Negative feedback, linearity and parameter invariance in linear electronics

Abstract

Negative feedback is a powerful approach capable of improving several aspects of a system. In linear electronics, it has been critical for allowing device invariance. Negative feedback is also known to enhance linearity in amplification, which is one of the most important foundations of linear electronics. At the same time, thousands of transistors types have been made available, suggesting that these devices, in addition to their assumed variability of parameters, have distinguishing properties. The current work reports a systematic approach to quantifying the potential of negative feedback, with respect to bipolar transistors, as a means of providing device invariance and linearity. Several approaches, including concepts and methods from signal processing, multivariate statistics and complex systems, are applied at the theoretical as well as experimental levels, and a number of interesting results are obtained. For instance, it has been verified that transistor types have well-defined characteristics which clearly segregate them into groups. The addition of feedback at moderate and intense levels promoted uniformization of the properties of these transistors when used in a class A common emitter configuration. However, such effect occurred with different efficiencies regarding the considered device features, and even intense feedback was unable to completely eliminate device dependence. This indicates that it would be interesting to consider the device properties in linear design even when negative feedback is applied. We also verified that the linearization induced in the considered experiments is relatively modest, with effects that depend on type of transfer function of the original devices.

Appendices

Appendix A: Theoretical analysis of negative feedback

The experimental circuit shown in Fig. 2 can be analyzed using a small signal hybrid h-parameter model [4]. This is done by first considering that the properties of a single transistor, shown in Fig. 21a, can be cast into an equivalent two-port network containing appropriate h-parameters, as shown in Fig. 21b. Likewise, we can represent the circuit used in the experiments (shown in Fig. 2) by a similar h-parameter circuit, which is shown in Fig. 22. Note that we considered $h_{ie}=0$ in the equivalent circuit. Using Kirchhoff’s circuit laws, the following system of equations can be defined for the equivalent circuit

Fig. 21

a Typical parameters used for describing the operation of a transistor. b Hybrid h-parameter model of a transistor

Fig. 22

Hybrid h-parameter model of the circuit considered in the current work (shown in Fig. 2). The shaded region represents the h-parameter model of the transistor

The voltage amplification of the circuit is given by

$$\begin{aligned} A_c^f=\frac{V_c}{V_{bb}}. \end{aligned}$$

(18)

Solving the system of equation for $V_c$ and $V_{bb}$ and replacing the result into Eq. 18 results in

$$\begin{aligned} A_c^f =\frac{R_c(\beta r_o - R_e - r_e)}{R_b(r_o - R_c)+((1+\beta )r_o + R_b - R_c)(R_e+r_e)}. \nonumber \\ \end{aligned}$$

(19)

In the absence of feedback, $R_e=0$ and we obtain the open loop amplification, given by

$$\begin{aligned} A_c =\frac{R_c(\beta r_o - r_e)}{R_b(r_o - R_c)+((1+\beta )r_o + R_b - R_c)r_e}. \end{aligned}$$

(20)

Equation 19 can be simplified if we specifically consider the parameters used in the present work. First, we rewrite the equation as

$$\begin{aligned} A_c^f =\frac{R_c\beta r_o(1 - \frac{R_e}{\beta r_o} - \frac{r_e}{\beta r_o})}{R_b(r_o - R_c)+\beta r_o(1+\frac{1}{\beta r_o} + \frac{R_b}{\beta r_o} - \frac{R_c}{\beta r_o})(R_e+r_e)}. \nonumber \\ \end{aligned}$$

(21)

According to the values presented in Table 7, we have that, on average, $\beta r_o = 449*14{,}822 = 6{,}655{,}078\,\Omega$. On the other hand, the maximum value among all considered circuit resistances and feedback situations is $M_p = \max \{R_b, R_c, R_e, r_e\} = 32{,}640\,\Omega$. Since $\beta r_o>> M_p$, we can consider that $M_p/(\beta r_o)\approx 0$. Therefore, the amplification can be approximated as

$$\begin{aligned} A_c^f \approx \frac{R_c\beta r_o}{R_b(r_o - R_c)+\beta r_o(R_e+r_e)}. \end{aligned}$$

(22)

It is also possible to associate Eq. 19 with the general expression of a feedback circuit, provided by Eq. 2. This is done by replacing $A^f$ and A in Eq. 2 by their respective values, $A_c^f$ and $A_c$, for the considered circuit:

$$\begin{aligned} A_c^f = \frac{A_c}{1+A_cf}. \end{aligned}$$

(23)

Thus,

$$\begin{aligned} f = \frac{A_c-A_c^f}{A_cA_c^f}. \end{aligned}$$

(24)

Replacing Eqs. 19 and 20 into Eq. 24 gives

$$\begin{aligned} f = \frac{R_e \left( R_b+\beta r_o\right) \left( (\beta +1) r_o-R_c\right) }{R_c \left( r_e-\beta r_o\right) \left( r_e+R_e-\beta r_o\right) }. \end{aligned}$$

(25)

