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APPLICATIONS AND DESIGN OF INTEGRATED CIRCUITS

Module by: Dinh Sy Hien

Summary: As an example of the application of op-amps in area of active filters, we will discuss the Butterwort filter. The discussion is only an introduction to the subject of the filter theory design. We will also discuss various types of oscillators, Schmitt trigger circuits, and nonsinusoidal oscillators.

ACTIVE FILTERS

An important application of op-amp is the active filter. The word filter refers to the process of removing undesired portion of the frequency spectrum. The word active implies the use of one or more active devices, usually an operational amplifier, in the filter circuit. As an example of the application of op-amps in area of active filters, we will discuss the Butterworth filter. The discussion is only an introduction to the subject of the filter theory design.
Two advantages of active filters over passive filters are:
  1. The maximum gain or the maximum value of the transfer function may be greater than unity.
  2. The loading effect is minimum, which means that the output response or the filter is essentially independent of the load driven by the filter.

Active Network Design

From our discussion of frequency response, we know that RC-networks form filters. Figure 1a is a simple example of a coupling capacitor circuit. The voltage transfer function for this circuit is
T ( s ) = V 0 ( s ) V i ( s ) = R R + 1 sC = sRC 1 + sRC T ( s ) = V 0 ( s ) V i ( s ) = R R + 1 sC = sRC 1 + sRC size 12{T \( s \) = { {V rSub { size 8{0} } \( s \) } over {V rSub { size 8{i} } \( s \) } } = { {R} over {R+ { {1} over { ital "sC"} } } } = { { ital "sRC"} over {1+ ital "sRC"} } } {} (1)
Figure 1: a) Simple high-pass filter and b) Bode plot of transfer function amplitude
The Bode plot of the voltage gain magnitude /T(jω)//T(jω)/ size 12{ lline T \( jω \) rline } {} is shown in Figure 1a. The circuit is called a high-pass filter.
Figure 2(a) is another example of a simple RC network. Here, the voltage transfer function is
T ( s ) = V 0 ( s ) V i ( s ) = 1 sC 1 sC + R = 1 1 + sRC T ( s ) = V 0 ( s ) V i ( s ) = 1 sC 1 sC + R = 1 1 + sRC size 12{T \( s \) = { {V rSub { size 8{0} } \( s \) } over {V rSub { size 8{i} } \( s \) } } = { { { {1} over { ital "sC"} } } over { { {1} over { ital "sC"} } +R} } = { {1} over {1+ ital "sRC"} } } {} (2)
Figure 2: a) Simple low pass filter and b) Bode plot of transfer function
The Bode plot of the voltage gain magnitude /T(jω)//T(jω)/ size 12{ lline T \( jω \) rline } {} for this circuit is shown in Figure 2(b). This circuit is called a low-pass filter.
Figure 3: a) High-pass filter with voltage follower and b) low pass filter with voltage follower.
Although these circuits both perform a basic filtering function, they may suffer from loading effects, substantially reducing the magnitude gain from the unity value shown in Figure 1(b) and Figure 2(b). Also, the cutoff frequency fLfL size 12{f rSub { size 8{L} } } {} and fHfH size 12{f rSub { size 8{H} } } {} may change when a load is connected to the output. The loading effect can essentially be eliminated by using a voltage follower as shown in Figure 3. In addition, a non-inverting amplifier configuration can be incorporated to increase the gain, as well as eliminate the loading effects.
These two filter circuits are called one-pole filters; the slope of the voltage gain magnitude curve outside the passband is 6 dB/octave or 20 dB/decade. This characteristic is called the rolloff. The rolloff becomes sharper or steeper with higher-order filters and is usually one of the specifications given for active filters.
Two other categories of filters are bandpass and band-reject. The desired ideal frequency characteristics are shown in Figure 4
Figure 4: Ideal frequency characteristics: a) band pass filter and b) band reject filter.

General Two-Pole Active Filter

Consider Figure 5 with admittances Y1Y1 size 12{Y rSub { size 8{1} } } {} through Y4Y4 size 12{Y rSub { size 8{4} } } {} and an ideal voltage follower. We will derive the transfer function for the general network and will then apply specific admittances to obtain particular filter characteristics.
A KCL equation at node VaVa size 12{V rSub { size 8{a} } } {} yields
( V i − V a ) Y 1 = ( V a − V b ) Y 2 + ( V a − V 0 ) Y 3 ( V i − V a ) Y 1 = ( V a − V b ) Y 2 + ( V a − V 0 ) Y 3 size 12{ \( V rSub { size 8{i} } - V rSub { size 8{a} } \) Y rSub { size 8{1} } = \( V rSub { size 8{a} } - V rSub { size 8{b} } \) Y rSub { size 8{2} } + \( V rSub { size 8{a} } - V rSub { size 8{0} } \) Y rSub { size 8{3} } } {} (3)
A KCL equation at node VbVb size 12{V rSub { size 8{b} } } {} produces
( V a − V b ) Y 2 = V b Y 4 ( V a − V b ) Y 2 = V b Y 4 size 12{ \( V rSub { size 8{a} } - V rSub { size 8{b} } \) Y rSub { size 8{2} } =V rSub { size 8{b} } Y rSub { size 8{4} } } {} (4)
From the voltage follower characteristics, we have Vb=V0Vb=V0 size 12{V rSub { size 8{b} } =V rSub { size 8{0} } } {}. Therefore, Equation 4 becomes
V a = V b ( Y 2 + Y 4 Y 2 ) = V 0 ( Y 2 + Y 4 Y 2 ) V a = V b ( Y 2 + Y 4 Y 2 ) = V 0 ( Y 2 + Y 4 Y 2 ) size 12{V rSub { size 8{a} } =V rSub { size 8{b} } \( { {Y rSub { size 8{2} } +Y rSub { size 8{4} } } over {Y rSub { size 8{2} } } } \) =V rSub { size 8{0} } \( { {Y rSub { size 8{2} } +Y rSub { size 8{4} } } over {Y rSub { size 8{2} } } } \) } {} (5)
Substituting Equation 5 into Equation 3 and again noting that Vb=V0Vb=V0 size 12{V rSub { size 8{b} } =V rSub { size 8{0} } } {}, we have
V i Y 1 + V 0 ( Y 2 + Y 3 ) = V a ( Y 1 + Y 2 + Y 3 ) = V 0 ( Y 2 + Y 4 Y 2 ) ( Y 1 + Y 2 + Y 3 ) V i Y 1 + V 0 ( Y 2 + Y 3 ) = V a ( Y 1 + Y 2 + Y 3 ) = V 0 ( Y 2 + Y 4 Y 2 ) ( Y 1 + Y 2 + Y 3 ) size 12{V rSub { size 8{i} } Y rSub { size 8{1} } +V rSub { size 8{0} } \( Y rSub { size 8{2} } +Y rSub { size 8{3} } \) =V rSub { size 8{a} } \( Y rSub { size 8{1} } +Y rSub { size 8{2} } +Y rSub { size 8{3} } \) =V rSub { size 8{0} } \( { {Y rSub { size 8{2} } +Y rSub { size 8{4} } } over {Y rSub { size 8{2} } } } \) \( Y rSub { size 8{1} } +Y rSub { size 8{2} } +Y rSub { size 8{3} } \) } {} (6)
Multiplying Equation 6 and rearranging terms, we get the following expression for the transfer function:
T ( s ) = V 0 ( s ) V i ( s ) = Y 1 Y 2 Y 1 Y 2 + Y 4 ( Y 1 + Y 2 + Y 3 ) T ( s ) = V 0 ( s ) V i ( s ) = Y 1 Y 2 Y 1 Y 2 + Y 4 ( Y 1 + Y 2 + Y 3 ) size 12{T \( s \) = { {V rSub { size 8{0} } \( s \) } over {V rSub { size 8{i} } \( s \) } } = { {Y rSub { size 8{1} } Y rSub { size 8{2} } } over {Y rSub { size 8{1} } Y rSub { size 8{2} } +Y rSub { size 8{4} } \( Y rSub { size 8{1} } +Y rSub { size 8{2} } +Y rSub { size 8{3} } \) } } } {} (7)
Figure 5: General two pole active filter.
To obtain a low-pass filter, both Y1Y1 size 12{Y rSub { size 8{1} } } {} and Y2Y2 size 12{Y rSub { size 8{2} } } {} must be conductances, allowing the signal to pass into the voltage follower at low frequencies. If element Y4Y4 size 12{Y rSub { size 8{4} } } {} is a capacitor, then the output rolloff at high frequencies.
To produce a two-pole function, element Y3Y3 size 12{Y rSub { size 8{3} } } {} must also be a capacitor. On the other hand, if elements Y1Y1 size 12{Y rSub { size 8{1} } } {} and Y2Y2 size 12{Y rSub { size 8{2} } } {} are capacitors, then the signal will be blocked at low frequencies but will be passed into the voltage follower at high frequencies resulting in a high-pass filter. Therefore, admittances Y3Y3 size 12{Y rSub { size 8{3} } } {} and Y4Y4 size 12{Y rSub { size 8{4} } } {} must both be conductances to produce a two-pole high-pass transfer function.

