Hybrid-pi model

<h2 id="bjt-parameters">BJT parameters</h2>
The hybrid-pi model is a linearized <a href="/facts/Two-port_network/cqUrBTYq">two-port network</a> approximation to the BJT using the small-signal base-emitter voltage, 
 
 
 
 
 
 v
 
 be
 
 
 
 
 
 {\displaystyle \textstyle v_{\text{be}}}
 
, and collector-emitter voltage, 
 
 
 
 
 
 v
 
 ce
 
 
 
 
 
 {\displaystyle \textstyle v_{\text{ce}}}
 
, as independent variables, and the small-signal base current, 
 
 
 
 
 
 i
 
 b
 
 
 
 
 
 {\displaystyle \textstyle i_{\text{b}}}
 
, and collector current, 
 
 
 
 
 
 i
 
 c
 
 
 
 
 
 {\displaystyle \textstyle i_{\text{c}}}
 
, as dependent variables.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

A basic, low-frequency hybrid-pi model for the <a href="/facts/Bipolar_transistor/9BMcGv5Q">bipolar transistor</a> is shown in figure 1. The various parameters are as follows.

g
          
            m
          
        
        =
        
          
            
            
              
                
                  i
                  
                    c
                  
                
                
                  v
                  
                    be
                  
                
              
            
            |
          
          
            
              v
              
                ce
              
            
            =
            0
          
        
        =
        
          
            
              I
              
                C
              
            
            
              V
              
                T
              
            
          
        
      
    
    {\displaystyle g_{\text{m}}=\left.{\frac {i_{\text{c}}}{v_{\text{be}}}}\right\vert _{v_{\text{ce}}=0}={\frac {I_{\text{C}}}{V_{\text{T}}}}}

is the <a href="/facts/Transconductance/lwh8krQa">transconductance</a>, evaluated in a simple model,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> where:

<ul><li>
 
 
 
 
 
 I
 
 C
 
 
 
 
 
 
 {\displaystyle \textstyle I_{\text{C}}\,}
 
 is the <a href="/facts/Quiescent_current/lK0e6hbK">quiescent</a> collector current (also called the collector bias or DC collector current)</li>
<li>
 
 
 
 
 
 V
 
 T
 
 
 =
 
 
 
 k
 T
 
 e
 
 
 
 
 
 {\displaystyle \textstyle V_{\text{T}}={\frac {kT}{e}}}
 
 is the <a href="/facts/Boltzmann_constant/5DIejyR3">thermal voltage</a>, calculated from the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>, 
 
 
 
 
 k
 
 
 
 {\displaystyle \textstyle k}
 
, the <a href="/facts/Elementary_charge/lPcj6CoQ">charge of an electron</a>, 
 
 
 
 
 e
 
 
 
 {\displaystyle \textstyle e}
 
, and the transistor temperature in <a href="/facts/Kelvin/E3trzC4C">kelvins</a>, 
 
 
 
 
 T
 
 
 
 {\displaystyle \textstyle T}
 
. At approximately <a href="/facts/Room_temperature/YHwUIUTE">room temperature</a> (295 K, 22 °C or 71 °F), 
 
 
 
 
 
 V
 
 T
 
 
 
 
 
 {\displaystyle \textstyle V_{\text{T}}}
 
 is about 25 mV.</li>
<li>
 
 
 
 
 r
 
 π
 
 
 =
 
 
 
 
 
 
 v
 
 be
 
 
 
 i
 
 b
 
 
 
 
 |
 
 
 
 v
 
 ce
 
 
 =
 0
 
 
 =
 
 
 
 V
 
 T
 
 
 
 I
 
 B
 
 
 
 
 =
 
 
 
 β
 
 0
 
 
 
 g
 
 m
 
 
 
 
 
 
 {\displaystyle r_{\pi }=\left.{\frac {v_{\text{be}}}{i_{\text{b}}}}\right\vert _{v_{\text{ce}}=0}={\frac {V_{\text{T}}}{I_{\text{B}}}}={\frac {\beta _{0}}{g_{\text{m}}}}}
 
</li></ul>
where:

<ul><li>
 
 
 
 
 
 I
 
 B
 
 
 
 
 
 {\displaystyle \textstyle I_{\text{B}}}
 
 is the DC (bias) base current.</li>
<li>
 
 
 
 
 
 β
 
 0
 
 
 =
 
 
 
 I
 
 C
 
 
 
 I
 
 B
 
 
 
 
 
 
 
 {\displaystyle \textstyle \beta _{0}={\frac {I_{\text{C}}}{I_{\text{B}}}}}
 
 is the current gain at low frequencies (generally quoted as hfe from the <a href="/facts/Bipolar_junction_transistor/9BMcGv5Q">h-parameter model</a>). This is a parameter specific to each transistor, and can be found on a datasheet.</li>
<li>
 
