# Chapter 48

## Electronics for design

This chapter is intended to introduce or refresh some useful electronics concepts with emphasis on design.

### 48.1 Feedback

Feedback is a very common technique that is used extensively in electronic circuits. The basic principle of feedback, as its name states it, is to feed the output of the circuit back to an input so as to compare the actual output to the target and use the difference information to modify the output. This is a powerful technique to improve output accuracy with respect to a circuit that would not use feedback. When feedback is used, the circuit looks like a loop and is often called servo-loop, feedback loop and sometimes simply loop.
When a loop is used, a concern always exist with stability. If the output phase shift is large enough, the correction may not drive the system in a stable state but it may enter into oscillation.
The general schematic for a loop is: OUT = A . (IN - u)

u = B . OUT

Then:

OUT = A . (IN - B . OUT)

OUT . (1 + A . B) = A . IN

OUT = IN . A / (1 + A . B)

If A.B >> 1,

OUT # IN . (1 / B)

If A . B = -1 OUT is undefined, the loop is unstable

### 48.2 Impedance matching circuits

When a generator exhibits an internal impedance Z, a question can be:

What is the best possible load impedance for that generator?

To answer this question, let's start with an experiment. Let's consider a circuit built from:

• A voltage source with open circuit voltage V0 =10V and with internal resistance RINT =10 Ω
• A load resistance RLOAD .

What about the voltage across load, the current through load, the power dissipated in the load?

VLOAD = V0 ⋅ RLOAD / (RINT + RLOAD)

ILOAD = V0 / (RINT + RLOAD)

PLOAD = VLOAD . ILOAD = V0^2 . RLOAD / (RINT + RLOAD)^2

Now, if we derive PLOAD with respect to RLOAD:

dPLOAD / dRLOAD = V0^2 . (RINT - RLOAD) / (RINT + RLOAD)^3

This derivative value is zero if RLOAD = RINT

At this point, PLOAD is:

PLOAD (RLOAD = RINT) = V0^2 . RINT / (RINT + RINT)^2 = V0^2  / (4 . RINT)

### 48.3 Simple passive circuits

The impedance and gain versus frequency characteristics of simple passive circuits are important as they allow building equivalent schematic and simplify circuit calculations.

#### 48.3.1 R-C High pass filter

This filter filters out low frequencies. Schematic is:

Figure 0.2:

Transfer function is:

Figure 0.3:

Cutoff frequency is:

f0= 1 2 ⋅ π ⋅ R ⋅ C

Input and output impedance expressions are:

Figure 0.4:

Group delay is:

Figure 0.5:

It must be noted that source and load impedance can significantly impact the transfer function.

#### 48.3.2 R-C Low pass filter

This filter filters out low frequencies. Schematic is:

Figure 0.6:

Transfer function is:

Figure 0.7:

Cutoff frequency is:

f0= 1 2 ⋅ π ⋅ R ⋅ C

Input and output impedance expressions are:

Figure 0.8:

Group delay is:

Figure 0.9:

It must be noted that source and load impedance can significantly impact the transfer function.

#### 48.3.3 R-C parallel impedance

When a resistor and a capacitor are connected in parallel, resulting impedance versus frequency is:

Figure 0.10:

If driven by a current source, this cell achieves a first order low-pass filter function. Cutoff frequency is:

f0= 1 2 ⋅ π ⋅ R ⋅ C

When a circuit element exhibits an impedance versus frequency that looks like that of a parallel R-C cell, it can be modeled as a parallel R-C cell and behaves the same.

### 48.4 Miller effect

This effect was originally a parasitic effect. In ICs, it is a common practice to use this effect.

If an amplifier with voltage gain A, infinite input impedance and negligible output impedance is shorted by an impedance Z between input and output:

What is the input impedance of the system? The input impedance is: ZIN= VIN IIN

Input current, by assumption, flows only in impedance Z:

IIN= VIN-VOUT Z

And we know that:

VOUT=A ⋅ VIN

Then: IIN= VIN ⋅ ( 1-A ) Z

So: ZIN= VIN IIN = Z 1-A

In case A is negative and large: ZIN ≃ Z |A|

Basically, the Miller effect dramatically reduces an inverting amplifier input impedance if some coupling exists between input and output that behaves like an impedance.

In particular, if coupling impedance Z is a capacitance C:

Z= 1 2 ⋅ π ⋅ f ⋅ C ZIN ≃ 1 2 ⋅ π ⋅ f ⋅ |A| ⋅ C

Which is equivalent to the impedance of a capacitor:

C'=|A| ⋅ C

The input capacitance appears multiplied by the amplifier gain! This is a concern for any common-emitter or common-source amplifying stage as they are inverting. The collector-base or drain-gate capacitance causes an increase in the input capacitance that can, in turn, reduce the bandwidth or the slew-rate by a load effect on the driving stage.

In ICs, Miller effect is often used to stabilize loops as it allows large apparent capacitors without using large capacitors, saving silicon area then minimizing cost. The loop architecture is generally tailored so as to create a dominant pole at the input of a large gain stage by using Miller effect. Sometimes, an extra amplifier is used just for the purpose of stabilizing the loop since the amplifier that multiplies the capacitance is often smaller than the equivalent capacitor. The drawback of this method is the amplifier current consumption.

### 48.5 Cascode amplifier

The so called “cascode” amplifier has been created long time ago to limit the parasitic Miller effect. A cascode amplifier is built by stacking a common emitter/source amplifier and a common base/gate stage:

Figure 0.12:

The common emitter/source amplifier would suffer from the Miller effect but since its input to output gain is -1, the effect is limited to an added equivalent input capacitance twice the input to output capacitance value.

The common base/gate stage does not exhibit any parasitic Miller effect since its input is driven with a constant voltage and can be bypassed suitably.

In addition, the common base/gate transistor can be made small as it does not impact the overall noise significantly and this improves the output cutoff frequency.

The two drawbacks of the cascode amplifier are:

• The need for two transistors instead of one, but this is not a concern in an IC.
• The need for a larger supply voltage or a smaller available swing. 