Appunti di lezione di Elettrotecnica

C

T

removing the dependence on , increases the accuracy of the ratio (the error on the implementation of

2

C 1

a capacitor is equal throughout a single silicon chip).

Another advantage in the use of switched capacitor filters consists in the possibility of obtaining a negative

resistance by means of the following modulation (Figure 2.11):

• phase 1: switches 1 and 3 closed, 2 and 4 open;

• phase 2: switches 2 and 4 closed, 1 and 3 open.

This allows to obtain a non-inverting stage, starting from an inverting topology.

Figure 2.11: Equivalent negative resistance

Chapter 3

Non-linear circuits

All the circuits analysed in this chapter makes use of the operational amplifier either with non linear elements

(i.e. diodes, transistors), or of its out of linearity (saturation) behaviour.

3.1 Rectifiers

These circuits can be used to approximate a non-linear transfer function through a custom piecewise linear

behaviour. In particular, different circuit implement the several piecewise linear characteristic with proper

slopes and transition points, and their result is added by means of a summing node. They are usually

implemented through operational amplifiers, so that negative feedback makes the transfer characteristic

independent on the direct conduction voltage drop on real diodes (that gets divided by the loop gain of the

amplifier). In this way, it is possible to rectify very small signals.

3.1.1 Single wave rectifier

Let us consider the circuit in Figure 3.9.

Figure 3.1: Theoretical single wave rectifier model

The operating principle of this architecture may be summarised as follows, considering that we are dealing

with an inverting stage:

• V >

when 0, the output of the amplifier "wants" to be negative. Therefore, the diode is in direct

i R

−

V V

conduction, and = .

2

u i

R 1

• V <

when 0, the output of the amplifier "wants" to be positive. It follows that the diode is not

i V

conducting, thus yielding the opening of the feedback loop, and as a consequence = 0 (output

u

directly connected to the virtual short circuit at the input pin).

However, the latter assumption is not valid, because with the opening of the feedback loop, the amplifier is

not in linearity anymore. Therefore, we cannot expect a perfectly zero voltage for negative input voltage.

This can be solved by avoiding, in any case, the absence of feedback, by designing two loops with diodes

19

CHAPTER 3. NON-LINEAR CIRCUITS 20

mounted in opposite directions, as shown in Figure 3.2. In this way, during each branch of the signal, there

will always be one diode in conduction, so that the op-amp always operates in linearity.

Figure 3.2: Real single wave rectifier

Another possible source of error consists in the limited slew rate of the amplifier: in fact, since the amplifier

must operate a step variation in voltage to overcome the threshold of the two diodes, this doesn’t happen

with infinite slope.

3.1.2 Double wave rectifier double

By combining a single wave rectifier and an inverting adder, one obtains the transfer function of a

wave rectifier, as shown in Figure 3.3.

Figure 3.3: Double wave rectifier C:

The transfer function of this device reads, neglecting the effects of the capacitor

−V −

V = 2V (3.1)

u i B

In particular:

• −V

V > V

when 0, = , thus yielding

i i

B V V

= (−1 + 2)V = (3.2)

u i i

• V < V

when 0, = 0, therefore

i B −V

V = (3.3)

u i

The function of the capacitor consists in implementing a low-pass filter in the second stage. If its cutoff

frequency is sufficiently small (large time constant of the circuit), we can extract the RMS value of the input

signal.

CHAPTER 3. NON-LINEAR CIRCUITS 21

3.1.3 Piecewise linear functions with saturation

It is possible to implement transfer functions with piecewise linear and constant behaviour by means of

Zener diodes in the feedback network. For example, let us consider the circuit in Figure 3.4.

Figure 3.4: Rectifier with Zener diode V >

The Zener diode behaves as a standard diode in direct conduction, therefore, for 0, the output voltage

i

V

is short circuited through the virtual ground, thus being = 0. For negative input voltages, the stage

u

R

−

V V V < V

is inverting, and its output reads = . This is valid only for ; afterwards, the required

2

u i u z

R 1

saturation is experienced. The voltage transfer characteristic can be summarised as follows:

 V >

0 0

i





 R

−

V = (3.4)

V V < V < V

0 and

2

u i i u z

R 1

 ≥



V V < V V

0 and

 z i u z

The introduction of other Zener diodes yields other non-linearity points. Let us consider the circuit in Figure

3.5: here, given the opposite polarity of the diodes, we can define further operating regions.