 
 
 
 
 r
 
 o
 
 
 =
 
 
 
 
 
 
 v
 
 ce
 
 
 
 i
 
 c
 
 
 
 
 |
 
 
 
 v
 
 be
 
 
 =
 0
 
 
  
 =
  
 
 
 1
 
 I
 
 C
 
 
 
 
 
 (
 
 
 V
 
 A
 
 
 +
 
 V
 
 CE
 
 
 
 )
 
  
 ≈
  
 
 
 
 V
 
 A
 
 
 
 I
 
 C
 
 
 
 
 
 
 
 {\displaystyle \textstyle r_{\text{o}}=\left.{\frac {v_{\text{ce}}}{i_{\text{c}}}}\right\vert _{v_{\text{be}}=0}~=~{\frac {1}{I_{\text{C}}}}\left(V_{\text{A}}+V_{\text{CE}}\right)~\approx ~{\frac {V_{\text{A}}}{I_{\text{C}}}}}
 
 is the output resistance due to the <a href="/facts/Early_effect/EdvBKu6W">Early effect</a> (
 
 
 
 
 
 V
 
 A
 
 
 
 
 
 {\displaystyle \textstyle V_{\text{A}}}
 
 is the Early voltage).</li></ul>
<h3>Related terms</h3>
The output <a href="/facts/Electrical_conductance/hxSt0UAj">conductance</a>, gce, is the reciprocal of the output resistance, ro:

g
 
 ce
 
 
 =
 
 
 1
 
 r
 
 o
 
 
 
 
 
 
 {\displaystyle g_{\text{ce}}={\frac {1}{r_{\text{o}}}}}
 
.
The <a href="/facts/Transresistance/lwh8krQa">transresistance</a>, rm, is the reciprocal of the transconductance:

r
 
 m
 
 
 =
 
 
 1
 
 g
 
 m
 
 
 
 
 
 
 {\displaystyle r_{\text{m}}={\frac {1}{g_{\text{m}}}}}
 
.
<h3>Full model</h3>

The full model introduces the virtual terminal, B′, so that the base spreading resistance, rbb, (the bulk resistance between the base contact and the active region of the base under the emitter) and rb′e (representing the base current required to make up for recombination of minority carriers in the base region) can be represented separately. Ce is the diffusion capacitance representing minority carrier storage in the base. The feedback components, rb′c and Cc, are introduced to represent the <a href="/facts/Early_effect/EdvBKu6W">Early effect</a> and Miller effect, respectively.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="mosfet-parameters">MOSFET parameters</h2>

A basic, low-frequency hybrid-pi model for the <a href="/facts/MOSFET/eTDJT2Km">MOSFET</a> is shown in figure 2. The various parameters are as follows.

g
          
            m
          
        
        =
        
          
            
            
              
                
                  i
                  
                    d
                  
                
                
                  v
                  
                    gs
                  
                
              
            
            |
          
          
            
              v
              
                ds
              
            
            =
            0
          
        
      
    
    {\displaystyle g_{\text{m}}=\left.{\frac {i_{\text{d}}}{v_{\text{gs}}}}\right\vert _{v_{\text{ds}}=0}}

is the <a href="/facts/Transconductance/lwh8krQa">transconductance</a>, evaluated in the Shichman–Hodges model in terms of the <a href="/facts/Q-point/lK0e6hbK">Q-point</a> drain current, 
 
 
 
 
 
 I
 
 D
 
 
 
 
 
 {\displaystyle \scriptstyle I_{\text{D}}}
 
:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

g
 
 m
 
 
 =
 
 
 
 2
 
 I
 
 D
 
 
 
 
 
 V
 
 GS
 
 
 −
 
 V
 
 th
 
 
 
 
 
 
 
 {\displaystyle g_{\text{m}}={\frac {2I_{\text{D}}}{V_{\text{GS}}-V_{\text{th}}}}}
 
,
where:

<ul><li>
 
 
 
 
 
 I
 
 D
 
 
 
 
 
 {\displaystyle \scriptstyle I_{\text{D}}}
 
 is the <a href="/facts/Quiescent_current/lK0e6hbK">quiescent</a> drain current (also called the drain bias or DC drain current)</li>
<li>
 
 
 
 
 
 V
 
 th
 
 
 
 
 
 {\displaystyle \scriptstyle V_{\text{th}}}
 
 is the <a href="/facts/Threshold_voltage/IparzT9A">threshold voltage</a> and</li>
<li>
 
 
 
 
 
 V
 
 GS
 
 
 
 
 
 {\displaystyle \scriptstyle V_{\text{GS}}}
 
 is the gate-to-source voltage.</li></ul>
The combination:

V
          
            ov
          
        
        =
        
          V
          
            GS
          
        
        −
        
          V
          
            th
          
        
      