Two-Pole Low-Pass Butterworth Filter

To form a low-pass filter, we set Y1=G1=1/R1Y1=G1=1/R1 size 12{Y rSub { size 8{1} } =G rSub { size 8{1} } = {1} slash {R rSub { size 8{1} } } } {}, Y2=G2=1/R2Y2=G2=1/R2 size 12{Y rSub { size 8{2} } =G rSub { size 8{2} } = {1} slash {R rSub { size 8{2} } } } {}, Y3=sC3Y3=sC3 size 12{Y rSub { size 8{3} } = ital "sC" rSub { size 8{3} } } {} and Y4=sC4Y4=sC4 size 12{Y rSub { size 8{4} } = ital "sC" rSub { size 8{4} } } {}, as shown in Figure 6. The transfer function, from Equation 7, becomes
T ( s ) = V 0 ( s ) V i ( s ) = G 1 G 2 G 1 G 2 + sC 4 ( G 1 + G 2 + sC 3 ) T ( s ) = V 0 ( s ) V i ( s ) = G 1 G 2 G 1 G 2 + sC 4 ( G 1 + G 2 + sC 3 ) size 12{T \( s \) = { {V rSub { size 8{0} } \( s \) } over {V rSub { size 8{i} } \( s \) } } = { {G rSub { size 8{1} } G rSub { size 8{2} } } over {G rSub { size 8{1} } G rSub { size 8{2} } + ital "sC" rSub { size 8{4} } \( G rSub { size 8{1} } +G rSub { size 8{2} } + ital "sC" rSub { size 8{3} } \) } } } {} (8)
At zero frequency, s = j ωω size 12{ω} {} = 0 and the transfer function is
T ( s = 0 ) = G 1 G 2 G 1 G 2 = 1 T ( s = 0 ) = G 1 G 2 G 1 G 2 = 1 size 12{T \( s=0 \) = { {G rSub { size 8{1} } G rSub { size 8{2} } } over {G rSub { size 8{1} } G rSub { size 8{2} } } } =1} {} (9)
In the high frequency limit, s=jω→∞s=jω→∞ size 12{s=jω rightarrow infinity } {} and the transfer function approaches zero. This circuit therefore acts as a low-pass filter.
Figure 6: General two pass filter.
A butterworth filter is a maximally flat magnitude filter. The transfer function is designed such that the magnitude of the transfer function is as flat as possible within the passband of the filter. This objective is achieved by taking the derivatives of the transfer function with respect to frequency and setting as many as possible equal to zero at the center of the passband, which is at zero frequency for the low-pass filter.
Let G1=G2=G=1/RG1=G2=G=1/R size 12{G rSub { size 8{1} } =G rSub { size 8{2} } =G= {1} slash {R} } {}. the transfer function is then
T ( s ) = 1 R 2 1 R 2 + sC 4 ( 2 R + sC 3 ) = 1 1 + sRC 4 ( 2 + sRC 3 ) T ( s ) = 1 R 2 1 R 2 + sC 4 ( 2 R + sC 3 ) = 1 1 + sRC 4 ( 2 + sRC 3 ) size 12{T \( s \) = { { { {1} over {R rSup { size 8{2} } } } } over { { {1} over {R rSup { size 8{2} } } } + ital "sC" rSub { size 8{4} } \( { {2} over {R} } + ital "sC" rSub { size 8{3} } \) } } = { {1} over {1+ ital "sRC" rSub { size 8{4} } \( 2+ ital "sRC" rSub { size 8{3} } \) } } } {} (10)
We define time constant as τ3=RC3τ3=RC3 size 12{τ rSub { size 8{3} } = ital "RC" rSub { size 8{3} } } {} and τ4=RC4τ4=RC4 size 12{τ rSub { size 8{4} } = ital "RC" rSub { size 8{4} } } {}. If we then set s = j ωω size 12{ω} {}, we obtain
T ( jω ) = 1 1 + j ωτ 4 ( 2 + j ωτ 3 ) = 1 ( 1 − ω 2 τ 3 τ 4 ) + j ( 2 ωτ 4 ) T ( jω ) = 1 1 + j ωτ 4 ( 2 + j ωτ 3 ) = 1 ( 1 − ω 2 τ 3 τ 4 ) + j ( 2 ωτ 4 ) size 12{T \( jω \) = { {1} over {1+j ital "ωτ" rSub { size 8{4} } \( 2+j ital "ωτ" rSub { size 8{3} } \) } } = { {1} over { \( 1 - ω rSup { size 8{2} } τ rSub { size 8{3} } τ rSub { size 8{4} } \) +j \( 2 ital "ωτ" rSub { size 8{4} } \) } } } {} (11)
The magnitude of the transfer function is therefore
/ T ( jω ) / = [ ( 1 − ω 2 τ 3 τ 4 ) 2 ] − 1 / 2 / T ( jω ) / = [ ( 1 − ω 2 τ 3 τ 4 ) 2 ] − 1 / 2 size 12{ lline T \( jω \) rline = \[ \( 1 - ω rSup { size 8{2} } τ rSub { size 8{3} } τ rSub { size 8{4} } \) rSup { size 8{2} } \] rSup { size 8{ - 1/2} } } {} (12)
For the maximally flat filter (that is, a filter with a minimum rate of change), which defines a Butterworth filter, we set
d / T / dω / ω = 0 = 0 d / T / dω / ω = 0 = 0 size 12{ { {d lline T rline } over {dω} } \rline rSub { size 8{ω=0} } =0} {} (13)
Taking the derivative, we find
d / T / dω = 1 2 [ ( 1 − ω 2 τ 3 τ 4 ) 2 + ( 2 ωτ 4 ) 2 ] − 3 / 2 [ − 4 ωτ 3 τ 4 ( 1 − ω 2 τ 3 τ 4 ) + 8 ωτ 4 2 ] d / T / dω = 1 2 [ ( 1 − ω 2 τ 3 τ 4 ) 2 + ( 2 ωτ 4 ) 2 ] − 3 / 2 [ − 4 ωτ 3 τ 4 ( 1 − ω 2 τ 3 τ 4 ) + 8 ωτ 4 2 ] size 12{ { {d lline T rline } over {dω} } = { {1} over {2} } \[ \( 1 - ω rSup { size 8{2} } τ rSub { size 8{3} } τ rSub { size 8{4} } \) rSup { size 8{2} } + \( 2 ital "ωτ" rSub { size 8{4} } \) rSup { size 8{2} } \] rSup { size 8{ - 3/2} } \[ - 4 ital "ωτ" rSub { size 8{3} } τ rSub { size 8{4} } \( 1 - ω rSup { size 8{2} } τ rSub { size 8{3} } τ rSub { size 8{4} } \) +8 ital "ωτ" rSub { size 8{4} } rSup { size 8{2} } \] } {} (14)
Setting the derivative equal to zero at ω=0ω=0 size 12{ω=0} {} yields
d / T / dω / ω = 0 = [ − 4 ωτ 3 τ 4 ( 1 − ω 2 τ 3 τ 4 ) + 8 ωτ 4 2 ] = 4 ωτ 4 [ − τ 3 ( 1 − ω 2 τ 3 τ 4 ) + 2τ 4 ] d / T / dω / ω = 0 = [ − 4 ωτ 3 τ 4 ( 1 − ω 2 τ 3 τ 4 ) + 8 ωτ 4 2 ] = 4 ωτ 4 [ − τ 3 ( 1 − ω 2 τ 3 τ 4 ) + 2τ 4 ] size 12{ { {d lline T rline } over {dω} } \rline rSub { size 8{ω=0} } = \[ - 4 ital "ωτ" rSub { size 8{3} } τ rSub { size 8{4} } \( 1 - ω rSup { size 8{2} } τ rSub { size 8{3} } τ rSub { size 8{4} } \) +8 ital "ωτ" rSub { size 8{4} } rSup { size 8{2} } \] =4 ital "ωτ" rSub { size 8{4} } \[ - τ rSub { size 8{3} } \( 1 - ω rSup { size 8{2} } τ"" lSub { size 8{3} } τ rSub { size 8{4} } \) +2τ rSub { size 8{4} } \] } {} (15)
Equation 15 is satisfied when 2τ4=τ32τ4=τ3 size 12{2τ rSub { size 8{4} } =τ rSub { size 8{3} } } {}, or
C 3 = 2C 4 C 3 = 2C 4 size 12{C rSub { size 8{3} } =2C rSub { size 8{4} } } {} (16)
Based on this condition, the transfer function magnitude is, from Equation 12,
/ T / = 1 [ 1 + 4 ( ωτ 4 ) 4 ] 1 / 2 / T / = 1 [ 1 + 4 ( ωτ 4 ) 4 ] 1 / 2 size 12{ lline T rline = { {1} over { \[ 1+4 \( ital "ωτ" rSub { size 8{4} } \) rSup { size 8{4} } \] rSup { size 8{1/2} } } } } {} (17)
The 3 dB, or cutoff, frequency occurs when /T/=1/2/T/=1/2 size 12{ lline T rline =1/ sqrt {2} } {}, or when 4(ω3dBτ4)=14(ω3dBτ4)=1 size 12{4 \( ω rSub { size 8{3 ital "dB"} } τ rSub { size 8{4} } \) =1} {}. We then find that
ω 3 dB = 2πf 3 dB = 1 τ 4 2 = 1 2 RC 4 ω 3 dB = 2πf 3 dB = 1 τ 4 2 = 1 2 RC 4 size 12{ω"" lSub { size 8{3 ital "dB"} } =2πf rSub { size 8{3 ital "dB"} } = { {1} over {τ rSub { size 8{4} } sqrt {2} } } = { {1} over { sqrt {2} ital "RC" rSub { size 8{4} } } } } {} (18)
In general, we can write the cutoff frequency in the form
ω 3 dB = 1 RC ω 3 dB = 1 RC size 12{ω"" lSub { size 8{3 ital "dB"} } = { {1} over { ital "RC"} } } {} (19)
Finally, comparing Equation 19, Equation 18 and Equation 16 yields
C 4 = 0 . 707 C C 4 = 0 . 707 C size 12{C rSub { size 8{4} } =0 "." "707"C} {} (20)
and
C 3 = 1 . 414 C C 3 = 1 . 414 C size 12{C rSub { size 8{3} } =1 "." "414"C} {} (21)
The two-pole low-pass Butterworth filter is shown in Figure 7(a). The Bode plot of the transfer function magnitude is shown in Figure 7(b). From Equation 17, the magnitude of the voltage transfer function for the two pole low-pass Butterworth filter, can be written as
/ T / = 1 1 + ( f f 3 dB ) 4 / T / = 1 1 + ( f f 3 dB ) 4 size 12{ lline T rline = { {1} over { sqrt {1+ \( { {f} over {f rSub { size 8{3 ital "dB"} } } } \) rSup { size 8{4} } } } } } {} (22)
Figure 7: a) Two-pole low-pass Butterworth filter and b)Bote plot transfer function magnitude.
Equation 15 shows that the derivative of the voltage transfer function magnitude at ω=0ω=0 size 12{ω=0} {} is zero even without setting 2τ4=τ32τ4=τ3 size 12{2τ rSub { size 8{4} } =τ rSub { size 8{3} } } {}. However, the added condition of 2τ4=τ32τ4=τ3 size 12{2τ rSub { size 8{4} } =τ rSub { size 8{3} } } {} produces the maximally flat transfer characteristics of the Butterworth filter.

