da multiple poles a jω-axis ko y-axis a pole-zero plot.

3. F(s) ba zan iya da poles a right half of the s-plane.

Hurwitz Polynomial

Idan duk dalilin da take daidaita a kan ma'adaniyar nau'in shugaban sistem an samun (kamar yadda aka yi) don haka denominator of the F(s) an kiranta Hurwitz polynomial.

A nan, Q(s) shine Hurwitz polynomial.

Al'amuran Hurwitz Polynomials

Ana da uku al'amuran muhimmiya a kan Hurwitz polynomials, kuma ana ambaci su a nan:

1. Don duk ukuwar s, value of the function P(s) zan iya zama real.

2. Real part of every root zan iya zama zero ko negative.

3. Idan a duba coefficients of denominator of F(s) is bn, b(n-1), b(n-2). . . . b0. A nan ita ce bn, b(n-1), b0 za su zama positive and bn and b(n-1) ba zan iya zama zero simultaneously.

4. Continued fraction expansion of even to the odd part of the Hurwitz polynomial zana iya tabbatawa da all positive quotient terms, idan even degree is higher or the continued fraction expansion of odd to the even part of the Hurwitz polynomial should give all positive quotient terms, if odd degree is higher.

5. In case of purely even or purely odd polynomial, we must do continued fraction with the of derivative of the purely even or purely odd polynomial and rest of the procedure is same as mentioned in the point number (4).

Daga baya, an samun wannan result, Idan duk coefficients of the quadratic polynomial are real and positive then that quadratic polynomial is always a Hurwitz polynomial.

Positive Real Functions

Koyar da yake canzawa da F(s) zan iya zama positive real function idan an samun waɗannan uku al'amuran muhimmiya:

1. F(s) zan iya tabbatar da ukuwar real values for all real values of s.

2. P(s) zan iya zama Hurwitz polynomial.

3. Idan a substitute s = jω then on separating the real and imaginary parts, the real part of the function should be greater than or equal to zero, means it should be non negative. This most important condition and we will frequently use this condition in order to find out the whether the function is positive real or not.

4. On substituting s = jω, F(s) should posses simple poles and the residues should be real and positive.

Al'amuran Positive Real Function

Ana da uku al'amuran muhimmiya a kan positive real functions kuma ana ambaci su a nan:

1. Both the numerator and denominator of F(s) should be Hurwitz polynomials.

2. The degree of the numerator of F(s) should not exceed the degree of denominator by more than unity. In other words (m-n) should be less than or equal to one.

3. If F(s) is positive real function then reciprocal of F(s) should also be positive real function.

4. Remember the summation of two or more positive real function is also a positive real function but in case of the difference it may or may not be positive real function.

Following are the four necessary but not the sufficient conditions for the functions to be a positive real function and they are written below:

1. The coefficient of the polynomial must be real and positive.

2. The degree of the numerator of F(s) should not exceed the degree of denominator by more than unity. In other words (m – n) should be less than or equal to one.

3. Poles and zeros on the imaginary axis should be simple.

4. Let us consider the coefficients of denominator of F(s) is bn, b(n-1), b(n-2). . . . b0.Here it should be noted that bn, b(n-1), b0 must be positive and b

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