RC Phase shift Amplifier

RC- Phaseshift Amplifier

Output of  \(1^{st}\) stage is coupled to the \(2^{nd}\) stage via RC network . So the the name coulpling capacitors
 \(C_{c1},C_{c2},C_{c3}\)  blocks dc components.
 \(R_{c1},R_{c2}\)→ collector resistance
\(R_{11},R_{12},R_{21},R_{22}\) →are biasing resistors
\(R_{e1},,R_{e2}\)→emitter resistors
\(C_{e1},C_{e2}\)→bypass capacitors prevents loss of amplification
These are 3 different frequency ranges
1) low frequency range
2) highr frequency range
3)  mid frequency range

Equivalent circuit (Single stage)

Middle frequency range

Current gain
\(A_{IM}=\frac{I_o}{I_b}\)
\(I_{o}=-g_mV_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}}\)
\(I_{o}=-g_mI_br_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}}\)
\(\frac{I_0}{I_b}=-g_mr_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}}\)
\(g_mr_{b^,e}=I_c\frac{r_{b^,e}}{V_{b^,e}}=\frac{I_c}{I_b}=h_{fe}\)
\(\frac{I_0}{I_b}=A_{IM}=-h_{fe}\frac{R_{co}}{R_{co}+R_{bc}}\)
$$ \bbox[5px,border:2px solid red]{ A_{IM}=-h_{fe}\frac{R_{co}}{R_{co}+R_{bc}} }$$ 

Voltage Gain
\(A_{VM}=\frac{V_o}{V_i}\)
\(V_{o}=I_oR_{bc}=-g_mV_{b^,e}\frac{R_{co}R_{bc}}{R_{co}+R_{bc}}=-g_mI_br_{b^,e}R_{cobc}\)
\(V_i=I_b(r_{b^,b}+{r_{b^,e}})=I_bh_{ie}\)
\(\frac{V_{o}}{V_i}=\frac{-g_mI_br_{b^,e}R_{cobc}}{I_bh_{ie}}=\frac{-g_mr_{b^,e}R_{cobc}}{h_{ie}}=\frac{-h_{fe}R_{cobc}}{h_{ie}}\) 
$$ \bbox[5px,border:2px solid red]{ A_{VM}=\frac{-h_{fe}R_{cobc}}{h_{ie}} }$$
Low frequence range

Current Gain
  
\(A_{IL}=\frac{I_o}{I_b}\)
\(I_{o}=-g_mV_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}\)  
\(I_{o}=-g_mI_br_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}\)
\(\frac{I_0}{I_b}=-g_mr_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}\)
\(g_mr_{b^,e}=I_c\frac{r_{b^,e}}{V_{b^,e}}=\frac{I_c}{I_b}=h_{fe}\)
\(\frac{I_0}{I_b}=A_{IL}=-h_{fe}\frac{R_{co}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}\) 
$$ \bbox[5px,border:2px solid red]{ A_{IL}=-h_{fe}\frac{R_{co}}{R_{co}+R_{bc}+\frac{1}{jωc_b}} }$$ 

Voltage Gain

 \(A_{VL}=\frac{V_o}{V_i}\)   
\(V_{o}=I_oR_{bc}=-g_mV_{b^,e}\frac{R_{co}R_{bc}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}=-g_mI_br_{b^,e}\frac{R_{co}R_{bc}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}\)
\(V_i=I_b(r_{b^,b}+{r_{b^,e}})=I_bh_{ie}\) 
\(\frac{V_{o}}{V_i}=\frac{-g_mI_br_{b^,e}\frac{R_{co}R_{bc}}{R_{co}+R_{bc}+\frac{1}{jωc_b}}}{I_bh_{ie}}=\frac{-g_mr_{b^,e}R_{co}R_{bc}}{h_{ie}(R_{co}+R_{bc}+\frac{1}{jωc_b})}=\frac{-h_{fe}R_{co}R_{bc}}{h_{ie}(R_{co}+R_{bc}+\frac{1}{jωc_b})}\) 
$$ \bbox[5px,border:2px solid red]{ A_{VL}=\frac{-h_{fe}R_{co}R_{bc}}{h_{ie}(R_{co}+R_{bc}+\frac{1}{jωc_b})} }$$ 

High Frequency Range

Current Gain

\(A_{IH}=\frac{I_o}{I_b}\)
\(I_{o}=-g_mV_{b^,e}\frac{R_{co}}{R_{co}+R_{bc}}\)
\(V_{b^,e}=\frac{I_br_{b^,e}\frac{1}{jωc}}{r_{b^,e}+\frac{1}{jωc}}=\frac{I_br_{b^,e}}{r_{b^,e}jωc+1}\) 

\(I_{o}=-g_m\frac{I_br_{b^,e}}{r_{b^,e}jωc+1}\frac{R_{co}}{R_{co}+R_{bc}}\)

\(\frac{I_o}{I_b}=\frac{-g_mr_b^,e}{r_{b^,e}jωc+1}\frac{R_{co}}{R_{co}+R_{bc}}=\frac{-h_{fe}R_{co}}{(r_{b^,e}jωc+1)(R_{co}+R_{bc})}\) 

$$ \bbox[5px,border:2px solid red]{ A_{IH}=\frac{-h_{fe}R_{co}}{(r_{b^,e}jωc+1)(R_{co}+R_{bc})} }$$ 

Volatage Gain

 \(A_{VH}=\frac{V_o}{V_i}\)
\(V_{o}=I_oR_{bc}=\frac{-I_bh_{fe}R_{co}R_{bc}}{(r_{b^,e}jωc+1)(R_{co}+R_{bc})}\)
\(V_i=(r_{b^,b}+{\frac{r_{b^,e}}{r_{b^,e}jωc+1}})I_b=\frac{I_b(r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e})}{r_{b^,e}jωc+1}\) 
 \(\frac{V_o}{V_i}=\frac{\frac{-I_bh_{fe}R_{co}R_{bc}}{(r_{b^,e}jωc+1)(R_{co}+R_{bc})}}{\frac{I_b(r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e})}{r_{b^,e}jωc+1}}=
\frac{\frac{-I_bh_{fe}R_{cobc}}{(r_{b^,e}jωc+1)}}{\frac{I_b(r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e})}{r_{b^,e}jωc+1}}=\frac{-h_{fe}R_{cobc}}{r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e}}\)
\(\frac{V_o}{V_i}=\frac{-h_{fe}R_{cobc}}{r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e}}=\) 
$$ \bbox[5px,border:2px solid red]{ A_{VH}=\frac{-h_{fe}R_{cobc}}{r_{b^,b}+r_{b^,b}r_{b^,e}jωc+r_{b^,e}} }$$