small signal common emitter amplifier | Small Signal CE Amplifier model analysis

Small Signal CE Amplifier , small signal common emitter amplifier , Small Signal CE Amplifier model analysis equivalent circuit of pdf .
CE amplifiers are very popular to amplify the small signal AC. After a transistor has been biased with a Q point near the middle of a DC load line, AC source can be coupled to the base. this produces fluctuations in the base current and hence in the collector current of the same shape and frequency. the output will be enlarged sine wave of same frequency.
The amplifier is called linear if it does not change the wave shape of the signal. As long as the input signal is small, the tranistor will use only a small part of the load line and the operation will be linear.
On the other hand, if the input signal is too large. the fluctuations along the load line will drive the transistor into either saturation or cut-off. this clips the peaks of the input and the amplifier is no longer linear.
The CE amplifier configuration is shown in fig. 24.
The coupling capacitor (C_C) passes an AC signal from one point to another. At the same time it does not allow the DC to pass through it. hence, it is also called blocking capaitor.
For example, in fig. 25 the AC voltage at point A is transmitted to point B. for this series reactance X_C should be very small compared in series resistance R_S. the circuit to the left of A may be a source and a series resistor or may be the thevenin equivalent of a complex circuit. similarly, R_L may be the load resistance or equivalent resistance of a complex network. the current in the loop is given by
I = V_in / (R_S + R_L)² + (X_C)²
= V_in / R² + X²
As a frequency increases X_C = (1/2 fc) decreases and current increases until it reaches to its maximum value V_in/R. therefore, the capacitor couples the signal properly from A to B then, X_C << R. the size of the coupling capacitor depends upon the lowest frequency to be coupled. normally, for lowest frequency X_C 0.1R is taken as design rule.
The coupling capaciitor acts like a switch, which is open to DC and shorted for AC.
The bypass capacitor C_B is similar to coupling capacitor except that it couples an ungrounded point to a grounded point. then C_B capacitor looks like a short to an AC signal and therefore, emitterb is said AC grounded. A bypass capacitor does not disturb the DC voltage at emitter because it looks open to DC current. As a design the rule X_Cb 0.1 R_E at lowest frequency.
Analysis of CE Ampliffier
In a transistor ampliffier, the DC source sets up quiescent current and voltages. the AC source then, produces fluctuations in these current and voltages. the simplest way to analyze this circuit is to split the analysis in two parts DC analysis and AC analysis. one can use superposition theorem for analysis.
AC and DC Equivalent Circuits
For DC equivalent circuit, reduce all AC voltage sources to zero and open all AC current sources and open all capacitors. with this reduced circuit shown in fig. 26 (a) DC current and voltages can be calculated.
For AC, equivalent circuit reduces DC voltage source to zero and open current source and short all capacitors. this circuit is used to calculate AC  currents and voltage as shown in fig. 26(b).
The total current in any branch is the sum of DC and AC currents through that branch. the total valtage across any branch is the sum of the DC voltage and AC voltage across that branch.
Phase Inversion
Because of the fluctuation is base current; collector current and collector voltage also swings above and below the quiescent voltage. the AC output voltage is inverted with respect to the AC  input voltage, means it is 180⁰ out of phase with input.
During the positive half cycle base current increases, causing the collector current to increase. this produces a large voltage drop across the collector resistor, therefore the voltage output decreases and negative half cycle of output voltage is obtained. conversely, on the negetive half cycle of input voltage less collector current flows and the voltage drop across the collector resistor decreases and hence collector voltage increases, we get the positive half cycle of output voltage as shown in fig.27.

3. Common-Collector Amplifier

If a high impedance source is connected to low impedance amplifier then, most of the signal is dropped across the internal impedance of the source. to avoid this problem common-collector amplifier is used in netween source and CE amplifier. it increases the input impedance of the CE amplifier without significant change in input voltage.
Fig. 28 shows a common-collector (CC) amplifier. since, there is no resistance in collector circuit therefore, collector circuit therefore, collector is AC grounded. if is also calld grounded collector amplifier. when input source drives the base, output appears across emitter resistor. A CC amplifier is like a heavily swamped CE amplifier with a collector resistor shorted and output taken across emitter resistor.
V_out = V_in – V_BE
Therefore, this circuit is also called emitter follower, because V_BE is very small. As V_in increases, V_out increases.
IfV_in is 2 V,                                      V_out = 1.3 V
IfV_in is 3 V,                                      V_out = 2.3 V
Since, V_out follows exactly the V_in therefore, there is no phase inversion between input and output. the output circuit voltage equation is given by
V_CE = V_CC – I_ER_E
Since,                I_E >> I_C
I_C = V_CC – V_CE / R_E
This is the equation of DC load line. the DC load line is shown in fig. 28.
