High Frequency Response Using Open Circuit Time Constant Analysis(OCTCA)

From the above equivalent circuit, we could use KVL and KCL to rigorously analyze the small signal response. However, once we have more than one transistor in a circuit, such an approach becomes impractical, and designers usually opt for approximate methods. As can be seen, $C_{\mu}$ is actually a Miller capacitance, and we can analyze it as such. However, another approach to analyze these effects is the Open Circuit Time Constant Analysis (OCTCA), which we will introduce here.

Very often a circuit contains a dominant pole which largely determines its high-frequency characteristics. The OCTCA gives a methodology for approximating the value of this dominant pole. With the OCTCA, you account for the effect of one high-frequency capacitor at a time. You do this by determining the resistance seen by $C_{\pi}$, while assuming $C_{\mu}$ is not there (open circuited). You then repeat the process by determining the resistance seen by $C_{\mu}$, while open circuiting $C_{\pi}$.If there were more high frequency capacitors in the circuit, you would repeat this process for each capacitor. The high frequency response is then approximated by a low pass filter with a single pole

$$p=\frac{-1}{\sum_i R_iC_i}$$ (87)

Where R_i is the resistance seen by C_i with all other high frequency capacitors shorted.

The OCTCA approach is probably best illustrated through example. Consider the CE amp in Fig. 4.11

  

Figure 4.11: Circuit for Illustrating OCTCA

Let's start by drawing the small signal equivalent circuit which is shown in Fig. 4.13.

  

Figure 4.12: Small Signal Equivalent Circuit Illustrating OCTCA

Notice that we have explicitly included the intrinsic capacitors $C_{\pi}$ and $C_{\mu}$. Now, we can divide the capacitors into two groups. The first group tend to give rise to higher voltage gain as the frequency increases. This group consists of C₁, C₂ and C_E. The second group causes the gain to decrease as frequency increases. This group consists of $C_{\pi}$ and $C_{\mu}$.Now, let's consider only frequencies which are so high that C₁, C₂ and C_E can be considered short circuits. Under these circumstances the small signal equivalent circuit is shown in Fig. 4.13

  

Figure 4.13: CE High Frequency Small Signal Equivalent Circuit

Of course we could now analyze this circuit formally using loop equations to obtain the voltage gain as a function of frequency. However, once circuits contain more than one transistor, such analyses become very tedious, and are usually best done numerically. Here, we illustrate the OCTCA which is easily extendible to complex circuits.

As mentioned above, with the OCTCA we open all capacitors and short the input. Then, we determine resistance seen by the capacitor under consideration. Let's start with $C_{\pi}$. First we open $C_{\mu}$. The resistance $C_{\pi}$ sees with $C_{\mu}$ open is then $R_S\vert\vert R_B\vert\vert r_{\pi}=R_{\pi o}$.So, the contribution due to capacitor $C_{\pi}$ can be approximated with the time constant $\tau_{\pi}=R_{\pi o}C_{\mu}$.

Determining the resistance seen by $C_{\mu}$ is not quite as obvious due to the dependent current source. To determine this resistance, which we will designate $R_{\mu o}$, we first open $C_{\pi}$, and then replace $C_{\mu}$with a current source and determine the voltage developed across it. The situation is illustrated in Fig. 4.14.

  

Figure 4.14: Circuit for determining resistance seen by $C_{\mu}$

In this figure, i_x is the value of the test current source which replaces $C_{\mu}$, and v₁-v₂ is the voltage developed across i_x. Also in the figure R_L'=R_C||R_L and $v_{\pi}$ is the voltage across $R_{\pi o}$.The resistance seen by $C_{\mu}$ is thus given by

$$R_{\mu o}=\frac{v_1-v_2}{i_x}$$ (88)

To find $R_{\mu o}$ in terms of known quantities apply Kirchoff's laws.

$$v_1=i_xR_{\pi o}$$ (89)

$$v_2=i_x(g_mR_{\pi o}+1)R_L'$$ (90)

Therefore, the resistance seen by $C_{\mu}$ is

$$R_{\mu o}=\frac{v_1-v_2}{i_x}= R_{\pi o}+g_mR_{\pi o}R_L'+R_L'$$ (91)

Thus, the time constant associated with $C_{\mu}$ is $\tau_{\mu}=C_{\mu} R_{\mu o}$.

Using the OCTCA, we approximate the high frequency response of the circuit with a first order low pass filter with a pole p at

$$p=\frac{-1}{\tau_{\pi}+\tau_{\mu}}$$ (92)

The voltage gain A(s) of the circuit at high frequencies is thus approximated by

$$A(s)=\frac{A_m}{1-\frac{s}{p}}$$ (93)

Where A_m is the midband gain.