Differential Input Signal Analysis in Symmetrical Circuits

Differential Input Signal Analysis

First, we analyze the circuit for a pure differential input signal. Thus, the input voltages are v_i1 = -v_i2 = v_d / 2. The analysis can be simplified considering that the equivalent circuit is symmetrical. Because of this symmetry and the opposite polarity independent generators, the voltage at point J is zero. Therefore, the behavior of the circuit remains invariant at point J shorting to ground from the middle of the differential circuit. From Figure 7, we can formulate the following equations:

• v_id / 2 = r_n i_b1 + (β + 1) I_B1
• v_o1 = -R_C β i_b1

Input Impedance and Voltage Gain

From these equations, we can find the input impedance and voltage gain. For example, the input impedance seen from the entry terminals can be found from the equation:
R_id = v_id / I_B1 = 2 [r_x + (β + 1) R_EF] (7.56)

Note that R_id is defined as the ratio between the differential input voltage v_id and the total input current. Therefore, R_id is the input impedance. From Equations (7.54) and (7.55), the voltage gain can be found as:
A_vds = v₁ / v_id = -R_C β / 2 [r_x + (β + 1) R_EF]

Where A_vds is the voltage gain for a differential input and asymmetric output (the subscript v indicates the voltage gain, d indicates the differential input signal, and s represents the asymmetric output).

Differential Output Voltage

Sometimes the circuit drives a differential load, and then the differential output voltage is v_d = v₁ - v₂. By symmetry, the output voltage of the right half of the circuit is v₂ = -v₁. Therefore, the differential output voltage is:
v_d = v₁ - v₂ = 2v₁

We define the gain of a differential output as:
A_vdb = v_d / v_id

Here, the subscript v denotes the voltage gain, d the differential input signal, and b the differential load. Using the equation to replace v_d, we get:
A_vdb = 2v₁ / v_id

Therefore:
A_vdb = 2 A_vds = -R_C β / [r_x + (β + 1) R_EF]

Output Impedance Calculation

To find the output impedance, we replace the input voltage sources and look for short circuits from the output terminals. Then, the input current I_B1 is zero, and the controlled source β I_B1 behaves like an open circuit. Therefore:

• For an asymmetrical output, the output impedance is R_os = R_C
• For a differential output, the output impedance is R_ob = 2 R_C