For analyzing circuits containing active devices such as transistors, it is more convenient to think of the input terminals of a four-terminal coupling network as a Thévenin-equivalent voltage source and the output terminals as a Norton-equivalent current source. We then describe the coupling network in terms of four hybrid parameters (h-parameters). We determine these parameters using the same measurement techniques as for z-parameters and y-parameters.
To find the open-circuit voltage of the Thévenin-equivalent source at input terminals (port 1) in Figure 1(a), we feed V2 into the output terminals (port 2). In this circuit, we consider the Thévenin-equivalent source to be a voltage-controlled voltage source. The parameter that represents the fraction of the output voltage appearing at the input terminals is V1/V2, which is a ratio without units. This parameter is the open-circuit reverse- voltage ratio, h12.
Since we are treating the dependent source as a voltage-controlled voltage source, we short-circuit the output terminals while we measure the input voltage and current, as shown in Figure 1(b). The parameter h11 is V1/I1, which is expressed in ohms and represents the short-circuit input impedance of the network. Since h12V2 is a voltage source, the equivalent input circuit for the coupling network shows the dependent voltage source and input impedance in series, as in Figure 1(c).
Figure 1 Finding the Thévenin-equivalent input circuit of a four-terminal network: (a) Open-circuit reverse voltage; (b) Internal input impedance; (c) Network input parameters
To determine the short-circuit current of the Norton-equivalent source at the output terminals (port 2) in Figure 2(a), we feed I1 into the input terminals and short-circuit the output terminals through the ammeter measuring I2. As long as the network impedances are linear (independent of voltage and current), I2 will be a constant fraction of the input current I1. The ratio I2/I1 is the short-circuit forward-current ratio, h21.
Figure 2 Finding the Norton-equivalent output circuit of a four-terminal network: (a) short-circuit forward current; (b) output admittance; (c) complete hybrid parameters.
The output impedance of a Norton-equivalent source is in parallel with the current source, so the fourth hybrid parameter is expressed as an admittance. Since we are treating this dependent source as a current-controlled current source, we leave the input terminals of the network open-circuit to make I1 zero while we measure I2 and V2. The parameter h22 is I2 /V2, which is expressed in Siemens and represents the open-circuit output admittance. These equations summarize the four hybrid parameters of a four-terminal coupling network:
Short-circuit input impedance:
\[\begin{matrix}{{\text{h}}_{\text{11}}}\text{=}\frac{{{\text{V}}_{\text{1}}}}{{{\text{I}}_{\text{1}}}}\left( \text{with }{{\text{V}}_{\text{2}}}\text{=0} \right) & {} & \left( 1 \right) \\\end{matrix}\]
Open-circuit reverse-voltage ratio:
\[\begin{matrix}{{\text{h}}_{\text{12}}}\text{=}\frac{{{\text{V}}_{\text{1}}}}{{{\text{V}}_{\text{2}}}}\left( \text{with }{{\text{I}}_{\text{1}}}\text{=0} \right)Open-Circuit & {} & \left( 2 \right) \\\end{matrix}\]
Short-circuit forward-current ratio:
\[\begin{matrix}{{\text{h}}_{\text{21}}}\text{=}\frac{{{\text{I}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}}\left( \text{with }{{\text{V}}_{\text{2}}}\text{=0} \right)Short-Circuit & {} & \left( 3 \right) \\\end{matrix}\]
Open-circuit output admittance:
\[\begin{matrix}{{\text{h}}_{\text{22}}}\text{=}\frac{{{\text{I}}_{\text{2}}}}{{{\text{V}}_{\text{2}}}}\left( \text{with }{{\text{I}}_{\text{1}}}\text{=0} \right) & {} & \left( 4 \right) \\\end{matrix}\]
Figure 2(c) shows the resulting h-parameter equivalent circuit. For the Thévenin-equivalent source for the network input, we can write a Kirchhoff’s voltage-law equation, as we did for z-parameters. For the Norton-equivalent source for the network output, we write a Kirchhoff’s current-law equation, as we did for y-parameters. The two unknowns in these equations are I1 and V2.
$\begin{align}& \begin{matrix}{{\text{h}}_{\text{11}}}{{\text{I}}_{\text{1}}}\text{+}{{\text{h}}_{\text{12}}}{{\text{V}}_{\text{2}}}\text{=}{{\text{E}}_{\text{1}}} & {} & \left( 5 \right) \\\end{matrix} \\& \begin{matrix}{{\text{h}}_{\text{21}}}{{\text{I}}_{\text{1}}}\text{+}{{\text{h}}_{\text{22}}}{{\text{V}}_{\text{2}}}\text{=}{{\text{I}}_{\text{2}}} & {} & \left( 6 \right) \\\end{matrix} \\\end{align}$
The transistor amplifier equivalent circuit of Figure 3 is a typical example of hybrid parameters.
Figure 3 Hybrid parameters of a simple transistor amplifier
We cannot use Thevenin’s theorem to find the equivalent internal resistance of a dependent source when the controlling element is included in the transformation. Therefore, when we want to determine the input and output impedances of coupling networks, we must calculate V/I. We determined h11 with a short circuit across the output terminals of the network. In the circuit of Figure 3, Zin differs slightly from h11 since the circuit has some reverse coupling.
