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

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