Low frequency response of single tube common emitter amplifier circuit
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Low frequency response of single tube common emitter amplifier circuit If there is no coupling capacitor and bypass capacitor in the single-tube common-emitter amplifier circuit, the amplitude-frequency characteristics of the frequency band can be horizontally extended to zero frequency, or f L = 0 , so there is no need to analyze the low-frequency response characteristics. Considering that the resistor-capacitor coupling method is often used in discrete device circuits or between the integrated circuit pins and external circuits, it is necessary to briefly discuss the low-frequency response characteristics of the resistor-capacitor coupling amplifier circuit. 1. Low frequency gain function The low - frequency response of the RC coupled common-emitter amplifier circuit is caused by the coupling capacitors C1 , C2 and the bypass capacitor CE , as shown in Figure 3-27 ( a ) . In the low frequency range, due to the voltage drop generated by C1 and C2, the actual signal voltage input to the transistor base and output to the load RL decreases as the frequency decreases. At the same time, since the capacitive reactance of CE increases as the frequency decreases , the parallel impedance Ze of capacitor CE and RE also increases , causing AC negative feedback, resulting in a decrease in low-frequency gain. In summary , the capacitors C1 , C2 and CE should be retained in the low-frequency equivalent circuit of the amplifier circuit , and 
The capacitance effect of the transistor itself is very small in the high-frequency range and can usually be ignored, so the simplified h model can be used instead. Taking into account the bias resistor RB ( = RB1 // RB2 ), its low - frequency equivalent circuit can be drawn as shown in Figure 3-27 ( b ) . According to the low-frequency equivalent circuit diagram 3-27 ( b ), the low-frequency current gain function AIS (s) (= Io/ Is ) of the circuit can be derived . For convenience, Rs, RB, RE and RC in Figure 3-27 ( b ) are represented by conductivity symbols Gs , GB , GE and GC , and then the complex frequency domain circuit equations of each node voltage U1 , U2 , U3 and U4 are listed .  ( 3 — 63 )
In addition, from the equivalent circuit of Figure 3-27 ( b ) , it can be seen that the output and input currents can be expressed as 
From the above equation, the low-frequency current gain function can be derived as  ( 3 — 64 )
In formula ( 3-64 ), the coefficients a m , a 2 and a 1 can be approximately expressed as follows under the condition that R B >> h ie , R s  ( 3 — 65 )
From the above formula we can see: ( 1 ) The low-frequency current gain function AIS (s) has three zeros and three poles. Two of the poles (p1 , p2 ) can be obtained from the roots of the second-order factor (s2 + a2s + a1 ) , which can be seen from the relationship between coefficients a2 and a1 in equation (3-65). They all contain C1 , CE and R B, RE , which means that the values of these two poles are related to both the circuit parameters and the emitter loop parameters, because h ie makes the two circuits interrelated. ( 2 ) The third pole p3 = -1 /( Rc + RL ) C2 is independent. Its value is determined only by the parameters of the output coupling circuit and has nothing to do with the input coupling circuit and the emitter loop . ( 3 ) Since the number of zeros and poles is equal, when s approaches infinity, the contributions of the zeros and poles cancel each other out, and the amplitude of the low-frequency current gain function AIS ( S) approaches the constant a m . It can be seen that the coefficient a m is the intermediate-frequency current gain. ( 4 ) From A IS (s) we can derive A US (s) 
Since the resistors Rs and Rl are pure resistances and have nothing to do with frequency, the amplitude-frequency characteristic of the low-frequency voltage gain function AUS(s) is the same as that of the low-frequency current gain function AIS ( s ) . 【Example 3-5 】 In the circuit of Figure 3-27 ( a ) , suppose RC = 2kW , RS = RE = RL = 1kW , RB = 10kW , C1 = 5mF , C2 = 10mF , CE = 100mF , hfe = 44, hie = 1.4kW of the transistor , try to find the expression of the low - frequency current gain function AISI ( s ) . Solution : Substituting the given values into equations (3-64) and (3-65) respectively , we can obtain the values of the coefficients and zeros . a m =12.5 a 1 =4500 a 2 =267 Z1 = Z2 = 0 Z3 = -10rad / s Substitute the coefficients a1 and a2 into the second- order factor ( s 2 + a 2 s + a 1 ) =( s 2 +267 s +4500)=( s +248.5) ( s +18.5) So the value of the extreme point is p 1 = 18.5 rad/s p2 = -248.5 rad /s p 3 = - 1/[(R c +R L )C 2 ]= - 33 rad/s The low-frequency current gain function of the circuit can be written from the above zero-pole values and coefficient values:  ( 3 — 66 )
Back to Top 2. Calculation of low frequency cutoff The low-frequency cutoff frequency can be calculated from the transfer function by either graphical or analytical methods. 1 ) Graphical method The relevant graphical method has been introduced in the previous section. Now we will solve the low- frequency current gain obtained from [ Example 3-5 ] 
The result of function AIS ( jw ) is plotted as an amplitude-frequency characteristic curve, as shown in Figure 3-28 . From the low-frequency asymptotic amplitude-frequency characteristics of Figure 5-28, it can be seen that point A is approximately the -3dB low-frequency cutoff frequency of the RC coupled common-emitter amplifier , that is, the low-frequency cutoff frequency wL and fL are w L =|p 2 |=248.5rad/s f L = w L /2 p = 39.6 Hz 2 ) Analytical method The analytical method is to derive the low-frequency cutoff from the low-frequency gain function. Next, we will first derive the calculation formula. The low-frequency gain function with n low-frequency poles can be written in the following general form  
When the operating frequency w = w L ,  , that is, the denominator of the above formula can be expressed as the modulus value  ( 3 — 68 )
Since the low-frequency cutoff frequency w L of the multi-pole amplifier is greater than any low-frequency pole value, that is,  ,  、× ××、 
Therefore, after the expansion of equation ( 3-68 ), only the quadratic term of w L can be retained and the other higher-order terms can be ignored, simplifying it into the following equation :  ( 3-69 )
From the above formula, we can get  ( 3 — 70 )
The above formula is an approximate calculation formula for finding the low-frequency cutoff frequency from multiple low-frequency poles. It can be seen that the low-frequency cutoff frequency w L is greater than any low-frequency pole. If the pole data obtained in [ Example 3-5 ] is used, the low-frequency cutoff f L of the circuit can be obtained using formula ( 3-70 ) : 
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