Two-Pole High-Pass Butterworth Filter

To perform a high-pass filter, the resistors and capacitors are interchanging from those in the low-pass filter. A two-pole high-pass Butterworth filter is shown in Figure 8(a). The analysis proceeds exactly the same as in the last section, except that the derivative is set equal to zero at s=jω=∞s=jω=∞ size 12{s=jω= infinity } {}. Also, the capacitors are set equal to each other. The 3dB or cutoff frequency can be written in the general form
ω 3 dB = 2πf 3 dB = 1 RC ω 3 dB = 2πf 3 dB = 1 RC size 12{ω rSub { size 8{3 ital "dB"} } =2πf rSub { size 8{3 ital "dB"} } = { {1} over { ital "RC"} } } {} (23)
Figure 8: a) Two-pole high-pass Butterworth filter and b) Bode plot transfer function magnitude.
We find that R3=0.707RR3=0.707R size 12{R rSub { size 8{3} } =0 "." "707"R} {} and R4=1.414RR4=1.414R size 12{R rSub { size 8{4} } =1 "." "414"R} {}. The magnitude of the voltage transfer function for the two-pole high-pass Butterworth is
/ T / = 1 1 + ( f 3 dB f ) 4 / T / = 1 1 + ( f 3 dB f ) 4 size 12{ lline T rline = { {1} over { sqrt {1+ \( { {f rSub { size 8{3 ital "dB"} } } over {f} } \) rSup { size 8{4} } } } } } {} (24)
The Bode plot of the transfer function magnitude for the two-pole high-pass Butterworth filter is shown in Figure 8(b).

High-Order Butterworth Filters

The filter order is the number of poles and is usually dictated by the application requirements. An N-pole active low-pass filter has a high-frequency rolloff rate of Nx6 dB/octave. Similarly, the response of an N-pole high-pass filter increases at a rate of Nx6 dB/octave, up to the cutoff frequency. In each case, the 3 dB frequency is defined as
f 3 dB = 1 2π RC f 3 dB = 1 2π RC size 12{f rSub { size 8{3 ital "dB"} } = { {1} over {2π ital "RC"} } } {} (25)
The magnitude of the voltage transfer function for a Butterworth Nth-order low-pass filter is
/ T / = 1 1 + ( f f 3 dB ) 2N / T / = 1 1 + ( f f 3 dB ) 2N size 12{ lline T rline = { {1} over { sqrt {1+ \( { {f} over {f rSub { size 8{3 ital "dB"} } } } \) rSup { size 8{2N} } } } } } {} (26)
For a Butterworth Nth-order high-pass filter, the voltage transfer function magnitude is
/ T / = 1 1 + ( f 3 dB f ) 2N / T / = 1 1 + ( f 3 dB f ) 2N size 12{ lline T rline = { {1} over { sqrt {1+ \( { {f rSub { size 8{3 ital "dB"} } } over {f} } \) rSup { size 8{2N} } } } } } {} (27)
Figure 9(a) shows a three-pole low-pass Butterworth filter. The three resistors are equal, and the relationship between the capacitors is found by taking the first and second derivatives of the voltage gain magnitude with respect to frequency and setting those derivatives equal to zero at s=jω=0s=jω=0 size 12{s=jω=0} {}. Figure 9(b) shows a three-pole high-pass Butterworth filter. In this case, the three capacitors are equal and the relationship between the resistors is also found through the derivatives.
Higher-order filters can be created by adding additional RC networks. However, the loading effect on each additional RC circuit becomes more severe. The usefulness of active filters is realized when two or more op-amp filter circuits are cascaded to produce one large higher-order active filter. Because of the low output impedance of the op-amp, there is virtually no loading effect between cascaded stages.
Figure 9: a) Three-pole low-pass Butterworth fielter and b)Three-pole high-pass Butterworth fielter.
Figure 10(a) shows a four-pole low-pass Butterworth filter. The maximally flat response of this filter is not obtained by simply cascading two two-pole filters. The relationship between the capacitors is found through the first three derivatives of the transfer function. The four-pole high-pass Butterworth filter is shown in Figure 10(b).
Higher-order filters can be designed but are not considered here. Bandpass and band-reject filters use similar circuit configurations.
Figure 10: Four-pole low pass Butterworth filter and b)four-pole high-pass Butterworth filter.

OSCILLATORS

In this section, we will look at the basic principles of sine-wave oscillators. In our study of feedback, we emphasized the need for negative feedback to provide a stable circuit. Oscillators, however, use positive feedback and, as such, are actually nonlinear circuits in some cases. The analysis and design of oscillator circuits are divided into two parts. In this first part, the condition and frequency for oscillation are determined; in the second part, means for amplitude control is addressed. We consider only the first step in this section to gain insight into the basic operation of oscillators.