Voltage Gain
Fig. 29(a) shows an emitter follower driven by a small AC voltage. the input is applied at the base of transistor and output is taken across the emitter resistor. fig. 29(b) shows the AC equivalent circuit of the amplifier. the emitter is replaced by AC resistance R’_E.
The AC output voltage is given by
and                V_out = R_E I_E
V_in = I_E (R_E + R’_E)
Therefore,            A = R_E / R_E + R’_E
Since                  R’_E >> R_E
A >> 1
Therefore, it is a unity gain amplifier. the practical emitter follower circuit is shown in fig. 29 (c).
The AC source (V_S) with a series resistance (R_S) drives the transistor base. because of the biasing resistor and input impedance of the base, some of the AC signal is lost across the source resistor.
The AC equivalent circuit is shown in fig. 29(d).
The input impedance at the base is given by
Z_{in (base)} = V_in/ I_B
= I_E (R’_E + R_E) / I_B
= BI_B (R’_E + R_E) / I_B
= B (R’_E + R_E)
Since, R’_E is very small in comparison with R_E.
Z_in(base) ~ BR_E
The total input impedance of an emitter follower includes biasing resistors in parallel with input impedance of the base.
Z_in = R₁||R₂||B(R’_E + R_E)
Since BR_E is very large as compared to R₁ and R₂.
Thus,                       Z_in~ R₁||R₂
Therefore, input impedance is very high.
Applying thevenin’s theorem to the base circuit of fig. 29(d). it becomes a source V_in and a series resistance (R₁||R₂||R_S) as shown in fig. 29(e).
V_in = (R₁||R₂||R_S) + I_E (R’_E + R_E)
Or      I_E = V_in / R_E + R’_E + R₁||R₂||R_S / B
The impedance of the amplifier seen from the output terminal is given by
V_out = AV_in = R_E    V_S
R_E +R’_E + R₁||R₂||R_S / B
~ V_in (if R_E is very large)
Example 8.  Find the Q point of the emitter follower circuit of given figure with R₁ = 10 K and R_E = 20 K. Assume the transistor has a B of 100 and input capacitor C is very-very large.
sol. We first find the thevenin’s equivalent of the base bias circuitry.
R_B = R₁||R₂ = 6.67 K
V_BB = R₁ V_CC / R₁ + R₂ = 12 (10⁴) / 30×10³ = 4 V
From the bias equation, we have,
I_C = I_CQ = V_BB – V_BE = 4 – 0.7  = 4.95 mA
R_B / B + R_E      6670/100 +600
Example 9. Find the output voltage swing of the circuit of given above figure.
Sol. The Q point location has already been calculated in example 1. We found that the quiescet collector current is 4.95 mA.
The output voltage swing = 2.I_{C Peak} (R_E||R_load)
= 2 (4.95 x 10-3) (300) = 2.97 V
This is less than the maximum possible output swing. continuing the analysis,
V_{CE, Q} = V_CC – V_CQR_E = 9.03 V
V’_CC = 10.5 / 300 = 35.1 mA
The load lines for this problem are shown in figure.
h-parameters
Small Singnal Low Frequency Transistor Models
All the transistor amplifiers are two port networks having two voltages and two currents. The positive directions of voltages and currents are shown in fig. 30 (a).
Out of four quantities two are independent and two are dependent. if the input current i₁ and output voltage v₂ are taken independent then other two quantities i₂ and v₁ can be expressed in terms of i₁ and v₂
v₁ = f₁ (i₁, v₂)
i₂ = f₂ (i₁, v₂)
The equations can be written as
v₁ = h₁₁ i₁ + h₁₂ v₂
i₂ = h₂₁ i₁ + h₂₂ v₂
Where, h₁₁, h₁₂, h₂₁ and h₂₂ are called h-parameters.
h₁₁ = v₁/ i₁   /v₂ = 0
= h_r
= fraction of output voltage at input with input open circuited or reverse voltage gain with input open circited to AC (dimensions)
h₂₁ = i₂/i₁  /v₂ = 0
= h_f
= negative of current gain with output short circuited to AC.
The current entering the load is neegative of I₂. This is also known as forward short circuit current gain.
h₂₂ = i₂/ i₁  / i₂ = 0
= h_o
= output admittance with input open circuited to AC
If these parameters are specified for a particular configuration, then suffixes e,b or c are also included, e.g., h_fre, h_ib and h-parameters of common-emitter and common-collector amplifiers. using two equations, the generalized model of the amplifier can be drewn as shown in fig. 30(b).
The Hybrid Model for a Transistor Amplifier
Let us consider CE configuration as shown in fig 31(a) the variables, i_b, i_c, v_c and v_b represent total instantaneous currents and voltages i_b and v_c can be taken as independent variables and v_b, i_c as dependest variables.
