Editing Two-port network (section)

==Impedance parameters (''z''-parameters)==
[[Image:Z-equivalent two port.png|thumbnail|300px| Figure&nbsp;2: z-equivalent two port showing independent variables {{math|''I''<sub>1</sub>}} and {{math|''I''<sub>2</sub>}}. Although resistors are shown, general impedances can be used instead.]]
{{main|Impedance parameters}}

:<math> \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} </math>

where

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

All the {{mvar|z}}-parameters have dimensions of [[ohm]]s.

For reciprocal networks {{math|1=''z''{{sub|12}} = ''z''{{sub|21}}}}.  For symmetrical networks {{math|1=''z''{{sub|11}} = ''z''{{sub|22}}}}.  For reciprocal lossless networks all the {{math|''z''{{sub|mn}}}} are purely imaginary.<ref name=Matt29>Matthaei et al, p. 29.</ref>

===Example: bipolar current mirror with emitter degeneration===
[[Image:Current mirror.png|thumbnail|left|200px| Figure&nbsp;3: Bipolar [[current mirror]]: {{math|''i''<sub>1</sub>}} is the ''reference current'' and {{math|''i''<sub>2</sub>}} is the ''output current''; lower case symbols indicate these are ''total'' currents that include the DC components]]
[[Image:Smal-signal mirror circuit.png|thumbnail|right|300px| Figure&nbsp;4: Small-signal bipolar current mirror: {{math|''I''{{sub|1}}}} is the amplitude of the small-signal ''reference current'' and {{math|''I''{{sub|2}}}} is the amplitude of the small-signal ''output current'']]
Figure&nbsp;3 shows a bipolar current mirror with emitter resistors to increase its output resistance.<ref group="nb">The emitter-leg resistors counteract any current increase by decreasing the transistor {{math|''V''<sub>BE</sub>}}. That is, the resistors {{math|''R''<sub>E</sub>}} cause negative feedback that opposes change in current. In particular, any change in output voltage results in less change in current than without this feedback, which means the output resistance of the mirror has increased.</ref> Transistor {{math|''Q''{{sub|1}}}} is ''diode connected'', which is to say its collector-base voltage is zero. Figure&nbsp;4 shows the small-signal circuit equivalent to Figure&nbsp;3. Transistor {{math|''Q''{{sub|1}}}} is represented by its emitter resistance {{math|''r''{{sub|E}}}}: 
:<math>r_\mathrm{E} \approx \frac{ \text{thermal voltage, } V_\mathrm{T} }{ \text{emitter current, } I_E},</math> 
a simplification made possible because the dependent current source in the hybrid-pi model for {{math|''Q''{{sub|1}}}} draws the same current as a resistor {{math|1 / ''g''<sub>m</sub>}} connected across {{math|''r''<sub>π</sub>}}. The second transistor {{math|''Q''{{sub|2}}}} is represented by its [[hybrid-pi model]]. Table 1 below shows the z-parameter expressions that make the z-equivalent circuit of Figure&nbsp;2 electrically equivalent to the small-signal circuit of Figure&nbsp;4.
{| class="wikitable" style="text-align:center; vertical-align:center; margin:1em auto 1em auto;"
|+ Table 1
! !! Expression !! Approximation
|-
| <math>R_{21} = \left. \frac{V_2}{I_1} \right|_{I_2=0} </math>
| <math>-(\beta r_\mathrm{O} - R_\mathrm{E}) \frac{r_\mathrm{E} + R_\mathrm{E}}{r_\pi + r_\mathrm{E} + 2R_\mathrm{E}} </math>
| <math>-\beta r_\mathrm{o} \frac{r_\mathrm{E} + R_\mathrm{E} }{r_\pi + 2R_\mathrm{E}} </math>
|-
| <math>R_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} </math> 
| <math>(r_\mathrm{E} + R_\mathrm{E}) \mathbin{\|} (r_\pi + R_\mathrm{E}) </math><ref group="nb">The double vertical bar denotes a [[parallel (operator)|parallel]] connection of the resistors: <math>R_1 \mathbin{\|} R_2 = 1/(1/R_1 + 1/R_2)</math>.</ref>
|
|-
| <math> R_{22} = \left. \frac{V_2}{I_2} \right|_{I_1=0} </math>
| &nbsp; <math> \left(1 + \beta \frac{R_\mathrm{E}}{r_\pi + r_\mathrm{E} + 2R_\mathrm{E}} \right) r_\mathrm{O} + \frac{r_\pi + r_\mathrm{E} + R_\mathrm{E}}{r_\pi + r_\mathrm{E} + 2R_\mathrm{E}} R_\mathrm{E}</math> &nbsp; 
| &nbsp; <math> \left(1 + \beta \frac{R_\mathrm{E}}{r_\pi + 2R_\mathrm{E}} \right) r_\mathrm{O} </math> &nbsp; 
|-
| <math>R_{12} = \left. \frac{V_1}{I_2} \right|_{I_1=0}</math>
| <math>R_\mathrm{E} \frac{r_\mathrm{E} + R_\mathrm{E}}{r_\pi + r_\mathrm{E} + 2R_\mathrm{E}}</math> 
| <math>R_\mathrm{E} \frac{r_\mathrm{E} + R_\mathrm{E}}{r_\pi + 2R_\mathrm{E}}</math> 
|}

The negative feedback introduced by resistors {{math|''R''<sub>E</sub>}} can be seen in these parameters. For example, when used as an active load in a differential amplifier, {{math|''I''<sub>1</sub> ≈ −''I''<sub>2</sub>}}, making the output impedance of the mirror approximately 
:<math>R_{22} - R_{21} \approx \frac{ 2\beta r_\mathrm{O}R_\mathrm{E} }{ r_\pi + 2R_\mathrm{E}}</math>
compared to only {{math|''r''<sub>O</sub>}} without feedback (that is with {{math|''R''<sub>E</sub>}} = 0{{nbsp}}Ω). At the same time, the impedance on the reference side of the mirror is approximately 
:<math>R_{11} - R_{12} \approx \frac{r_\pi}{r_\pi + 2R_\mathrm{E}} (r_\mathrm{E} + R_\mathrm{E}),</math>
only a moderate value, but still larger than {{math|''r''<sub>E</sub>}} with no feedback. In the differential amplifier application, a large output resistance increases the difference-mode gain, a good thing, and a small mirror input resistance is desirable to avoid [[Miller effect]].