Figure 3.5: Rectifier with two Zener diodes

Assuming that the diodes have the same threshold:

• ∈

V , V

if [−V ], the branch with the diodes can be neglected, because one diode is off, and the other

u z z

is not yet in Zener condition;

• ≤ −V

V > V

when 0 and , D2 is in Zener condition and limits the transfer characteristic;

i u z

• ≥

V < V V

when 0 and , D1 in in Zener mode.

i u z

The transfer function can be summarised as follows, with general thresholds for the Zener diodes:

 R

− ∈

V V , V

[−V ]

2 i u z2 z1

 R

 1

 −V ≤ −V

V = (3.5)

V > V

0 and

u z2 i u z2

 ≥

 V V < V V

0 and

 z1 i u z1

CHAPTER 3. NON-LINEAR CIRCUITS 22

3.2 Schmidt-Trigger: operational amplifier in comparator mode

comparing

When the operational amplifier is employed without feedback (Figure 3.6), it performs a function

among the voltage levels at the non-inverting and inverting pin:

−

+

• − ≈

V V V > V V

if = 0, the output voltage is ;

U AL

d

• ≈ −V

V < V

if 0, the output voltage is U AL

d

In this way, we are able to know whether a signal (connected to one input pin) has crossed or not a given

treshold (connected to the other input pin).

Figure 3.6: Non-inverting comparator

This simple circuit has a non-negligible issue: every signal is affected by noise, that can cross the treshold

multiple times in an interval, if the operational amplifier is fast enough. Multiple transitions will cause an

unwanted oscillation in the output voltage, that can be problematic in some application.

two references;

A possible solution consists in comparing the input signal with thus the use of the Schmidt-

trigger, or comparator with hysteresis, so that the comparing function is able to discriminate between the

, V

signal and the noise. This yields the comparison of the input voltage with an interval ∆V = [V ]:

T H+ T H−

only

the output voltage is switched when the signal comes out of the whole window. In order for this

solution to be effective, the amplitude of ∆V must be at least equal to the peak-to-peak amplitude of the

noise. The corresponding circuit for an inverting Schmidt-trigger is found in Figure 3.7, where one can notice

positive feedback,

the presence of that makes the amplifier operate in strongly non-linear conditions.

Figure 3.7: Inverting Schmidt-trigger, circuit

The operation of this circuit may be described as follows, assuming that no current enters the operational

amplifier. −

+

• − ≤

V V V

If = +V , this means that 0. In particular, since the resistive network implements a

u AL

voltage divider, we can write: R 1

+

V V

= (3.6)

AL

R R

+

1 2 +

increase V

As a consequence, an in the input voltage that overcomes yields a switching of the output

−V

voltage to .

AL

CHAPTER 3. NON-LINEAR CIRCUITS 23

• −V

V

If, instead = , the reference voltage at the non-inverting pin reads:

u AL R

1

+ −

V V

= (3.7)

AL

R R

+

1 2

+

decreases V V

Therefore, when the input voltage up to , the output of the device switches to .

AL

One can notice that the effect of the positive feedback is to provide the system with a variable reference

voltage, to which the input signal is compared. The overall transfer characteristic is represented in Figure

3.8. Figure 3.8: Inverting Schmidt-trigger, input-output characteristic

The width of the window between the two reference voltages is a design parameters, that can be set by

R R

properly choosing the values of the resistances and .

1 2

3.3 Signal generators

Positive feedback may be employed to produce specific waveforms, starting from electrical noise through

components and by successive amplification.

3.3.1 Astable multivibrator: square wave generator

Let us consider the circuit in Figure 3.9, that contains an RC series network in negative feedback. The input

R

signal is here auto generated by the output voltage, since the presence of the resistor allows us to draw

V

current, thus charging and discharging the capacitor, that now experiences a voltage across it equal to .

C

The behaviour of the circuit may be analysed as follows, assuming that the capacitor is initially discharged.

Figure 3.9: Square wave generator

CHAPTER 3. NON-LINEAR CIRCUITS 24

• V V τ RC. V

When = , the resistance charges the capacitor with time constant equal to = When

u OH C

reaches R

1

+ V

V = OH

R R

+

1 2

V V

toggles to .

u OL

• Now, the output voltage charges the capacitor with opposite polarity, until

R

1

+

V V

= OL

R R

+

1 2

This behaviour is periodic, thus yielding the square wave behaviour whose period can be calculated by

substitution in the voltage across the capacitor, and its expression reads:

R

β

1+

1

β

T with = (3.8)

= 2RC ln − β R R

1 +

1 2

Triangular wave generator

An approximation of a triangular wave generator can be obtained by reducing the amplitude of the hysteresis

V

window to the point where the swing in may be approximated to a linear function of time, thus obtaining,

C

indeed, a triangular waveform.

3.3.2 Sine wave generators: oscillators

Also in this case, a signal can be generated through a hybrid configuration with positive and negative

feedback. In particular, the negative feedback branch sets the gain of the circuit, while the positive feedback

frequency selective network.

branch is composed of a In Figure 3.10 an exampl