Editing Two-port network (section)

==Hybrid parameters (''h''-parameters) {{anchor|h-parameters}}== <!-- Other articles link to this section. Fix these links if moving or renaming.  New links are advised to use the anchors rather than heading names -->
[[Image:H-equivalent circuit.PNG|thumbnail|300px|Figure&nbsp;6: H-equivalent two-port showing independent variables {{math|''I''{{sub|1}}}} and {{math|''V''{{sub|2}}}}; {{math|''h''<sub>22</sub>}} is reciprocated to make a resistor]]
:<math> \begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} </math>

where

:<math>\begin{align}
  h_{11} &\mathrel{\stackrel{\text{def}}{=}} \left. \frac{V_1}{I_1} \right|_{V_2 = 0} &
  h_{12} &\mathrel{\stackrel{\text{def}}{=}} \left. \frac{V_1}{V_2} \right|_{I_1 = 0} \\
  h_{21} &\mathrel{\stackrel{\text{def}}{=}} \left. \frac{I_2}{I_1} \right|_{V_2 = 0} &
  h_{22} &\mathrel{\stackrel{\text{def}}{=}} \left. \frac{I_2}{V_2} \right|_{I_1 = 0}
\end{align}</math>

This circuit is often selected when a current amplifier is desired at the output. The resistors shown in the diagram can be general impedances instead.

Off-diagonal {{mvar|h}}-parameters are [[dimensionless]], while diagonal members have dimensions the reciprocal of one another.

For reciprocal networks {{math|1=''h''{{sub|12}} = –''h''{{sub|21}}}}. For symmetrical networks {{math|1=''h''{{sub|11}}''h''{{sub|22}} – ''h''{{sub|12}}''h''{{sub|21}} = 1}}. For reciprocal lossless networks {{math|''h''{{sub|12}}}} and {{math|''h''{{sub|21}}}} are real, while {{math|''h''{{sub|11}}}} and {{math|''h''{{sub|22}}}} are purely imaginary.

===Example: common-base amplifier===
[[Image:Common base hybrid pi current follower.PNG|thumbnail|200px|Figure&nbsp;7: Common-base amplifier with AC current source {{math|''I''{{sub|1}}}} as signal input and unspecified load supporting voltage {{math|''V''{{sub|2}}}} and a dependent current {{math|''I''{{sub|2}}}}.]]
'''Note:''' Tabulated formulas in Table 2 make the {{mvar|h}}-equivalent circuit of the transistor from Figure&nbsp;6 agree with its small-signal low-frequency [[hybrid-pi model]] in Figure&nbsp;7. Notation: {{math|''r''<sub>&pi;</sub>}} is base resistance of transistor, {{math|''r''<sub>O</sub>}} is output resistance, and {{math|''g''<sub>m</sub>}} is mutual transconductance. The negative sign for {{math|''h''<sub>21</sub>}} reflects the convention that {{math|''I''{{sub|1}}, ''I''{{sub|2}}}} are positive when directed ''into'' the two-port. A non-zero value for {{math|''h''<sub>12</sub>}} means the output voltage affects the input voltage, that is, this amplifier is '''bilateral'''. If {{math|1=''h''<sub>12</sub> = 0}}, the amplifier is '''unilateral'''.

{| class="wikitable" style="text-align:center; vertical-align:center; margin: 1em auto 1em auto"
|+ Table 2
! !! Expression !! Approximation
|-
| <math>h_{21} = \left. \frac{ I_{2} }{ I_1 } \right|_{V_2=0} </math>
| <math> -\frac{ \frac{\beta}{\beta + 1} r_\mathrm{O} + r_\pi }{ r_\mathrm{O} + r_\pi} </math>
| <math> -\frac{ \beta }{ \beta + 1 } </math>
|-
| <math>h_{11} = \left. \frac{V_1}{I_1} \right|_{V_2=0}  </math> 
| <math> r_\pi \mathbin{\|} r_\mathrm{O} </math>
| <math>r_\pi</math>
|-
| <math>  h_{22} = \left. \frac{I_2}{V_2} \right|_{I_1=0} </math>
| <math> \frac{1}{(\beta + 1)(r_\mathrm{O} + r_\pi)} </math>
| <math> \frac{1}{(\beta + 1)r_\mathrm{O} } </math>
|-
| <math> h_{12} = \left. \frac{V_1}{V_2} \right|_{I_1=0} </math>
| <math> \frac{r_\pi}{r_\mathrm{O} + r_\pi}</math> 
| <math> \frac{r_\pi}{r_\mathrm{O}} \ll 1</math>
|}

===History===
The {{mvar|h}}-parameters were initially called ''series-parallel parameters''.  The term ''hybrid'' to describe these parameters was coined by D. A. Alsberg in 1953 in "Transistor metrology".<ref>56 IRE 28.S2, p. 1543</ref>  In 1954 a joint committee of the [[Institute of Radio Engineers|IRE]] and the [[AIEE]] adopted the term {{mvar|h}}-''parameters'' and recommended that these become the standard method of testing and characterising transistors because they were "peculiarly adaptable to the physical characteristics of transistors".<ref>AIEE-IRE committee report, p. 725</ref>  In 1956, the recommendation became an issued standard; 56 IRE 28.S2.  Following the merge of these two organisations as the [[IEEE]], the standard became Std 218-1956 and was reaffirmed in 1980, but has now been withdrawn.<ref>IEEE Std 218-1956</ref>