Hybrid Parameter or h parameter

Transistors is a 2 part network.Input and Output of a transistor are related through 2 equation
\[V_1=h_{11}I_1+h_{12}V_2\]

\[I_2=h_{21}I_1+h_{22}V_2\]


\( \left . \frac{V_1}{I_1}  \right |_{V_2=0} \) derived point impedance when output is short circuited $$h_{i},h_{ie},h_{ib}h_{ic}$$
\( \left . \frac{V_1}{V_2}  \right |_{I_1=0} \) Reverse transfer voltage ratio when input is open circuit
 $$h_{r},h_{re},h_{rb}h_{rc}$$
\( \left . \frac{I_2}{I_1}  \right |_{V_2=0} \) forward current ratio when output is short circuited $$h_{f},h_{fe},h_{fb}h_{fc}$$
\( \left . \frac{I_2}{V_2}  \right |_{I_1=0} \) driving point output admittance when input is open circuit $$h_{o},h_{oe},h_{ob}h_{oc}$$

Equivalent circuit

For CE

\[V_b=h_{ie}I_b+h_{re}V_c\]

\[I_c=h_{fe}I_b+h_{oe}V_c\]

For CC

\[V_b=h_{ic}I_b+h_{rc}V_e\]
\[I_e=h_{fc}I_b+h_{oc}V_e\]

For CB

\[V_e=h_{ib}I_e+h_{rb}V_c\]
\[I_c=h_{fb}I_e+h_{ob}V_c\]


Analysis (using without considering source)

Current Gain
\(A_i=\frac {I_L}{I_I}=\frac {-I_2}{I_1}\)

\(I_2=h_fI_1+h_oV_2\)
\( = h_fI_1+h_oI_LR_L\)
\(I_2 = h_fI_1-h_oI_LR_L\)
\(I_2(1+h+oR_L) = h_fI_1\)
\( {\frac{I_2}{I_1} }= {\frac{h_f}{1+h_oR_L}}\)
\( A_I={\frac{-I_2}{I_1} }= {\frac{-h_f}{1+h_oR_L}}\)
cc
\( A_I= {\frac{-h_{fc}}{1+h_{oc}R_L}}\)
ce
\( A_I= {\frac{-h_{fe}}{1+h_{oe}R_L}}\)
cb
\( A_I= {\frac{-h_{fb}}{1+h_{ob}R_L}}\)
\( 2)Input Resistance(R_i) \)
\( R_i= \frac{v_1}{I_1} \)
\( V_1=h_iI_1+h_rV_2\)
\( V_1=h_iI_1-h_rI_2R_L\)
\( \frac{V_1}{I_1}=h_i-\frac{h_rI_2R_L}{I_1}\)
$$ \bbox[5px,border:2px solid red]
{
 R_i=h_i+h_rA_IR_L
}$$

ce
\( R_i=h_{ie}+h_{re}A_IR_L\)
cc
\( R_i=h_{ic}+h_{rc}A_IR_L\)
cb
\( R_i=h_{ib}+h_{rb}A_IR_L\)

3) \( Voltage Gain (A_v)\)
\( A_v=\frac{V_2}{V_1}\)
\(V_2=-I_2R_L\)
\( A_I=\frac{-I_2}{I_1}\)
\( -I_2={A_II_1}\)
=>\( V_2=A_II_1R_L\)
\( A_v=\frac{V_2}{V_1}=\frac{A_II_1R_L}{V_1}=\frac{A_IR_L}{R_i}\)

$$ \bbox[5px,border:2px solid red]
{
 A_v=\frac{A_II_1R_L}{V_1}=\frac{A_IR_L}{R_i}
}$$

4 )  \( Output Resistance(R_o)\)
\( R_o=\frac{V_2}{I_2}\)
\(I_2= h_fI_1+h_oV_2\)
\( \frac{I_2}{V_2} = \frac{h_fI_1}{V_2}+h_o\)
\( Y_o= \frac{h_fI_1}{V_2}+h_o\)

when \( V_s=0\)

\( (R_s+h_i)I_1+h_rV_2=0\)
\( \frac{I_1}{V_2}=\frac{-h_r}{h_i+R_s}\)
\( Y_o= \frac{-h_fh_r}{h_i+R_s}+h_o\)
\( Y_o= h_o-\frac{h_fh_r}{h_i+R_s}\)   
\( R_o=\frac{1}{Y_o}\) 
Power Gain
\( A_p=A_vA_I\)   

Analysis (using with considering source)

Voltage gain

\( A_{vs}=\frac{V_2}{V_s}=\frac{V_2}{V_1}\frac{V_1}{V_s}\)
\( A_{vs}=A_v\frac{V_1}{V_s}\)

\( V_1=\frac{V_sR_i}{R_i+R_s}\)
=>\( \frac{V_1}{V_s}=\frac{R_i}{R_i+R_s}\)

$$ \bbox[5px,border:2px solid red]{  A_{vs}=\frac{A_vR_i}{R_i+R_s}}$$

Current Gain


\(A_{is}=\frac {I_L}{I_s}=\frac {-I_2}{I_s}=\frac {-I_2}{I_1}\frac {I_1}{I_s}\)
\(A_{is}=A_I\frac {I_1}{I_s}\)

\( I_1=I_s\frac {R_s}{R_s+R_L}\)
\(\frac {I_1}{I_s} = \frac {R_s}{R_s+R_L}\)
$$ \bbox[5px,border:2px solid red]{ A_{is} = A_I \frac {R_s}{R_s+R_L} }$$