Basic Principle of Oscillation

The basic oscillator consists of an amplifier and a frequency-selective network connected in a feedback loop. Figure 11 shows a block diagram the fundamental feedback circuit, in which we are implicitly assuming that negative feedback is employed. Although actual oscillator circuits do not have an input signal, we initially include one here to help in the analysis. In previous feedback circuits, we assumed the feedback transfer function ββ size 12{β} {} was independent of frequency. In oscillator circuits, however, ββ size 12{β} {} is the principal portion of the loop gain that is dependent on frequency.
Figure 11: Block diagram of the fundamental feedback circuit.
For the circuit shown, the ideal closed-loop transfer function is given by
A f ( s ) = A ( s ) 1 + A ( s ) β ( s ) A f ( s ) = A ( s ) 1 + A ( s ) β ( s ) size 12{A rSub { size 8{f} } \( s \) = { {A \( s \) } over {1+A \( s \) β \( s \) } } } {} (28)
And the loop gain of the feedback circuit is
T ( s ) = A ( s ) β ( s ) T ( s ) = A ( s ) β ( s ) size 12{T \( s \) =A \( s \) β \( s \) } {} (29)
We know that the loop gain T(s) is positive for negative feedback, which means that the feedback signal vfbvfb size 12{v rSub { size 8{ ital "fb"} } } {} subtracts from the input signal vsvs size 12{v rSub { size 8{s} } } {}. If T(s) = - 1, the closed-loop transfer function goes to infinity, which means that the circuit can have a finite output for a zero input signal.
As T(s) approaches -1, an actual circuit becomes nonlinear, which means that the gain does not go to infinity. Assume that T(s) = -1 so that positive feedback exists over a particular frequency range. If a spontaneous signal (due to noise) is created at vsvs size 12{v rSub { size 8{s} } } {}, and the error signal vsvs size 12{v rSub { size 8{s} } } {} is reinforced and increased. This reinforcement process continues at only those frequencies for which the total phase shift around the feedback loop is zero. Therefore, the condition for oscillation is that, at a specific frequency, we have
T ( jω 0 ) = A ( jω 0 ) β ( jω 0 ) = − 1 T ( jω 0 ) = A ( jω 0 ) β ( jω 0 ) = − 1 size 12{T \( jω rSub { size 8{0} } \) =A \( jω rSub { size 8{0} } \) β \( jω rSub { size 8{0} } \) = - 1} {} (30)
The condition that T(jω0)=−1T(jω0)=−1 size 12{T \( jω rSub { size 8{0} } \) = - 1} {} is called the Barkhausen criterion.
Equation 30 shows that two conditions must be satisfied to sustain oscillation:
  1. The total phase shift through the amplifier and feedback network must be Nx 36003600 size 12{"360" rSup { size 8{0} } } {}, where N = 0, 1, 2, …,
  2. The magnitude of the loop gain must be unity.
In the feedback circuit block diagram shown in Figure 11, we implicitly assume negative feedback. For an oscillator, the feedback transfer function, or the frequency-selective network, must introduce an additional 180 degree phase shift such that the net phase around the entire loop is zero. For the circuit to oscillate at a single frequency ω0ω0 size 12{ω rSub { size 8{0} } } {}, the condition for oscillation, from Equation 30, should be satisfied at only that one frequency.

Phase-Shift Oscillator

An example of an op-amp oscillator is the phase-shift oscillator. One configuration of this oscillator circuit is shown in Figure 12. The basic amplifier of the circuit is the op-amp A3A3 size 12{A rSub { size 8{3} } } {}, which is connected as an inverting amplifier with its output connected to a three-stage RC filter. The voltage followers in the circuit eliminate loading effects between each RC filter stage.
Figure 12: Phase-shift oscillator circuit with voltage follower buffer stages.
The inverting amplifier introduces a – 180 degree phase shift, which means that each RC network must provide 60 degrees of phase shift to produce 180 degrees required of the frequency-sensitive feedback network in order to produce positive feedback. Note that the inverting terminal of op-amp A3 is at virtual ground; therefore, the RC network between op-amps A2A2 size 12{A rSub { size 8{2} } } {} and A3A3 size 12{A rSub { size 8{3} } } {} functions exactly as the other two RC networks. We assume that the frequency effects of the op-amps themselves occur at much higher frequencies than the response due to the RC networks. Also, to aid in the analysis, we assume an input signal ( vivi size 12{v rSub { size 8{i} } } {}) exists at one node as shown in the figure.
The transfer function of the first RC network is
v 1 = ( sRC 1 + sRC ) ( v i ) v 1 = ( sRC 1 + sRC ) ( v i ) size 12{v rSub { size 8{1} } = \( { { ital "sRC"} over {1+ ital "sRC"} } \) \( v rSub { size 8{i} } \) } {} (31)
Since the RC networks are assumed to be identical, and since there is no loading effect of one RC stage on another, we have
v 3 v i = ( sRC 1 + sRC ) 3 = β ( s ) v 3 v i = ( sRC 1 + sRC ) 3 = β ( s ) size 12{ { {v rSub { size 8{3} } } over {v rSub { size 8{i} } } } = \( { { ital "sRC"} over {1+ ital "sRC"} } \) rSup { size 8{3} } =β \( s \) } {} (32)
where β(s)β(s) size 12{β \( s \) } {} is the feedback transfer function. The amplifier gain A(s) in Equation 28 and Equation 29 is actually the magnitude of the gain, or
A ( s ) = / v 0 v i / = R 2 R A ( s ) = / v 0 v i / = R 2 R size 12{A \( s \) = lline { {v rSub { size 8{0} } } over {v rSub { size 8{i} } } } rline = { {R rSub { size 8{2} } } over {R} } } {} (33)
The loop gain is then
T ( s ) = A ( s ) β ( s ) = R 2 R ( sRC 1 + sRC ) 3 T ( s ) = A ( s ) β ( s ) = R 2 R ( sRC 1 + sRC ) 3 size 12{T \( s \) =A \( s \) β \( s \) = { {R rSub { size 8{2} } } over {R} } \( { { ital "sRC"} over {1+ ital "sRC"} } \) rSup { size 8{3} } } {} (34)
From Equation 31, the condition for oscillation is that /T(jω0)/=1/T(jω0)/=1 size 12{ lline T \( jω rSub { size 8{0} } \) rline =1} {} and the phase of T(jω0)T(jω0) size 12{T \( jω rSub { size 8{0} } \) } {} must be 180 degrees. When these requirements are satisfied, then v0 will equal ( vivi size 12{v rSub { size 8{i} } } {}) and a separate input signal will not be required.
If we set s = j, Equation 33 becomes
T ( jω ) = R 2 R ( jω RC 1 + jω RC ) 3 = − ( R 2 R ) ( jω RC ) ( ω RC ) 2 [ 1 − 3ω 2 R 2 C 2 ] + jω RC [ 3 − ω 2 R 2 C 2 ] T ( jω ) = R 2 R ( jω RC 1 + jω RC ) 3 = − ( R 2 R ) ( jω RC ) ( ω RC ) 2 [ 1 − 3ω 2 R 2 C 2 ] + jω RC [ 3 − ω 2 R 2 C 2 ] size 12{T \( jω \) = { {R rSub { size 8{2} } } over {R} } \( { {jω ital "RC"} over {1+jω ital "RC"} } \) rSup { size 8{3} } = - \( { {R rSub { size 8{2} } } over {R} } \) { { \( jω ital "RC" \) \( ω ital "RC" \) rSup { size 8{2} } } over { \[ 1 - 3ω rSup { size 8{2} } R rSup { size 8{2} } C rSup { size 8{2} } \] +jω ital "RC" \[ 3 - ω rSup { size 8{2} } R rSup { size 8{2} } C rSup { size 8{2} } \] } } } {} (35)
To satisfy the condition T(jω0)=−1T(jω0)=−1 size 12{T \( jω rSub { size 8{0} } \) = - 1} {}, the imaginary component of Equation 32 must equal zero. Since the numerator is purely imaginary, the denominator must become purely imaginary, or {1−3ω2R2C2}=0{1−3ω2R2C2}=0 size 12{ lbrace 1 - 3ω rSup { size 8{2} } R rSup { size 8{2} } C rSup { size 8{2} } rbrace =0} {} which yields
ω = 1 3 RC ω = 1 3 RC size 12{ω= { {1} over { sqrt {3} ital "RC"} } } {} (36)
where ωω size 12{ω} {} is the oscillation frequency. At this frequency, Equation 35 becomes
T ( jω ) = − R 2 R ( j / 3 ) ( 1 / 3 ) 0 + ( j / 3 ) [ 3 − ( 1 / 3 ) ] = − R 2 R ( 1 8 ) T ( jω ) = − R 2 R ( j / 3 ) ( 1 / 3 ) 0 + ( j / 3 ) [ 3 − ( 1 / 3 ) ] = − R 2 R ( 1 8 ) size 12{T \( jω \) = - { {R rSub { size 8{2} } } over {R} } { { \( j/ sqrt {3} \) \( 1/3 \) } over {0+ \( j/ sqrt {3} \) \[ 3 - \( 1/3 \) \] } } = - { {R rSub { size 8{2} } } over {R} } \( { {1} over {8} } \) } {} (37)
Consequently, the condition T(jω)==1T(jω)==1 size 12{T \( jω \) "=="1} {} is satisfied when
R 2 R = 8 R 2 R = 8 size 12{ { {R rSub { size 8{2} } } over {R} } =8} {} (38)
Equation 38 implies that if the magnitude of the inverting amplifier gain is greater than 8, the circuit will spontaneous begin oscillating and will sustain oscillation.
Using Equation 31, we can determine the effect of each RC network in the phase-shift oscillator. At the oscillation frequency ω0ω0 size 12{ω rSub { size 8{0} } } {}, the transfer function of each RC network stage is
jω 0 RC 1 + jω 0 RC = ( j / 3 ) 1 + ( j / 3 ) = j 3 + j jω 0 RC 1 + jω 0 RC = ( j / 3 ) 1 + ( j / 3 ) = j 3 + j size 12{ { {jω rSub { size 8{0} } ital "RC"} over {1+jω rSub { size 8{0} } ital "RC"} } = { { \( j/ sqrt {3} \) } over {1+ \( j/ sqrt {3} \) } } = { {j} over { sqrt {3} +j} } } {} (39)
which can be written in terms of the magnitude and phase, as follows:
1 2 × [ < 90 0 − 30 0 ] = 1 2 × 60 0 1 2 × [ < 90 0 − 30 0 ] = 1 2 × 60 0 size 12{ { {1} over {2} } times \[ <"90" rSup { size 8{0} } - <"30" rSup { size 8{0} } \] = { {1} over {2} } times <"60" rSup { size 8{0} } } {} (40)
or
1 2 × [ < 90 0 − 30 0 ] = 1 2 × 60 0 1 2 × [ < 90 0 − 30 0 ] = 1 2 × 60 0 size 12{ { {1} over {2} } times \[ <"90" rSup { size 8{0} } - <"30" rSup { size 8{0} } \] = { {1} over {2} } times <"60" rSup { size 8{0} } } {} (41)
as required each RC network introduces a 60 degree phase shift, but they each also introduce an attenuation factor of (1/2) for which the amplifier must compensate.
The two voltage followers shown in the circuit in Figure 12 need not be included in a practical phase shift oscillator. Figure 13 shows a phase shift oscillator without the voltage-follower buffer stages. The three RC network stages and the inverting amplifier are still included. The loading effect of each successive RC network complicates the analysis, but the same principle of operation applies. The analysis shows that the oscillation frequency is
ω 0 = 1 6 RC ω 0 = 1 6 RC size 12{ω rSub { size 8{0} } = { {1} over { sqrt {6} ital "RC"} } } {} (42)
Figure 13: Phase shift oscillator circuit.
and the amplifier resistor ratio must be
R 2 R = 29 R 2 R = 29 size 12{ { {R rSub { size 8{2} } } over {R} } ="29"} {} (43)
in order to sustain oscillation.

