Circuit Theory II · 120 h · Topic 3 of 8

Two-port networks

Two attenuators of 6 and 10 dB in cascade give 16 dB only if each one sees the impedance it was designed for. The transmission matrix tells you what happens with any other load.

z and y parameters h parameters ABCD matrix Cascade Reciprocity

01A box with two pairs of terminals

Many circuits are used as a block with one input and one output: an amplifier, a filter, an attenuator, a transformer, a line. A two-port network describes them solely by what happens at their two pairs of terminals, regardless of how they are built inside. It has four variables, \( V_1, I_1, V_2, I_2 \), and since it is linear, two of them are determined by the other two. Depending on which are chosen as independent there are six sets of parameters; four are in constant use.

02Impedance and admittance parameters

\[ V_1 = z_{11} I_1 + z_{12} I_2 \] \[ V_2 = z_{21} I_1 + z_{22} I_2 \] Each z parameter is measured by leaving one pair of terminals open-circuited: \( z_{11} = V_1/I_1 \) with \( I_2 = 0 \). The y parameters, the inverses of these, are measured with a short circuit.

If the two-port is reciprocal —any network of resistors, capacitors, inductors and transformers is—, \( z_{12} = z_{21} \): the transfer is the same in both directions. An amplifier is not reciprocal.

03Hybrid parameters and the transistor

\[ V_1 = h_{11} I_1 + h_{12} V_2 \] \[ I_2 = h_{21} I_1 + h_{22} V_2 \] They mix an impedance, a reverse voltage gain, a current gain and an admittance. In the common-emitter bipolar transistor they are called \( h_{ie} \), \( h_{re} \), \( h_{fe} \) and \( h_{oe} \), and they are the ones given in datasheets.

The h parameters are natural for the transistor because they are measured under easy conditions: \( h_{ie} \) and \( h_{fe} \) with the output short-circuited for the signal, and \( h_{re} \) and \( h_{oe} \) with the input open. The small-signal model of the bipolar transistor amplifier is exactly this two-port.

04Transmission parameters and the cascade connection

\[ \begin{pmatrix} V_1 \\ I_1 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} V_2 \\ -I_2 \end{pmatrix} \] They relate the input to the output. Their great advantage: the matrix of two two-ports in cascade is the product of their matrices, in the order in which they are connected.

Block	ABCD matrix
Series impedance \( Z \)	\( \begin{pmatrix} 1 & Z \\ 0 & 1 \end{pmatrix} \)
Shunt admittance \( Y \)	\( \begin{pmatrix} 1 & 0 \\ Y & 1 \end{pmatrix} \)
Ideal transformer 1 : n	\( \begin{pmatrix} 1/n & 0 \\ 0 & n \end{pmatrix} \)
Lossless line of length \( \ell \)	\( \begin{pmatrix} \cos\beta\ell & jZ_0\sin\beta\ell \\ j\sin\beta\ell/Z_0 & \cos\beta\ell \end{pmatrix} \)

A ladder network is then analyzed by multiplying simple matrices, which is what RF circuit simulation programs do. The transmission line fits into the same calculation.

05The two-port with source and load

With a load \( R_L \) at the output, the transmission parameters directly give what matters:

\[ Z_{in} = \frac{A R_L + B}{C R_L + D} \] \[ \frac{V_2}{V_1} = \frac{R_L}{A R_L + B} \] The input impedance depends on the load, and the gain on what the two-port has connected on both sides. That is why a block measured with one load does not perform the same with another.

Lab · two-ports in cascade

A 50 Ω generator drives two two-ports in cascade with a load \( R_L \). The ABCD matrices are multiplied and the input impedance and the gains are calculated. The attenuators are designed for 50 Ω; the transistor has \( h_{ie} = 2.5 \) kΩ, \( h_{re} = 2 \times 10^{-4} \), \( h_{fe} = 100 \) and \( h_{oe} = 20\ \mu \)S.

Load R_L =

06In the lab

Activity 1 · Measuring the z parameters

Build a T network of resistors, measure \( z_{11}, z_{12}, z_{21}, z_{22} \) with the terminals open and verify reciprocity.

Activity 2 · Cascade

Connect two attenuators in cascade, measure the total attenuation and compare it with the product of the matrices.

Activity 3 · h parameters of a transistor

Measure \( h_{ie} \) and \( h_{fe} \) of a biased transistor, with the output short-circuited for the signal through a capacitor, and compare with the datasheet.

07Common mistakes

Multiplying the matrices in reverse order. Matrix multiplication does not commute.
Forgetting the sign of \( I_2 \). In the ABCD parameters the output current is taken as flowing out.
Using parameters measured at a different bias. The transistor’s h parameters change with the operating point.
Assuming the input impedance does not depend on the load. That is only so if \( C R_L \ll D \) and \( B \ll A R_L \).

08Self-assessment

What ABCD matrix does a 100 Ω series resistor followed by a 100 Ω shunt resistor have?

\( \begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0.01 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 100 \\ 0.01 & 1 \end{pmatrix} \).

What input impedance does an ideal 1:2 transformer have with a 200 Ω load?

\( 200 / 2^2 = 50\ \Omega \).

When is a two-port reciprocal?

When \( z_{12} = z_{21} \); all passive networks of linear elements are.

What voltage gain does a common emitter with h_fe = 100, h_ie = 2.5 kΩ and a 2 kΩ load have, neglecting h_re and h_oe?

\( -h_{fe} R_L / h_{ie} = -80 \).

09Further reading

James W. Nilsson and Susan A. Riedel. Electric Circuits. 10th ed., Pearson, 2014 (Spanish edition, “Circuitos eléctricos”, 2015). The six sets of parameters, their conversions and the interconnection of two-ports.
Mac E. Van Valkenburg. Network Analysis. 3rd ed., Prentice Hall, 1974 (Spanish edition, “Análisis de redes”, Limusa). Two-ports within the general theory of networks.
David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. ABCD and scattering parameters in radio-frequency networks.

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