Using taylor’s series expression and neglecting higher order tems, we obtain
v_b = f_i/ i_b / v_c              i_b + f_i/v_c / i_b    v_c
i_c = f₂/i_b / v_c               i_b + f₂/v_c / i_b   v_c
The partial derivatives are taken keeping the collector voltage or base current constant. the v_b, v_c, i_b, i_c represent the small signal (incremental) base and collector current and voltage and can be represented as
v_b, i_b, v_c, i_c.
v_b = h_ie i_b + h_rev_c
i_c = h_fei_b + h_oe v_b
Where,       h_ie = f₁/i_b /_vc       = v_b/i_b /_vc
h_re = f₁/v_c /_ib    = v_b/v_c /_ib
h_fe = f₂ /i_b /_vc        = i_c/i_b /_vc
h_oe = f₂/v₂ /_ib        = v_b/v_c /_ib
The model for CE configuration is shown in fig. 31(b).
Determination of h-Parameters
To determine the four h-parameters of transistor amplifier, input and output characteristics are used. input characteristic depicts the relationship between input voltage and input current with output voltage as parameter. the output characteristic depicts the relationship between output voltage and output current with input current as parameter. fig.32(a). shows the output characteristics of CE amplifier.
h_fe = i_c/i_b /_vc    = i_c2 – i_c1 / i_b2 – i_b1
The current increments are taken around the quiescent point Q which corresponds to i_b = I_B and to the collector voltage V_CE = V_C
h_oe = i_c / v_c /_ib
The value of h_oe at the quiescent operating point is given by the slope of the output characteristic at the operating point (i.e., slope of tangent AB).
h_ie = v_b/i_b = v_b/i_b /_vc
h_ie is the slope of the appropriate input on fig. 21(b) at the operating point (slope of tangent EFat Q).
h_re = V_B/V_C    = V_B/V_C  /_IB       = V_B2 – V_B1 / V_C2 – V_C1
A vertical line on the input characteristic represents constant base current. the parameter h_re can be obtained from the ratio (V_B2 – V_B1) and (V_C2 – V_C1) for at Q.
Typical CE h-parameters of transistor 2N1573 are given below.
h_ie   = 1000
h_re = 2.5 *10-4
h_fe  = 50
h_oe = 25 mA/V
H-Parameters Analysis of a Transistor Amplifier Using h-Parameters
To form a transistor amplifier, if is only necessary to connect an external load and signal source as indicated in fig. 33(a) and to bias the transistor properly.
Consider the two-port network of CE amplifier. R_s is the source resistance and Z_L is the load impedance h-parameters are assumed to be constant over the operating range, the AC equivalent circuit is shown in fig. 33(b) (phasor notations are used assuming sinusoidal voltage input). the quantities of interest are the current gain, input impedance, voltage gain and output impedance.
Current Gain
For the transistor amplifier stage A, is defined as the ratio of output to input currents.
A_f  = I_L /I_B = -I_C / I_B
(I_L + I_C = 0,   I_L = – I_C)
I_C = h_feI_{B +} h_oeV_C
V_C = I_L Z_L = – I_C Z_L
I_C = h_fe I_B + h_oe (-I_C Z_L)
Or                  I_C / I_B =  h_fe/ 1+ h_oe Z_l
A_l = – h_fe / 1 + h_oe Z_L
Input Impedance
The impedance looking into the amplifier input terminals (1, 1′) is the input impedance Z_i
Z_I = V_B/ I_B
V_B = h_ieI_B + h_reV_E
V_B / I_B = h_ie + h_re V_C/I_B
= h_ie – h_reI_CZ_L / I_B
Z_I = h_ie + h_reA_IZ_L
h_ie – h_reh_feZ_L / 1 + h_oeZ_l
Z_i = h_ie – h_reh_fe / y_l + h_oe   (since, Y_L = 1 /Z_L)
Voltage Gain
The ratio of output voltage to input voltage gives the gain of the transistors.
A_V  = V_C/V_B = I_CZ_L / V_B
A_V = I_B A_IZ_L / V_B = A_IZ_L /Z_I
Output Admittance
It is defined as ,
Y_O = I_C/V_C /_VS = 0
I_C = h_feI_B + h_oeV_C
I_C /V_C = h_fe I_B / V_C + h_oe
when,       V_S = 0, R_S I_B + H_IE I_B + H_REV_C = 0
I_B/V_C = – h_re / R_S + h_IE
Y_O = h_oe – h_reh_fe / R_S + H_IE
Voltage amplification taking into account source impedance (R_S) IS given by
A_VS = V_C/V_S = V_C /V_B + V_B/ V_S                 (V_B = V_S / R₂ + Z_I = Z_I)
= A_V   Z_I / Z_I + R_S
= A_IZ_L / Z_I + R_S
A_v is the voltage gain for an ideal voltage source (R_V = 0) Consider input source to be a current soure I_S in parallel with a resistance R_S as shown in Fig . 34 (a).