Wien-Bridge Oscillator

Another basic oscillator is the Wien-bridge circuit, shown in Figure 14. The circuit consists of an op-amp connected in a non-inverting configuration and two RC networks connected as the frequency-selecting feedback circuit.
Again, we initially assume that an input signal exists at the non-inverting terminal of the op-amp. Since the non-inverting amplifier introduces zero phase shift, the frequency-selective feedback circuit must also introduce zero phase shift to create the positive feedback condition.
Figure 14: Wien-bridge oscillation.
The loop gain is the product of the amplifier gain and the feedback transfer function, or
T ( s ) = ( 1 + R 2 R 1 ) ( Z p Z p + Z s ) T ( s ) = ( 1 + R 2 R 1 ) ( Z p Z p + Z s ) size 12{T \( s \) = \( 1+ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } \) \( { {Z rSub { size 8{p} } } over {Z rSub { size 8{p} } +Z rSub { size 8{s} } } } \) } {} (44)
where ZpZp size 12{Z rSub { size 8{p} } } {} and ZsZs size 12{Z rSub { size 8{s} } } {} are the parallel and series RC network impedances, respectively. These impedances are
Z p = R 1 + sRC Z p = R 1 + sRC size 12{Z rSub { size 8{p} } = { {R} over {1+ ital "sRC"} } } {} (45)
and
Z s = 1 + sRC sRC Z s = 1 + sRC sRC size 12{Z rSub { size 8{s} } = { {1+ ital "sRC"} over { ital "sRC"} } } {} (46)
Combining Equation 45(a), Equation 46(b) and Equation 44, we get expression for the loop gain function,
T ( s ) = ( 1 + R 2 R 1 ) ( 1 3 + sRC + 1 3 + sRC + ( 1 / sRC ) ) T ( s ) = ( 1 + R 2 R 1 ) ( 1 3 + sRC + 1 3 + sRC + ( 1 / sRC ) ) size 12{T \( s \) = \( 1+ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } \) \( { {1} over {3+ ital "sRC"+ left ( { {1} over {3+ ital "sRC"+ \( 1/ ital "sRC" \) } } right )} } \) } {} (47)
Since this circuit has no explicit negative feedback, as was assumed in the general network shown in Figure 11, the condition for oscillation is given by
T ( jω 0 ) = 1 = ( 1 + R 2 R 1 ) ( 1 3 + jω 0 RC + ( 1 / jω 0 RC ) ) T ( jω 0 ) = 1 = ( 1 + R 2 R 1 ) ( 1 3 + jω 0 RC + ( 1 / jω 0 RC ) ) size 12{T \( jω rSub { size 8{0} } \) =1= \( 1+ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } \) \( { {1} over {3+jω rSub { size 8{0} } ital "RC"+ \( 1/jω rSub { size 8{0} } ital "RC" \) } } \) } {} (48)
Since T(jω0)T(jω0) size 12{T \( jω rSub { size 8{0} } \) } {} must be real, the imaginary component of Equation 48 must be zero; therefore,
jω 0 RC + 1 jω 0 RC = 0 jω 0 RC + 1 jω 0 RC = 0 size 12{jω rSub { size 8{0} } ital "RC"+ { {1} over {jω rSub { size 8{0} } ital "RC"} } =0} {} (49)
which gives the frequency of oscillation as
ω 0 = 1 RC ω 0 = 1 RC size 12{ω rSub { size 8{0} } = { {1} over { ital "RC"} } } {} (50)
The magnitude condition is then
1 = ( 1 + R 2 R 1 ) ( 1 3 ) 1 = ( 1 + R 2 R 1 ) ( 1 3 ) size 12{1= \( 1+ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } \) \( { {1} over {3} } \) } {} (51)
or
R 2 R 1 = 2 R 2 R 1 = 2 size 12{ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } =2} {} (52)
Equation 52 (b) states that to ensure the startup of oscillation, we must have (R2/R1) > 2.

