As stated in Remark 2.3(1) (Section 2.9.5.2), a positive feedback path (from the output to the positive input terminal) of an OP Amp destabilizes the output so that the output voltage will have the alternative of þVom (positive saturation voltage þVsat) or Vom (negative saturation voltageVsat). As will be illustrated in this section, positive/negative feedback can be combined to generate a periodic waveform such as a rectangular or triangular wave, which seems to be a wonderful harmony of two antagonistic entities each causing instability/stability.
3.5.2.1 Square-Wave Generator
Consider the OP Amp circuit with positive/negative feedback in Figure 3.15(a). Noting that:
(a) its output voltage will be either vo¼ þVom orVom depending on which one of the two input terminals is of higher voltage, and
(b) the voltage of the positive input terminal is vþ¼ bvo¼ bVomwith b¼ R1=ðR1þ R2Þ,
suppose that the output voltage at some instant is vo¼ þVom. Then the positive input terminal of the OP Amp has the node voltage of
vþ¼ þb Vom with b¼ R1
R1þ R2
(which is possible only when vC¼ v< vþ) and the voltage vC¼ v(at the input terminal) of the capacitor (connected via R3to the output terminal with vo¼ þVom) rises exponentially towardþVomtill it catches up with vþ¼ þb Vom. As soon as vC¼ vgoes above vþ¼þb Vom (so that v> vþ), the voltages at the output andþ input terminal will go down to
vo¼ Vom and vþ¼ bVom
respectively. Then vC¼ v falls exponentially towardVomtill it touches down at vþ¼ b Vom. As soon as vC¼ v goes below vþ¼ b Vom(so that v< vþ), the voltages at the output andþ input terminal will go up to
vo¼ þVom and vþ¼ þb Vom
respectively. This process repeats itself periodically, generating a rectangular wave at the output, as depicted in Figure 3.15(b).
Now, in order to find the period of the rectangular wave, we start from observing that the voltage vC¼ vwill go up and down repetitively in the range limited by the following two threshold values:
VH¼ þb Vom and VL¼ b Vom with b¼ R1
R1þ R2
ð3:60Þ and its waveform can be described by the formula (3.39) as
vðtÞ ¼ð3:39Þ½vðt0þÞ vð1Þeðtt0Þ=T þ vð1Þ; T¼ R3C ð3:61Þ Complying with this formula, the voltage vC¼ vduring the rising interval is described by
vðtÞ ¼ð3:61ÞðVL VomÞeðtt0Þ=Tþ Vom
and the time TRtaken for vC¼ vto rise from VLto VHcan be obtained as
VH¼ ðVL VomÞeTR=Tþ Vom; eTR=T¼Vom VH
Vom VL
ð3:60Þ¼ 1 b 1þ b TR¼ T ln1 b
1þ b¼ R3C ln1þ b
1 b ð3:62Þ
In the same manner, the time TFtaken for vC¼ vto fall from VHto VLcan be obtained as
VL¼ ðVHþ VomÞeTF=T Vom; eTF=T¼ Vomþ VL
Vomþ VH
ð3:60Þ¼ 1 b 1þ b TF¼ T ln1 b
1þ b¼ R3C ln1þ b
1 b ð3:63Þ
Consequently, the period of the rectangular/sawtooth waves turns out to be P¼ TRþ TF¼ 2T ln1þ b
1 b¼ 2R3C ln2R1þ R2
R2
ð3:64Þ Note. This oscillatory phenomenon may be interpreted as the result of negative feedback, having the node voltage at the negative input terminal try to catch up with that at the positive input terminal, which runs away to the other side every time it is about to be caught.
Figure 3.15 An RC OP Amp circuit with positive/negative feedback as a square-wave generator 3.5 Analysis of First-Order OP Amp Circuits 141
Note. The formula (3.39) seems to be just right for this problem. Why are we not happy to have it at the right place at the right time?
Note. In fact, Figure 3.15(a) is the PSpice schematic, which can be created in the Schematic Editor window, where R1¼ R2¼ R3¼ 1 kO and C ¼ 1 mF. This has been run with the Analysis type of ‘Time Response (Transient)’ to yield the waveforms depicted in Figure 3.15(b). To your mind, is the period of the voltage waveforms close to what you get from Equation (3.64)? Who could restrain himself/herself from shouting in admiration of the PSpice software developers at this time?