This equation can be approximated through a similar procedure used for deriving Eq. 22. First, Eq. 25 is rewritten as

$$\begin{aligned} f = \frac{R_e \left( \frac{R_b}{\beta r_o}+1\right) \left( 1+\frac{1}{\beta r_o}-\frac{R_c}{\beta r_o}\right) }{R_c \left( \frac{r_e}{\beta r_o}-1\right) \left( \frac{r_e}{\beta r_o}+\frac{R_e}{\beta r_o}-1\right) }. \end{aligned}$$

(26)

Since $\beta r_o$ is significantly larger than all other parameters, we obtain

$$\begin{aligned} f \approx \frac{R_e}{R_c}. \end{aligned}$$

(27)

Another interesting quantity that can be calculated is the variation of amplification $A_c^f$ respective to a variation of the current gain $\beta$ of the transistor. This is done by calculating the derivative of Eq. 19 with respect to $\beta$, resulting in

$$\begin{aligned} \frac{\partial A_c^f}{\partial \beta } = \frac{r_oR_c(R_b+R_e+r_e)(r_o-R_c+R_e+r_e)}{{\mathcal {D}}^2}, \end{aligned}$$

(28)

where $\mathcal {D} = R_b(r_o - R_c)+((1+\beta )r_o + R_b - R_c)(R_e+r_e)$. The sensitivity $S_{A_c^f}(\beta )$ of $A_c^f$ with respect to $\beta$ is defined as [4]

$$\begin{aligned} S_{A_c^f}(\beta ) \equiv \frac{\beta }{A_c^f}\frac{\partial A_c^f}{\partial \beta }, \end{aligned}$$

(29)

Thus,

$$\begin{aligned} S_{A_c^f}(\beta ) = \frac{\beta r_o \left( R_b+R_e+r_e\right) \left( r_o-R_c+R_e+r_e\right) }{ \mathcal {D}\left( \beta r_o -R_e-r_e\right) }. \end{aligned}$$

(30)

Since $\beta r_o$ is large, the sensitivity can be approximated as

$$\begin{aligned} S_{A_c^f}(\beta )\approx \frac{\left( R_b+R_e+r_e\right) \left( r_o-R_c+R_e+r_e\right) }{R_b \left( r_o-R_c\right) +\beta r_o(R_e + r_e)}. \end{aligned}$$

(31)

The variation of the amplification with respect to $r_o$ can also be calculated:

$$\begin{aligned} \frac{\partial A_c^f}{\partial r_o} = \frac{R_c \left( R_b+r_e+R_e\right) \left( -\beta R_c+(\beta +1)(R_e + r_e)\right) }{\mathcal {D}^2}\nonumber \\ \end{aligned}$$

(32)

The respective sensitivity of $A_c^f$ with respect to variations of $r_o$ is given by

$$\begin{aligned} S_{A_c^f}(r_o) \equiv \frac{r_o}{A_c^f}\frac{\partial A_c^f}{\partial r_o} \end{aligned}$$

(33)

Thus,

$$\begin{aligned} S_{A_c^f}(r_o) = \frac{r_o \left( R_b+r_e+R_e\right) \left( -\beta R_c+(\beta +1) (R_e+r_e)\right) }{ \mathcal {D}\left( \beta r_o - r_e - R_e\right) },\nonumber \\ \end{aligned}$$

(34)

which can be approximated as

$$\begin{aligned} S_{A_c^f}(r_o) \approx \frac{\left( R_b+r_e+R_e\right) \left( -R_c+r_e+R_e\right) }{R_b \left( r_o-R_c\right) +\beta r_o(R_e+r_e)} \end{aligned}$$

(35)

Fig. 23

Three examples, respective to each of the performed experiments—a no feedback, b moderate feedback and c intense feedback—of experimental curves of $V_a$ in terms of $V_{bb}$

We can also derive an expression for the sensitivity of amplification $A_c^f$ to small variations in open loop amplification $A_c$. This is done by calculating the derivative of $A_c^f$ with respect to $A_c$ using Eq. 23:

$$\begin{aligned}&\frac{\partial A_c^f}{\partial A_c} = \frac{1}{(1+fA_c)^2} \nonumber \\&\quad = \frac{1}{\left( \frac{R_e \left( R_b+\beta r_o\right) \left( (\beta +1) r_o-R_c\right) }{\left( r_e+R_e-\beta r_o\right) \left( r_e \left( R_b-R_c+(\beta +1) r_o\right) +R_b \left( r_o-R_c\right) \right) }-1\right) {}^2} \end{aligned}$$

(36)

Note that this derivative can also be written as

$$\begin{aligned} \frac{\partial A_c^f}{\partial A_c} = \left( \frac{A_c^f}{A_c}\right) ^2. \end{aligned}$$

(37)

Therefore, the sensitivity of $A_c^f$ with respect to $A_c$ is given by

$$\begin{aligned}&S_{A_c^f}(A_c) = \frac{A_c}{A_c^f}\frac{\partial A_c^f}{\partial A_c} = \frac{A_c^f}{A_c} = \nonumber \\&\frac{\left( r_e+R_e-\beta r_o\right) \left( r_e \left( R_b-R_c+(\beta +1) r_o\right) +R_b \left( r_o-R_c\right) \right) }{\left( r_e-\beta r_o\right) \mathcal {D}} \end{aligned}$$