Additional Oscillator Configurations

Oscillators that use transistors and LC tuned circuits or crystals in their feedback networks can be used in the hundreds of kHz to hundreds of MHz frequency range. Although these oscillators do not typically contain an op-amp, we include a brief discussion of such circuits for completeness. We will examine the Colpitts, Harley, and crystal oscillators.
Colpitts Oscillators
The ac equivalent circuit of the Colpitts oscillator with an FET is shown in Figure 15. A circuit with BJT can be designed. A parallel LC resonant circuit is used to establish the oscillator frequency, and feedback is provided by a voltage divider between capacitors C1C1 size 12{C rSub { size 8{1} } } {} and C2C2 size 12{C rSub { size 8{2} } } {}. Resistor R in conjunction with the transistor provides the necessary gain at resonance. We assume that the transistor frequency response occurs at a high enough frequency that the oscillation frequency is determined by the external elements only.
Figure 16 shows the small-signal equivalent circuit of the Colpitts oscillator. The transistor output resistance r0 can be included in R. A KCL equation at the output node yields
V 0 ( 1 sC 1 ) + V 0 R + g m V gs + V 0 sL + 1 sC 2 = 0 V 0 ( 1 sC 1 ) + V 0 R + g m V gs + V 0 sL + 1 sC 2 = 0 size 12{ { {V rSub { size 8{0} } } over { \( { {1} over { ital "sC" rSub { size 8{1} } } } \) } } + { {V rSub { size 8{0} } } over {R} } +g rSub { size 8{m} } V rSub { size 8{ ital "gs"} } + { {V rSub { size 8{0} } } over { ital "sL"+ { {1} over { ital "sC" rSub { size 8{2} } } } } } =0} {} (53)
Figure 15: The ac equivalent circuit, MOSFET Colpitts oscillator.
Figure 16: Small-signal equivalent circuit, MOSFET Colpitts oscillator.
And a voltage divider produces
V gs = 1 sC 2 1 sC 2 + sL V 0 V gs = 1 sC 2 1 sC 2 + sL V 0 size 12{V rSub { size 8{ ital "gs"} } = { { { {1} over { ital "sC" rSub { size 8{2} } } } } over { { {1} over { ital "sC" rSub { size 8{2} } } } + ital "sL"} } V rSub { size 8{0} } } {} (54)
Substituting Equation 54 into Equation 53, we find that
V 0 [ g m + sC 2 + ( 1 + s 2 LC 2 ) ( 1 R + sC 1 ) ] = 0 V 0 [ g m + sC 2 + ( 1 + s 2 LC 2 ) ( 1 R + sC 1 ) ] = 0 size 12{V rSub { size 8{0} } \[ g rSub { size 8{m} } + ital "sC" rSub { size 8{2} } + \( 1+s rSup { size 8{2} } ital "LC" rSub { size 8{2} } \) \( { {1} over {R} } + ital "sC" rSub { size 8{1} } \) \] =0} {} (55)
If we assume that oscillation has started, then V0≠0V0≠0 size 12{V rSub { size 8{0} } <> 0} {} and can be eliminated from Equation 55. We then have
s 3 LC 1 C 2 + s 2 LC 2 R + s ( C 1 + C 2 ) + ( g m + 1 R ) = 0 s 3 LC 1 C 2 + s 2 LC 2 R + s ( C 1 + C 2 ) + ( g m + 1 R ) = 0 size 12{s rSup { size 8{3} } ital "LC" rSub { size 8{1} } C rSub { size 8{2} } + { {s rSup { size 8{2} } ital "LC" rSub { size 8{2} } } over {R} } +s \( C rSub { size 8{1} } +C rSub { size 8{2} } \) + \( g rSub { size 8{m} } + { {1} over {R} } \) =0} {} (56)
Letting s = j ωω size 12{ω} {}, we obtain
( g m + 1 R − ω 2 LC 2 R ) + jω [ ( C 1 + C 2 ) − ω 2 LC 1 C 2 ] = 0 ( g m + 1 R − ω 2 LC 2 R ) + jω [ ( C 1 + C 2 ) − ω 2 LC 1 C 2 ] = 0 size 12{ \( g rSub { size 8{m} } + { {1} over {R} } - { {ω rSup { size 8{2} } ital "LC" rSub { size 8{2} } } over {R} } \) +jω \[ \( C rSub { size 8{1} } +C rSub { size 8{2} } \) - ω rSup { size 8{2} } ital "LC" rSub { size 8{1} } C rSub { size 8{2} } \] =0} {} (57)
The condition for oscillation implies that both the real and imaginary components of Equation 57 must be zero. From the imaginary component, the oscillation frequency is
ω 0 = 1 L ( C 1 C 2 C 1 + C 2 ) ω 0 = 1 L ( C 1 C 2 C 1 + C 2 ) size 12{ω rSub { size 8{0} } = { {1} over { sqrt {L \( { {C rSub { size 8{1} } C rSub { size 8{2} } } over {C rSub { size 8{1} } +C rSub { size 8{2} } } } \) } } } } {} (58)
which is the resonant frequency of the LC circuit. From the real part of Equation 57, the condition for oscillation is
ω 0 2 LC 2 R = g m + 1 R ω 0 2 LC 2 R = g m + 1 R size 12{ { {ω rSub { size 8{0} } rSup { size 8{2} } ital "LC" rSub { size 8{2} } } over {R} } =g rSub { size 8{m} } + { {1} over {R} } } {} (59)
Combining Equation 58 and Equation 59 yields
C 2 C 1 = g m R C 2 C 1 = g m R size 12{ { {C rSub { size 8{2} } } over {C rSub { size 8{1} } } } =g rSub { size 8{m} } R} {} (60)
where gmRgmR size 12{g rSub { size 8{m} } R} {} is the magnitude of the gain. Equation 61 states that to initiate oscillations spontaneously, we must have gmRgmR size 12{g rSub { size 8{m} } R} {} > (C2/C1).
Harley Oscillator
Figure 17 shows the ac equivalent circuit of the Harley oscillator with a BJT. An FET can also be used. Again, a parallel LC resonant circuit establishes the oscillation frequency, and feedback is provided by a voltage divider between inductors L1L1 size 12{L rSub { size 8{1} } } {} and L2L2 size 12{L rSub { size 8{2} } } {}.
Figure 17: The ac equivalent circuit, BJT Harley oscillator.
The analysis of the Harley oscillator is essentially identical to that of the Colpitts oscillator. The frequency of oscillation, neglecting transistor frequency effects, is
ω 0 = 1 ( L 1 + L 2 ) C ω 0 = 1 ( L 1 + L 2 ) C size 12{ω rSub { size 8{0} } = { {1} over { sqrt { \( L rSub { size 8{1} } +L rSub { size 8{2} } \) C} } } } {} (61)
Crystal Oscillator
A piezoelectric crystal, such as quartz, exhibits electromechanical resonance characteristics in response to a voltage applied across the crystal. The oscillations are very stable over time and temperature coefficients on the order of 1 ppm per 0C0C size 12{"" lSup { size 8{0} } C} {}. The oscillation frequency is determined by the crystal dimensions. This means that crystal oscillators are fixed-frequency devices.
Figure 18: a) Piezoelectric crystal circuit symbol and b) piezoelectric crystal equivalent circuit.
The circuit symbol for the piezoelectric crystal is shown in Figure 18(a), and the equivalent circuit is shown in Figure 18(b). The inductance L can be as high as a few hundred Henrys, the capacitance C, can be on the order of 0.001 pF, and the capacitance CpCp size 12{C rSub { size 8{p} } } {} can be on the order of a few pF. Also, the Q-factor can be on the order of 104104 size 12{"10" rSup { size 8{4} } } {}, which means that the series resistance r can be neglected.
The impedance of the equivalent circuit in Figure 18(b) is
Z ( s ) = 1 sC p s 2 + ( 1 / LC s ) s 2 + [ ( C p + C s ) / ( LC s C p ) ] Z ( s ) = 1 sC p s 2 + ( 1 / LC s ) s 2 + [ ( C p + C s ) / ( LC s C p ) ] size 12{Z \( s \) = { {1} over { ital "sC" rSub { size 8{p} } } } { {s rSup { size 8{2} } + \( 1/ ital "LC" rSub { size 8{s} } \) } over {s rSup { size 8{2} } + \[ \( C rSub { size 8{p} } +C rSub { size 8{s} } \) / \( ital "LC" rSub { size 8{s} } C rSub { size 8{p} } \) \] } } } {} (62)
Equation 62 indicates that the crystal has two resonant frequencies, which are very close together. At the series-resonant frequency fsfs size 12{f rSub { size 8{s} } } {}, the reactance of the series branch is zero; at the parallel-resonant frequency fpfp size 12{f rSub { size 8{p} } } {}, the reactance of the crystal approaches infinity.
Between the resonant frequencies fsfs size 12{f rSub { size 8{s} } } {} and fpfp size 12{f rSub { size 8{p} } } {}, the crystal reactance is inductive, so the crystal can be substituted for an inductance, such as that in a Colpitts oscillator. Figure 19 shows the ac equivalent circuit of a Pierce oscillator, which is similar to the Colpitts oscillator shown in Figure 15 but with the inductor replaced by the crystal. Since the crystal reactance is inductive over a very narrow frequency range, the frequency of oscillation is also confined to this narrow range and is quite constant relative to changes in bias current of temperature. Crystal oscillator frequencies are usually in the range of tens of kHz to tens of MHz.

SCHMITT TRIGGER CIRCUITS

In this section, we will analyze another class of circuits that utilize positive feedback. The basic circuit is commonly called a Schmitt trigger, which can be used in the class of waveform generators called multivibrators. The three general types of multivibrators are: bistable, monostable, and astable. In this section, we will examine the bistable multivibrator, which has a comparator with positive feedback and has two stable states. We will discuss the comparator first, and will then describe various applications of the Schmitt trigger.