3.5.2.2 Rectangular/Triangular-Wave Generator
Consider the circuit of Figure 3.16(a) in which the left OP Amp U1 with negative feedback makes an inverting integrator (Section 3.5.1) and the right OP Amp U2 with positive feedback forms a noninverting trigger (Section 2.9.5.2). Noting that:
(a) the output (vo1) of the inverting integrator is applied to the input of the noninverting trigger, (b) the output (vo2) of the noninverting trigger is fed back into the input of the inverting integrator,
and
(c) the two threshold values at which the noninverting trigger changes its output voltage vo2are VH¼ þbVom; VL¼ bVom with b¼ R2=R3 ðEquation ð2:40Þ or Figure 2:36Þ ð3:65Þ suppose the output voltage vo2of the OP Amp U2 at some instant is
vo2¼ þVom
Then this constant positive voltage is fed back into the input (vi1) of the inverting integrator to make its output (vo1) decrease linearly till vo1<bVom. When vo1<bVom, the voltage vþ2(at the positive input terminal of U2) goes below v2¼ 0 (at the negative input terminal of U2) as
vþ2¼ vo1 R2
vo1 Vom
R2þ R3
¼ R3
R2þ R3
ðvo1þ bVomÞ < 0 ¼ v2! vþ2 v2< 0 so that the output voltage vo2of U2 changes fromþVomtoVom:
vo2¼ Vom
This constant negative voltage is fed back into the input (vi1) of the inverting integrator to make its output (vo1) increase linearly till vo1> bVom. When vo1> bVom, the voltage vþ2goes above v2¼ 0:
vþ2¼ vo1 R2
vo1 ðVomÞ R2þ R3
¼ R3
R2þ R3
ðvo1 bVomÞ > 0 ¼ v2! vþ2 v2> 0 so that the output voltage vo2of U2 changes fromVomback toþVom:
vo2¼ þVom
This process repeats itself periodically, generating a triangular wave at the output (vo1) of U1 and a rectangular wave at the output (vo2) of U2, as depicted in Figure 3.16(b).
Now, in order to find the period of the triangular/rectangular wave, we start from observing that the output voltage vo1of the inverting integrator is described by Equation (3.56) as
vo1ðtÞ ¼ð3:56Þvo1ðt0Þ 1 R1C
ðt t0
vo2ðtÞdt ð3:66Þ
Using this equation, the time taken for vo1to rise from VLto VHand the time taken for vo1to fall from VH
to VLcan be obtained as
VH¼ VL 1
R1CðVomÞTR; TR¼R1CðVH VLÞ Vom
ð3:67Þ VL¼ VH 1
R1CðþVomÞTF; TF¼R1CðVH VLÞ Vom
ð3:68Þ
Consequently, the period of the triangular/rectangular waves turns out to be
P¼ TRþ TF¼ 2R1CVH VL
Vom ð3:65Þ¼
4b R1C¼ 4R2
R3
R1C ð3:69Þ
Note. Although the circuits in Figures 3.15(a) and 3.16(a) are called astable (unstable) circuits, they have two ‘quasi-stable’ states in each of which they stay for a time interval determined by the time constant.
Note. A circuit like those of Figures 3.15(a) and 3.16(a) that repeatedly alternates between two states with a period depending on the charging of a capacitor is called a relaxation oscillator.
Note. Figure 3.16(a) is the PSpice schematic of a triangular/rectangular wave generator consisting of an inverting integrator (realized by an RC OP Amp circuit with negative feedback) and a noninverting trigger (realized by a positive feedback OP Amp), where R1¼ R2¼ 1 kO, R3¼ 2 kO, and C ¼ 1 mF. This has been run with the Analysis type of
‘Time Response (Transient)’ to yield the waveforms depicted in Figure 3.16(b). To your eyes, is the period of the voltage waveforms close to what you get from Equation (3.69)?
Figure 3.16 A combination of an inverting integrator and a noninverting trigger
3.5 Analysis of First-Order OP Amp Circuits 143