In this case, overall current gain A_IS is defined as
A_IS = I_L / I_S
= – I_C / I_S
= – I_C / I_B * I_B / I_S                   (I_B = I_S* R_S / R_S + Z_I)
= A_I * R_S / R_S + Z_I
R_S — _OO, A_IS –A_I
To analyze multistage amplifier, the h-parameters of the transistor used are obtained from manufacture data sheet. the manufacture data sheet usually provides h-parameters in CE configuration. these parameters may be converted into CC and CB values. for example, fig. 34(b) h_re in terme of CE parameter can be obtained as follows:
For CE transistor configuration,
V_BE + h_ieI_B  + h_reV_CE
I_C = h_feI_B + h_oe V_CE
The circuit can be redrawn like CC transistor configuration as shown in fig 35.
V_BE = h_ie I_B + h_RE V_EC
I_C = h_FE I_B + h_OE V_EC
h_rc  = V_BE / V_EC / I_B = 0
= V_BE + V_EC/ V_EC  / I_E = 0
= (V_BE / V_EC + 1 )  / I_B = 0
Since , I_B = 0,    V_BE = H_RE, V_C = – h_REV_EC
 
h_rc = 1 + (h_reV_EC / V_EC)
= 1 – h_re
Similarly,  h_fe = I_E / I_B / V_EC = 0   = – (I_B + I_C) / I_B  / V_EC = 0
= – ( 1 + h_fe)
given below.
h_ie = 2k, h_fe = 50, h_re = 6*10^-4, h_oc = 25 mA/V
h_ic = 2k, h_fe = -51, h_re = 1, h_re = 25 mA/V
The small signal model of the transistor amplifier is shown in figure below.
Sol. In the circuit, the collector resistance of first stage is shunted by the input impedance of last stage. therefore, the analysis is started with last stage. it is convenient to first compute current gain, input impedance and voltage gain. then, output impedance is calculated starting from first stage and moving towards end.
A_i2 = -h_fe / 1 + h_oc Z_L = 51 / 1 + 25*10^-6 * 5 * 10³
= 45.3
R_i2 = h_ic + h_reA_i2Z_L
               = 2 * 10³ + 1 * 45.3 *5 *10³
= 228.5 K               (high input impedance)
A_V2 = V_O / V₂ = A_IZ_L / Z_I2
= 45.3 * 5 / 228.5 = 0.99 = 1
R_L1 = R_C1||R_I2
= 5 * 228.5 / 5 + 228.5 = 4.9K
A_I1 = – h_fe / 1 + h_oeR_L =  -50 / 1+25*10^-6*4.9*10³ = 44.5
R_I1 = h_ie + h_reA_I1R_L1
= 2-6*10^-4*44.5*4.9 = 1.87K
Voltage gain of first state is
A_V1 = A_I1R_L1 / R_I1 = -44.5*4.9/1.87 =-116.6
Y_o1 = h_oe – h_feh_re / h_ie + R_S
= 25*10^-6 – 50*6*10^-4/2*10³+1*10³
= 15*10^-6 mho
R_o1 = 1/ y_o1 =66.7k
R’_O1 = R_O1||R_C1
= 66.7||5=4.65K
The effective source resistance R’_S2 For the second stage is R₀₁||R_C1. Thus, R_S2 = R’₀₁ = 4.65 K
Frequency Response
Y_o2 = h_oe – h_feh_rc / h_ic + R_s2
=25*10^-8 – (-51) (1) /2*10³ + 4.65*10³
= 7.70*10^-3 A/V
R_o2 = 1/y₀₂ = 130
R_o2 = R_o2||R_C2
= 0.13||5K =127
Overall current gain of the amplifier is A_i and is given by
A_i = – I_E2 / I_b1
= – I_e2 / I_b2 * I_b2 / I_e1 * I_C1 / I_B1
= – A_I2 . I_B2/I_C1 * A_I1
The equivalent circuit of the amplifier is showe in fig. 3. from the circuit, it clear that current i_c1 is divided into two parts.
Therefore, I_B2 / I_C1 = -R_C1 / R_C1 + R_I2
A_I = A_I2 . A_I1 . R_C1 / R_C1 + R_I2
= 45. 3* (-44.5) * 5 / 228.5 + 5 = – 43.2
A_V = V_O / V₁ = V_O / V₂ * V₂ / V₁
= A_V2 . A_V1
= 0.99*(-11.6) = 115
Overall voltage gain of the amplifier is given by
A_VS = – V_O / V_S = A_V . R_I1 / R_I1 + R_S
= – 115* 1.87 / 1.87 +1 = – 75.3