Comparator

The comparator is essentially an op-amp operated in an open-loop configuration, as shown in Figure 19(a). As the name implies, a comparator compares two voltages to determine which is larger. The comparator is usually biased at voltages +VS+VS size 12{+V rSub { size 8{S} } } {} and −VS−VS size 12{ - V rSub { size 8{S} } } {}, although other biases are possible.
Figure 19: A) Open-loop comparator and b) voltage transfer characteristics of voltage comparator.
The voltage transfer characteristics, neglecting any offset voltage effects, are shown in Figure 19(b). When v2v2 size 12{v rSub { size 8{2} } } {} is slightly greater than v1v1 size 12{v rSub { size 8{1} } } {}, the output is driven to a high saturated state VHVH size 12{V rSub { size 8{H} } } {}; when v2v2 size 12{v rSub { size 8{2} } } {} is slightly less than v1v1 size 12{v rSub { size 8{1} } } {}, the output is driven to a low saturated state VLVL size 12{V rSub { size 8{L} } } {}. The saturated output voltages VHVH size 12{V rSub { size 8{H} } } {} and VLVL size 12{V rSub { size 8{L} } } {} may be close to the supply voltages +VS+VS size 12{+V rSub { size 8{S} } } {} and −VS−VS size 12{ - V rSub { size 8{S} } } {}, respectively, which means that VLVL size 12{V rSub { size 8{L} } } {} may be negative. The transition region is the region in which the output voltage is in neither of its saturation states. This region occurs when the input differential voltage is in the range −δ<(v2−v1)<+δ−δ<(v2−v1)<+δ size 12{ - δ< \( v rSub { size 8{2} } - v rSub { size 8{1} } \) "<+"δ} {}. If, for example, the open-loop gain is 105 and the difference between the two output states is (VH−VL)=10V(VH−VL)=10V size 12{ \( V rSub { size 8{H} } - V rSub { size 8{L} } \) ="10"V} {}, then
2 δδ size 12{δ} {} = 10/ 105105 size 12{"10" rSup { size 8{5} } } {} = 10−410−4 size 12{"10" rSup { size 8{ - 4} } } {} V = 0.1 mV
The range of input differential voltage in the transition region is normally very small.
One major difference between a comparator and op-amp is that a comparator need not to be frequency compensated. Frequency stability is not a consideration since the comparator is being driven into one of two states. Since a comparator does not contain a frequency compensation capacitor, it is not slew-rate-limited by the compensation capacitor as is the op-amp. Typical response times for the comparator output to change states are in the range of 30 to 200 ns. An expected response time for a 741 op-amp with a slew rate of 0.7 V/ μμ size 12{μ} {}s would be on the order of 30 μμ size 12{μ} {}s, which is factor of 1000 times greater.
Figure 20 shows two comparator configurations along with their voltage transfer characteristics. In both, the input transition region width is assumed to be negligibly small. The reference voltage may be either positive or negative, and the output saturation voltages are assumed to be symmetrical about zero. The crossover voltage is defined as the input voltage at which the output changes states.
Figure 20: a) Non-inverting comparartor circuit and b) inverting circuit.
Two other comparator configurations, in which the crossover voltage is a function of resistor ratios, are shown in Figure 21. Input bias current compensation is also included in this figure. From Figure 21(a), we use superposition to obtain
v + = ( R 2 R 1 + R 2 ) V REF + ( R 1 R 1 + R 2 ) v i v + = ( R 2 R 1 + R 2 ) V REF + ( R 1 R 1 + R 2 ) v i size 12{v rSub { size 8{+{}} } = \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{ ital "REF"} } + \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) v rSub { size 8{i} } } {} (63)
The ideal crossover voltage occurs when v+=0v+=0 size 12{v rSub { size 8{+{}} } =0} {}, or
R 2 V REF + R 1 v i = 0 R 2 V REF + R 1 v i = 0 size 12{R rSub { size 8{2} } V rSub { size 8{ ital "REF"} } +R rSub { size 8{1} } v rSub { size 8{i} } =0} {} (64)
Which can be written as
v i = − R 2 R 1 V REF v i = − R 2 R 1 V REF size 12{v rSub { size 8{i} } = - { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } V rSub { size 8{ ital "REF"} } } {} (65)
Figure 21: Other comparator circuits: a) non-inveting and b) inverting.
The output goes high when v+>0v+>0 size 12{v rSub { size 8{+{}} } >0} {}. From Equation 63, we see that v0 = high when vIvI size 12{v rSub { size 8{I} } } {} is greater than the crossover voltage. A similar analysis produces the characteristics shown in Figure 21(b).
Figure 22 shows one application of a comparator, to control street lights. The input signal is the output of a photodetector circuit. Voltage vIvI size 12{v rSub { size 8{I} } } {} is directly proportional to the amount of light incident on the photodetector. During the night, vI<VREFvI<VREF size 12{v rSub { size 8{I} } <V rSub { size 8{ ital "REF"} } } {}, and v0v0 size 12{v rSub { size 8{0} } } {} is on the order of VS=+15VVS=+15V size 12{V rSub { size 8{S} } "=+""15"V} {}; the transistor turns on. The current in the relay switch then turns the street lights on. During the day, the light incident on the photodetector produces an output signal such that vI>VREFvI>VREF size 12{v rSub { size 8{I} } >V rSub { size 8{ ital "REF"} } } {}. In this case, v0 is on the order of −VS=−15V−VS=−15V size 12{ - V rSub { size 8{S} } = - "15"V} {}, and the transistor turns off.
Figure 22: Comparator application.
Diode D1D1 size 12{D rSub { size 8{1} } } {} is used as a protection device, preventing reverse bias breakdown in the B-E junction. With zero output current, the relay switch is open and the street lights are off. At dusk and dawn, vI=VREFvI=VREF size 12{v rSub { size 8{I} } =V rSub { size 8{ ital "REF"} } } {}.
Figure 23: a) Comparator circuit including input noise source, b) input signal, and c) output signal,showing chatter effect.
The open-loop comparator circuit shown in Figure 22 may exhibit unacceptable behavior in response to noise in the system. Figure 23(a) shows the same comparator circuit, but with a variable light source, such as clouds causing the light intensity to fluctuate over a short period of time. A variable light intensity would be equivalent to a noise source vnvn size 12{v rSub { size 8{n} } } {} in series with the signal source vIvI size 12{v rSub { size 8{I} } } {}. If we assume that vIvI size 12{v rSub { size 8{I} } } {} is increasing linearly with time (corresponding to dawn), then the total input signal vI'vI' size 12{v rSub { size 8{I} } rSup { size 8{'} } } {} versus time is shown in Figure 23(b). When vI'>VREFvI'>VREF size 12{v rSub { size 8{I} } rSup { size 8{'} } >V rSub { size 8{ ital "REF"} } } {}, the output switches low; when vI'<VREFvI'<VREF size 12{v rSub { size 8{I} } rSup { size 8{'} } <V rSub { size 8{ ital "REF"} } } {}, the output switches high, producing a chatter effect in the output signal as shown in Figure 23(c). This effect would turn the street lights off and on over a relatively short time period. If the amplitude of the noise signal increases, the chatter becomes more severe. This chatter can be eliminated by using a Schmitt trigger.

