#27 Port references, finite parameter representations, measurement conditions and reciprocity

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Replace internal detail with a valid two-port model

   

   Change from internal elements to terminal behavior.

   A block model must preserve the chosen port voltage and current relations.

   For a linear two-port in its valid operating range, two terminal pairs can be described by suitable parameter matrices.

   Narration transcript

   So far, most circuits were drawn from the inside: every resistor, capacitor, inductor, op-amp, and connection. Now we change the viewpoint. A complicated circuit can be treated as a two-port block: one input port, one output port, and a small set of measured parameters.

2. 2. State voltage polarity and entering-current references

   

   Define each port voltage from its upper terminal to its lower terminal.

   Both reference currents enter the upper, positive-reference terminals and return through the paired lower terminals.

   The same entering-current convention gives a consistent equation form; it does not make every parameter matrix symmetric.

   Narration transcript

   A two-port has four terminal variables: voltage and current at port one, and voltage and current at port two. The standard convention is that both currents enter the network. This may feel a little strange at the output port, but it makes the equations symmetric and reusable.

3. 3. Predict voltages from independent port currents

   

   Use an impedance matrix when independent port currents determine finite, unique port voltages.

   First voltage equation:

   $\displaystyle V_{1}=z_{11} I_{1}+z_{12} I_{2}$

   Second voltage equation:

   $\displaystyle V_{2}=z_{21} I_{1}+z_{22} I_{2}$

   The diagonal terms are open-circuit driving-point impedances; the off-diagonal terms are transfer impedances.

   Every impedance parameter has units of ohms; the matrix entries may be complex and frequency-dependent.

   Narration transcript

   The first description is the z-parameter model. It writes port voltages as functions of port currents. The diagonal terms are driving-point impedances. The off-diagonal terms are transfer impedances. Every z value has the unit ohm.

4. 4. Extract z parameters under open-circuit conditions

   

   Extract each impedance-matrix column with the other port open; the driven current must be nonzero.

   With port two open, so its current is zero:

   $\displaystyle z_{11}=\frac{V_{1}}{I_{1}}, z_{21}=\frac{V_{2}}{I_{1}}$

   With port one open, so its current is zero:

   $\displaystyle z_{12}=\frac{V_{1}}{I_{2}}, z_{22}=\frac{V_{2}}{I_{2}}$

   Narration transcript

   To measure z parameters, use open-circuit tests. If port two is open, its current is zero, so z eleven and z twenty one come directly from voltage divided by the driven current. If port one is open, its current is zero, so z twelve and z twenty two are found the same way from the other side.

5. 5. Predict currents from independent port voltages

   

   Use an admittance matrix when independent port voltages determine finite, unique port currents.

   First current equation:

   $\displaystyle I_{1}=y_{11} V_{1}+y_{12} V_{2}$

   Second current equation, with both reference currents still entering:

   $\displaystyle I_{2}=y_{21} V_{1}+y_{22} V_{2}$

   Diagonal entries are short-circuit driving-point admittances; cross entries are transfer admittances. All have units of siemens.

   Narration transcript

   The second description is the y-parameter model. It writes port currents as functions of port voltages. The diagonal terms are driving-point admittances, and the off-diagonal terms are transfer admittances. Every y value has the unit siemens.

6. 6. Extract y parameters under short-circuit conditions

   

   Extract each admittance-matrix column with the other port shorted; the driven voltage must be nonzero.

   With port two shorted, so its voltage is zero:

   $\displaystyle y_{11}=\frac{I_{1}}{V_{1}}, y_{21}=\frac{I_{2}}{V_{1}}$

   With port one shorted, so its voltage is zero:

   $\displaystyle y_{12}=\frac{I_{1}}{V_{2}}, y_{22}=\frac{I_{2}}{V_{2}}$

   Narration transcript

   To measure y parameters, use short-circuit tests. If port two is shorted, its voltage is zero, so y eleven and y twenty one are obtained by driving port one. If port one is shorted, its voltage is zero, so y twelve and y twenty two are obtained by driving port two.

7. 7. Choose an existing finite parameter representation

   

   Choose a parameter representation that exists and is convenient for the task.

   Impedance form is convenient for open-circuit conditions and suitable series interconnections.

   Admittance form is convenient for short-circuit conditions and suitable parallel or nodal descriptions.

   When both matrices exist and are nonsingular, conversion is a matrix inverse, not entrywise reciprocals:

   $\displaystyle Y=Z^{-1}$

   Narration transcript

   Which model should you choose? Use z when open-circuit tests are natural, or when series behavior and impedances are easier to see. Use y when short-circuit tests are natural, or when shunt behavior and nodal equations are easier. The model is a tool, not a religion.

8. 8. Distinguish reciprocity from port interchange symmetry

   

   Keep reciprocity, passivity and port-interchange symmetry distinct.

   For a reciprocal two-port, wherever these parameter matrices exist:

   $\displaystyle z_{12}=z_{21}, y_{12}=y_{21}$

   Port-interchange symmetry also requires equal diagonal terms, not only those cross-term conditions:

   $\displaystyle z_{11}=z_{22}, y_{11}=y_{22}$

   Narration transcript

   Two quick checks help catch mistakes. For a reciprocal passive network, the transfer parameters match: z twelve equals z twenty one, and y twelve equals y twenty one. For a symmetric network, the two driving-point parameters match: z eleven equals z twenty two, and y eleven equals y twenty two.

9. 9. Reuse the block with correct source and load conditions

   

   Choose references, verify the model exists, and specify the operating frequency or transform variable.

   Derive or simulate the required test ratios with the right open or short condition; test definitions do not require physically shorting a powered amplifier.

   Use the matrix with the actual source and load; a load current leaving port two has the opposite sign:

   $\displaystyle I_{L}=-I_{2}$

   Narration transcript

   The workflow is simple. Draw or simulate the circuit, choose the port convention, run the required open or short tests, fill the matrix, and then use the matrix instead of redrawing the inside every time. That is why two-port models are so useful in amplifiers, filters, transmission lines, and cascaded stages.

10. 10. Summarize z and y models and their limitations

    

    Summary: a two-port model compresses linear terminal behavior, not every internal detail.

    Retain the port polarity, entering-current convention, operating conditions and model-existence assumptions.

    Impedance form predicts port voltages from currents using open-circuit parameter tests.

    Admittance form predicts port currents from voltages using short-circuit parameter tests.

    For cascades, a suitable transmission matrix is often more convenient; do not simply multiply z or y matrices.

    Narration transcript

    Summary. A two-port block replaces internal complexity with port variables and a parameter matrix. z parameters use currents to predict voltages and are measured with open circuits. y parameters use voltages to predict currents and are measured with short circuits. Next, we can extend this view to hybrid and ABCD parameters, where cascaded stages become much easier.

Source video: Circuit Theory-2 #27 | Two-Port Networks: z and y Parameters (3:45)