    
    {\displaystyle V_{\text{ov}}=V_{\text{GS}}-V_{\text{th}}}

is often called overdrive voltage.

r
          
            o
          
        
        =
        
          
            
            
              
                
                  v
                  
                    ds
                  
                
                
                  i
                  
                    d
                  
                
              
            
            |
          
          
            
              v
              
                gs
              
            
            =
            0
          
        
      
    
    {\displaystyle r_{\text{o}}=\left.{\frac {v_{\text{ds}}}{i_{\text{d}}}}\right\vert _{v_{\text{gs}}=0}}

is the output resistance due to <a href="/facts/Channel_length_modulation/X0LJ5039">channel length modulation</a>, calculated using the Shichman–Hodges model as

r
                  
                    o
                  
                
              
              
                
                =
                
                  
                    1
                    
                      I
                      
                        D
                      
                    
                  
                
                
                  (
                  
                    
                      
                        1
                        λ
                      
                    
                    +
                    
                      V
                      
                        DS
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  
                    1
                    
                      I
                      
                        D
                      
                    
                  
                
                
                  (
                  
                    
                      V
                      
                        E
                      
                    
                    L
                    +
                    
                      V
                      
                        DS
                      
                    
                  
                  )
                
                ≈
                
                  
                    
                      
                        V
                        
                          E
                        
                      
                      L
                    
                    
                      I
                      
                        D
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}r_{\text{o}}&={\frac {1}{I_{\text{D}}}}\left({\frac {1}{\lambda }}+V_{\text{DS}}\right)\\&={\frac {1}{I_{\text{D}}}}\left(V_{E}L+V_{\text{DS}}\right)\approx {\frac {V_{E}L}{I_{\text{D}}}}\end{aligned}}}

using the approximation for the channel length modulation parameter, λ:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

λ
 =
 
 
 1
 
 
 V
 
 E
 
 
 L
 
 
 
 
 
 {\displaystyle \lambda ={\frac {1}{V_{\text{E}}L}}}
 
.
Here VE is a technology-related parameter (about 4 V/μm for the <a href="/facts/65_nm_process/yqoIVRgR">65 nm technology node</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>) and L is the length of the source-to-drain separation.
The drain conductance is the reciprocal of the output resistance:

g
 
 ds
 
 
 =
 
 
 1
 
 r
 
 o
 
 
 
 
 
 
 {\displaystyle g_{\text{ds}}={\frac {1}{r_{\text{o}}}}}
 
.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bipolar_junction_transistor/9BMcGv5Q">h-parameter model</a></li>
<li><a href="/facts/Small-signal_model/P3VYrx4K">Small-signal model</a></li></ul>
<h2 id="references-and-notes">References and notes</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Giacoletto, L.J. "Diode and transistor equivalent circuits for transient operation" IEEE Journal of Solid-State Circuits, Vol 4, Issue 2, 1969 [1] <a href="https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1049963&contentType=Journals+%26+Magazines&sortType%3Dasc_p_Sequence%26filter%3DAND%28p_IS_Number%3A22508%29" target="_blank">https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1049963&contentType=Journals+%26+Magazines&sortType%3Dasc_p_Sequence%26filter%3DAND%28p_IS_Number%3A22508%29</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (Second ed.). New York: McGraw-Hill. pp. Section 13.5, esp. Eqs. 13.19. ISBN 978-0-07-232099-2. <a href="978-0-07-232099-2" target="_blank">978-0-07-232099-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">R.C. Jaeger and T.N. Blalock (2004). Eq. 5.45 pp. 242 and Eq. 13.25 p. 682. McGraw-Hill. ISBN 978-0-07-232099-2. <a href="978-0-07-232099-2" target="_blank">978-0-07-232099-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dhaarma Raj Cheruku, Battula Tirumala Krishna, Electronic Devices And Circuits, pages 281-282, Pearson Education India, 2008 ISBN 8131700984. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">R.C. Jaeger and T.N. Blalock (2004). Eq. 4.20 pp. 155 and Eq. 13.74 p. 702. McGraw-Hill. ISBN 978-0-07-232099-2. <a href="978-0-07-232099-2" target="_blank">978-0-07-232099-2</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">W. M. C. Sansen (2006). Analog Design Essentials. Dordrechtμ: Springer. p. §0124, p. 13. ISBN 978-0-387-25746-4. <a href="978-0-387-25746-4" target="_blank">978-0-387-25746-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">W. M. C. Sansen (2006). Analog Design Essentials. Dordrechtμ: Springer. p. §0124, p. 13. ISBN 978-0-387-25746-4. <a href="978-0-387-25746-4" target="_blank">978-0-387-25746-4</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Hybrid-pi model open-in-new

Hybrid-pi model