Basic Inverting Schmitt Trigger

The Schmitt trigger or bistable multivibrator uses positive feedback with a loop-gain greater than unity to produce a bistable characteristic. Figure 24(a) shows one configuration of a Schmitt trigger. Positive feedback occurs because the feedback resistor is connected between the output and noninverting input terminals. Voltage v2v2 size 12{v rSub { size 8{2} } } {}, in terms of the output voltage, can be found by using a voltage divider equation, to yield
v ± = ( R 1 R 1 + R 2 ) v 0 v ± = ( R 1 R 1 + R 2 ) v 0 size 12{v rSub { size 8{ +- {}} } = \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) v rSub { size 8{0} } } {} (66)
Voltage v±v± size 12{v rSub { size 8{ +- {}} } } {} does not remain constant, it is a function of the output voltage. Input signal v1v1 size 12{v rSub { size 8{1} } } {} is applied to the inverting terminal.
Figure 24: a) Schmiit trigger circuit, b) voltage trnasfer characteristics as input voltage increase, c) voltage transfer characteristic as input voltage decreases, and d) net voltage transfer characteristics, showing hysteresis effect.
Additional Schmitt Trigger Configurations
Voltage transfer characteristics
To determine the voltage transfer characteristics, we assume that the output of the comparator is in one state, namely v0=VHv0=VH size 12{v rSub { size 8{0} } =V rSub { size 8{H} } } {}, which is the high state. Then
v ± = ( R 1 R 1 + R 2 ) V H v ± = ( R 1 R 1 + R 2 ) V H size 12{v rSub { size 8{ +- {}} } = \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{H} } } {} (67)
As long as the input signal is less than v±v± size 12{v +- {}} {}, the output remains in its high state. The crossover voltage occurs when vi=v±vi=v± size 12{v rSub { size 8{i} } =v +- {}} {}, and is defined as VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {}. We have
v TH = ( R 1 R 1 + R 2 ) V H v TH = ( R 1 R 1 + R 2 ) V H size 12{v rSub { size 8{ ital "TH"} } = \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{H} } } {} (68)
When vivi size 12{v rSub { size 8{i} } } {} is greater than VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {}, the voltage at the inverting terminal is greater than that at the noninverting terminal. The differential input voltage ( v±v± size 12{v +- {}} {} = VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {}) is amplified by the open-loop gain of the comparator, and the output switches to its low state, or v0=VLv0=VL size 12{v rSub { size 8{0} } =V rSub { size 8{L} } } {}. Voltage v±v± size 12{v +- {}} {} then becomes
v ± = ( R 1 R 1 + R 2 ) V L v ± = ( R 1 R 1 + R 2 ) V L size 12{v rSub { size 8{ +- {}} } = \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{L} } } {} (69)
Since VL<VHVL<VH size 12{V rSub { size 8{L} } <V rSub { size 8{H} } } {}, the input voltage vivi size 12{v rSub { size 8{i} } } {} is still greater than v±v± size 12{v +- {}} {} , and the output remains in its low state as vivi size 12{v rSub { size 8{i} } } {} continues to increase. This voltage transfer characteristic is shown in Figure 24(b). Implicit in these transfer characteristics is the assumption that VHVH size 12{V rSub { size 8{H} } } {} is positive and VLVL size 12{V rSub { size 8{L} } } {} is negative.
Now consider the transfer characteristic as vivi size 12{v rSub { size 8{i} } } {} decreases. As long as vi is larger than (R1+R2)R1/VLv±(R1+R2)R1/VLv± size 12{v +- = lbrace {R rSub { size 8{1} } } slash { \( R rSub { size 8{1} } +R rSub { size 8{2} } \) rbrace V rSub { size 8{L} } } } {}, the output remains in its low saturation state. The crossover voltage now occurs when vi=v±vi=v± size 12{v rSub { size 8{i} } =v +- {}} {} , and is defined as VTLVTL size 12{V rSub { size 8{ ital "TL"} } } {}. We have
V TL = ( R 1 R 1 + R 2 ) V L V TL = ( R 1 R 1 + R 2 ) V L size 12{V rSub { size 8{ ital "TL"} } = \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{L} } } {} (70)
As vivi size 12{v rSub { size 8{i} } } {} drops below this value, the voltage at the noninverting terminal is greater than that at the inverting terminal. The differential voltage at the comparator terminals is amplified by the open-loop gain, and the output switches to its high state, or v0=VHv0=VH size 12{v rSub { size 8{0} } =V rSub { size 8{H} } } {}. As vivi size 12{v rSub { size 8{i} } } {} continues to decrease, it remains less than v±v± size 12{v +- {}} {} , therefore, v0v0 size 12{v rSub { size 8{0} } } {} remains in its high state. This voltage transfer characteristic is shown in Figure 24(c).
Complete voltage transfer and bistable characteristics
The complete voltage transfer characteristics of the Schmitt trigger shown in Figure 24(a) combine the characteristics shown in Figure 24(b) and Figure 24(c). These complete characteristics are shown in Figure 24(d). As shown, the crossover voltages depend on whether the input voltage is increasing or decreasing. The complete transfer characteristics therefore show a hysteresis effect. The width of the hysteresis is the difference between the two crossover voltages VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {} and VTLVTL size 12{V rSub { size 8{ ital "TL"} } } {}.
The bistable characteristic of the circuit occurs around the point vi=0vi=0 size 12{v rSub { size 8{i} } =0} {}, at which the output may be in either its high or low state. The output remains in either state as long as vivi size 12{v rSub { size 8{i} } } {} remains in the range VTL<vi<VTHVTL<vi<VTH size 12{V rSub { size 8{ ital "TL"} } <v rSub { size 8{i} } <V rSub { size 8{ ital "TH"} } } {}. The output switches states only if the input increases above VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {} or decreases below VTLVTL size 12{V rSub { size 8{ ital "TL"} } } {}.
The complete voltage transfer characteristics given in Figure 24(d) show the inverting characteristics of this particular Schmitt trigger. When the input signal becomes sufficiently positive, the output is in its low state; when the input signal is sufficiently negative, the output is its high state. Since the input signal is applied to the inverting terminal of comparator, this characteristic is as expected.

Additional Schmitt Trigger Configurations

A noninverting Schmitt trigger can be designed by applying the input signal to the network connected to the comparator noninverting terminal. Also, both crossover voltages of a Schmitt trigger circuit can be shifted in either a positive or negative direction by applying a reference voltage. We will study these general circuit configurations, the resulting voltage transfer characteristics, and an application of a Schmitt trigger circuit in this section.
Noninverting Schmitt trigger circuit
Consider the circuit shown in Figure 25(a). The inverting terminal is held essentially at ground potential, and the input signal is applied to resistor R1R1 size 12{R rSub { size 8{1} } } {}, which is connected to the comparator noninverting terminal. Voltage v±v± size 12{v +- {}} {} at the noninverting terminal then becomes a function of both the input signal vivi size 12{v rSub { size 8{i} } } {} and the output voltage v0v0 size 12{v rSub { size 8{0} } } {}. using superposition, we find that
v ± = ( R 2 R 1 + R 2 ) v i + ( R 1 R 1 + R 2 ) v 0 v ± = ( R 2 R 1 + R 2 ) v i + ( R 1 R 1 + R 2 ) v 0 size 12{v rSub { size 8{ +- {}} } = \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) v rSub { size 8{i} } + \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) v rSub { size 8{0} } } {} (71)
Figure 25: Noninverting Schmitt trigger circuit and b) voltage transfer characteristics.
If vivi size 12{v rSub { size 8{i} } } {} is negative and the output is in its low state, then v0=VLv0=VL size 12{v rSub { size 8{0} } =V rSub { size 8{L} } } {} (assumed to be negative), v+v+ size 12{v rSub { size 8{+{}} } } {} is negative, and the output remains in its low saturation state. Crossover voltage vi=VTHvi=VTH size 12{v rSub { size 8{i} } =V rSub { size 8{ ital "TH"} } } {} occurs when v+v+ size 12{v rSub { size 8{+{}} } } {}= 0 and v0=VLv0=VL size 12{v rSub { size 8{0} } =V rSub { size 8{L} } } {}, or, from Equation 71,
0 = R 2 V TH + R 1 V L 0 = R 2 V TH + R 1 V L size 12{0=R rSub { size 8{2} } V rSub { size 8{ ital "TH"} } +R rSub { size 8{1} } V rSub { size 8{L} } } {} (72)
Which can be written
V TH = − ( R 1 R 2 ) V L V TH = − ( R 1 R 2 ) V L size 12{V rSub { size 8{ ital "TH"} } = - \( { {R rSub { size 8{1} } } over {R rSub { size 8{2} } } } \) V rSub { size 8{L} } } {} (73)
Since VLVL size 12{V rSub { size 8{L} } } {} is negative, VTHVTH size 12{V rSub { size 8{ ital "TH"} } } {} is positive.
If we let vi=VTH+δvi=VTH+δ size 12{v rSub { size 8{i} } =V rSub { size 8{ ital "TH"} } +δ} {}, where δδ size 12{δ} {} is a small positive voltage; the input voltage is just greater than the crossover voltage and Equation 71 becomes
v + = ( R 2 R 1 + R 2 ) ( V TH + δ ) + ( R 1 R 1 + R 2 ) V L v + = ( R 2 R 1 + R 2 ) ( V TH + δ ) + ( R 1 R 1 + R 2 ) V L size 12{v rSub { size 8{+{}} } = \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) \( V rSub { size 8{ ital "TH"} } +δ \) + \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{L} } } {} (74)
Equation 74 then becomes
v + = ( R 2 R 1 + R 2 ) ( − R 1 R 2 ) V L + ( R 2 R 1 + R 2 ) δ + ( R 1 R 1 + R 2 ) V L v + = ( R 2 R 1 + R 2 ) ( − R 1 R 2 ) V L + ( R 2 R 1 + R 2 ) δ + ( R 1 R 1 + R 2 ) V L size 12{v rSub { size 8{+{}} } = \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) \( { { - R rSub { size 8{1} } } over {R rSub { size 8{2} } } } \) V rSub { size 8{L} } + \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R"" lSub { size 8{2} } } } \) δ+ \( { {R rSub { size 8{1} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) V rSub { size 8{L} } } {} (75)
Or
v + = ( R 2 R 1 + R 2 ) δ > 0 v + = ( R 2 R 1 + R 2 ) δ > 0 size 12{v rSub { size 8{+{}} } = \( { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } \) δ>0} {} (76)
When v+v+ size 12{v rSub { size 8{+{}} } } {} > 0, the output switches to its high saturation state.
The lower crossover voltage vi=VTLvi=VTL size 12{v rSub { size 8{i} } =V rSub { size 8{ ital "TL"} } } {} occurs when v+v+ size 12{v rSub { size 8{+{}} } } {} = 0 and v0=VHv0=VH size 12{v rSub { size 8{0} } =V rSub { size 8{H} } } {}. From Equation 71, we have
0 = R 2 V TL + R 1 V H 0 = R 2 V TL + R 1 V H size 12{0=R rSub { size 8{2} } V rSub { size 8{ ital "TL"} } +R rSub { size 8{1} } V rSub { size 8{H} } } {} (77)
Which can be written
V TL = − ( R 1 R 2 ) V H V TL = − ( R 1 R 2 ) V H size 12{V rSub { size 8{ ital "TL"} } = - \( { {R rSub { size 8{1} } } over {R rSub { size 8{2} } } } \) V rSub { size 8{H} } } {} (78)
Since VHVH size 12{V rSub { size 8{H} } } {} > 0, then VTLVTL size 12{V rSub { size 8{ ital "TL"} } } {}< 0.
The complete voltage transfer characteristics are shown in Figu