正實函數

正實函數（Positive-real functions）的縮寫是PR函數或是PRF，是在電路分析中會出現的一種數學函數。正實函數是複數函數Z(s)，其變數s也是複數。有理函數若在複平面的右半邊都有正的實部，且可解析，在實軸上都為實數，就是正實函數。

其定義可以表示為下式：

${\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \Re (s)>0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \Im (s)=0\end{aligned}}$

在電路分析中Z(s)表示阻抗，而s為S平面變數，也常用其實部及虛部表示：

$s=\sigma +i\omega \,\!$

則正實函數的定義會改為下式：

${\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \sigma >0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \omega =0\end{aligned}}$

正實函數在電路分析的重要性在於正實函數的條件也就是電路可實現性的條件。Z(s)可實現為單埠有理阻抗若且唯若其符合正實函數的條件。此情形下的可實現表示可以用有限個分立理想的被動線性元件（以電路來說就是電阻器、电感元件、电容器）來實現。

定義

「正實函數」最早是由Otto Brune所定義，描述符合以下條件的函數Z(s) ：

• 是有理函數（二個多項式的商）
• s為實數時，函數有實數值。
• s的實部為正時，函數的實數也為正值。

許多作者嚴格依照上述定義，包括明確要求是有理函數 .。不過Cauer之前就有提出類似，但要求較寬的條件，也有些作者將「正實函數」的定義認為是Cauer提出的這一種，其他作者則認為Cauer的定義是基本定義的擴展版本。

歷史

正實函數的條件最早是由Wilhelm Cauer(1926)提出，他確定了這些是必要條件 Otto Brune(1931)開始使用「正實」（positive-real）一詞，並且證明是可實現的充份條件及必要條件。

性質

已隱藏部分未翻譯内容，歡迎參與翻譯。

• The sum of two PR functions is PR.
• The 复合函数 of two PR functions is PR. In particular, if Z(s) is PR, then so are 1/Z(s) and Z(1/s).
• All the zeros and poles of a PR function are in the left half plane or on its boundary of the imaginary axis.
• Any poles and zeroes on the imaginary axis are 极点 (复分析) (have a 重覆度 of one).
• Any poles on the imaginary axis have real strictly positive 留数, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.
• Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a 调和函数 over the plane, and therefore satisfies the maximum principle).
• For a 有理函數 PR function, the number of poles and number of zeroes differ by at most one.

擴展版本

正實函數有許多的擴展版本，希望用導抗函數來處理更大範圍的被動線性電路。

無理函數

已隱藏部分未翻譯内容，歡迎參與翻譯。

The impedance Z(s) of a network consisting of an infinite number of components (such as a semi-infinite Ladder_network), need not be a rational function of s, and in particular may have branch points on the negative real s-axis. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all real s, and only require this when s is positive. Thus, a possibly irrational function Z(s) is PR if and only if

• Z(s) is analytic in the open right half s-plane (Re[s] > 0)
• Z(s) is real when s is positive and real
• Re[Z(s)] ≥ 0 when Re[s] ≥ 0

Some authors start from this more general definition, and then particularize it to the rational case.

矩陣值函數

已隱藏部分未翻譯内容，歡迎參與翻譯。

Linear electrical networks with more than one port may be described by impedance parameters admittance parameters. So by extending the definition of PR to matrix-valued functions, linear multi-port networks which are passive may be distinguished from those that are not. A possibly irrational matrix-valued function Z(s) is PR if and only if

• Each element of Z(s) is analytic in the open right half s-plane (Re[s] > 0)
• Each element of Z(s) is real when s is positive and real
• The 埃尔米特矩阵 part (Z(s) + Z^†(s))/2 of Z(s) is 正定矩阵 when Re[s] ≥ 0

參考資料

• Wilhelm Cauer (1932) The Poisson Integral for Functions with Positive Real Part, Bulletin of the American Mathematical Society 38:713–7, link from Project Euclid.
• W. Cauer (1932) "Über Funktionen mit positivem Realteil", Mathematische Annalen 106: 369–94.