(38)

Considering again the fact that $\beta r_o$ is much larger than other terms, we have that

$$\begin{aligned} S_{A_c^f}(A_c) \approx \frac{r_e \beta r_o+R_b \left( r_o-R_c\right) }{\beta r_o(R_e + r_e)+R_b \left( r_o-R_c\right) }. \end{aligned}$$

(39)

The output resistance of the circuit, $R_{o}=V_c/I_c$, can be found by setting $V_{bb}=0$ in the equivalent h-parameter model of the circuit. In other words, the input becomes short-circuited. Equations 13–16 can then be used to find

$$\begin{aligned} R_{o} = \frac{R_b r_o + (R_b + (1+\beta )r_o)(R_e+r_e)}{R_b+R_e+r_e}. \end{aligned}$$

(40)

This equation can be rewritten as

$$\begin{aligned} R_{o} = \frac{R_b r_o + \beta r_o(\frac{R_b}{\beta r_o} + \frac{1}{\beta r_o}+1)(R_e+r_e)}{R_b+R_e+r_e}. \end{aligned}$$

(41)

If $\beta r_o>>R_b$, we have that

$$\begin{aligned} R_{o} \approx r_o\frac{R_b + \beta R_e + \beta r_e}{R_b+R_e+r_e} \end{aligned}$$

(42)

Fig. 24

a A typical curve $V_a$ in terms of $V_{bb}$, with the respectively obtained values $V_{a,\min }$ and $V_{bb,\min }$. b The space $I_c$ in terms of $V_c$ and some of the isolines parametrized by $V_{bb}$, including that corresponding to ${\min }$, as well as the respective $\Delta V_{bb}$ and $\Delta I_{c}$. c The same isolines as in b, but with the setting necessary to estimate $R_o$

In a similar fashion, the input resistance, $R_{i}=V_{bb}/I_b$, is found by setting $V_c=0$ and solving the system of equations for $V_{bb}$ and $I_b$, giving

$$\begin{aligned}&R_{i} = \frac{R_b r_o + (R_b + (1+\beta )r_o)(R_e+r_e)}{R_e+r_e+r_o}. \end{aligned}$$

(43)

$$\begin{aligned}&R_{i} \approx r_o \frac{R_b + \beta R_e+\beta r_e}{R_e+r_e+r_o}. \end{aligned}$$

(44)

Appendix B: Transistor parameters estimation method

Finding a representative and practical estimation of the transistor parameters ($R_o$, $\beta$ and $V_a$) is an important task [13]. While some datasheets traditionally present the transistors parameters in terms of average, maximum and minimum, the transistor parameters can vary substantially depending on the circuit configuration space ($I_c \times V_c \times V_{bb}$) [24], which complicates the estimation of the parameters. We developed a method to estimate the representative parameters of a transistor, in which we employed the voltage (see Sect. 2.4) as a measurement of configuration stability, i.e., we try to identify the $V_{bb}$ interval where $V_a$ is more stable. This section describes this methodology.

To estimate the voltage, we start by considering all the isolines (corresponding to constant $V_{bb}$) obtained experimentally for a transistor. For each isoline, the voltage $V_a(V_{bb})$ is estimated by applying least squares method [7] along its last 100 data points. This constraint is necessary in order to avoid the nonlinear saturation region. Because this estimation is still susceptive to noise, we improve the signal-to-noise ratio of $V_a(V_{bb})$ by applying the Savitzky–Golay filter (S–G) [21], which approximates the data to a set of smooth polynomial functions.

Figure 23 shows the curves of $V_a(V_{bb})$ estimated for a transistor under the three feedback conditions considered in this work: no feedback (a), moderate (b) and intense feedback (c). In all cases, the curves start with a prominent depression region followed by a fast increase in $V_a$ with $V_{bb}$. Interestingly, this result holds for all the considered transistors, but the minimum attained value $V_{a, \min }$ and the corresponding isoline $V_{bb, \min }$ can vary from case to case. Since the depression region indicates a region of well-behaved $V_a$, we use the minimum of the curve, $V_{a, \min }$, as a representative value for the voltage.

Having defined a stable reference for $V_a$ (as shown in Fig. 24a), a working interval can be imposed upon $V_{bb}$ and $I_c$, spanning from $V_{bb, \min }$ to two previous isolines (the experiments always take 64 isolines corresponding to a $V_{bb}$ step of 5 / 64 V), as indicated in Fig. 24b. Now, $\beta$ can be estimated as $\Delta I_c / \Delta I_b$, where $\Delta I_b = \Delta V_{bb} / R_b$. This approach assumes small $V_b$ variation along the working interval. The estimation of $R_o$ can be performed by numerically calculating (by using least square approximation) the slope of the $V_{bb,\min }$ isoline considering its last 100 points, as illustrated in Fig. 24c.