Operational Amplifier and Oscillator

In a non-inverting amplifier the input signal is applied to the non-inverting (+) terminal, and part of the output is fed back to the inverting (−) terminal through the divider Rf and R1 (negative feedback). The output is in phase with the input.

Circuit

              Rf
        +---/\/\/\---+
        |            |
  R1    |   |\       |
 +-/\/\-+---|-\      |
 |      V−  |  >-----+---o Vo
GND   Vi o--|+/
        V+  |/

Assumptions (ideal op-amp)

1. Infinite open-loop gain, so V+ = V− (virtual short).
2. Infinite input resistance, so no current enters either input terminal.

Derivation

Because of the virtual short:

V− = V+ = Vi

No current enters the (−) terminal, so the same current I flows through Rf and R1. KCL at the inverting node:

(Vo − V−)/Rf = (V− − 0)/R1
(Vo − Vi)/Rf = Vi/R1
Vo − Vi      = Vi (Rf/R1)
Vo           = Vi (1 + Rf/R1)

Equivalently, Rf and R1 form a voltage divider: V− = Vo × R1/(R1 + Rf) = Vi.

Closed-loop voltage gain:

A_CL = Vo/Vi = 1 + Rf/R1

Points to note

• The gain is always ≥ 1 and positive (no phase inversion).
• Gain depends only on the external resistors, not on the op-amp's open-loop gain.
• Input resistance is very high (ideally infinite), so it does not load the source.
• Example: Rf = 9 kΩ, R1 = 1 kΩ gives A_CL = 1 + 9 = 10; Vi = 0.5 V gives Vo = 5 V.
• Special case: Rf = 0 or R1 = ∞ gives A_CL = 1, the voltage follower (buffer).

With finite open-loop gain A: A_CL = A/(1 + Aβ), β = R1/(R1 + Rf), which tends to 1/β = 1 + Rf/R1 when Aβ ≫ 1.

Barkhausen criteria

For sustained oscillations in a feedback circuit with amplifier gain A and feedback factor β:

1. The loop gain magnitude must be unity: |Aβ| = 1.
2. The total phase shift around the loop must be 0° or 360° (positive feedback).

(In practice |Aβ| is made slightly greater than 1 to start oscillations.)

Relaxation oscillator (op-amp square wave generator)

A relaxation oscillator produces a non-sinusoidal output by repeatedly charging and discharging a capacitor. The op-amp works as a comparator (Schmitt trigger) with positive feedback through R1–R2 and an RC timing circuit on the inverting input.

              R
        +---/\/\/\-----+
        |              |
        |    |\        |
 Vc  +--+----|-\       |
     |       |  >------+---o Vo (±Vsat)
    C    +---|+/       |
     |   |   |/        |
    GND  +----/\/\/\---+
         |     R2
         R1
         |
        GND

Operation

• Positive feedback gives V+ = β Vo, where β = R1/(R1 + R2). The output is always at +Vsat or −Vsat.
• Suppose Vo = +Vsat. Then V+ = +β Vsat. C charges through R towards +Vsat.
• When Vc just exceeds +β Vsat, V− > V+ and the output switches to −Vsat. Now V+ = −β Vsat.
• C now discharges and charges in the negative direction through R towards −Vsat.
• When Vc falls below −β Vsat, the output switches back to +Vsat. The cycle repeats.

The output is a square wave (±Vsat) and the capacitor voltage is an exponential triangle-like wave between ±β Vsat.

 Vo  +Vsat |‾‾‾‾|    |‾‾‾‾|
         0 +----+----+----+---- t
     -Vsat      |____|    |___
 Vc  +βVsat   /\        /\
     -βVsat  /  \__/   /  \__/

Frequency of oscillation

T = 2RC ln[(1 + β)/(1 − β)]
f = 1/T = 1/(2RC ln[(1 + β)/(1 − β)])

If R2 = 1.16 R1 (β ≈ 0.462), ln[(1 + β)/(1 − β)] ≈ 1, so f ≈ 1/(2RC). If R1 = R2 (β = 0.5), T = 2RC ln 3 ≈ 2.2RC.

A summing amplifier (adder) is an op-amp circuit whose output is proportional to the (weighted) sum of several input voltages. The common form is the inverting summing amplifier.

Circuit

 V1 o--[R1]--+
             |        Rf
 V2 o--[R2]--+-----/\/\/\-----+
             |                |
 V3 o--[R3]--+   |\           |
             +---|-\          |
              V− |  >---------+--o Vo
           GND --|+/
                 |/

Assumptions (ideal op-amp)

1. V+ = 0 (grounded), and because of the virtual short V− = V+ = 0 (virtual ground).
2. No current enters the op-amp input.

Derivation

Input currents:

I1 = (V1 − 0)/R1 = V1/R1
I2 = V2/R2
I3 = V3/R3

No current enters the (−) terminal, so by KCL all these currents flow through Rf:

If = I1 + I2 + I3

The output voltage, with the inverting node at 0 V:

0 − Vo = If × Rf
Vo     = −Rf (V1/R1 + V2/R2 + V3/R3)

Special cases

• Equal resistors R1 = R2 = R3 = Rf = R: Vo = −(V1 + V2 + V3) (inverting adder).
• R1 = R2 = R3 = R, Rf = R/3: Vo = −(V1 + V2 + V3)/3 (averaging amplifier).
• Unequal resistors give a scaling (weighted) summer, Vo = −(a1V1 + a2V2 + a3V3) with ai = Rf/Ri.

Example: V1 = 1 V, V2 = 2 V, V3 = −0.5 V with all resistors 10 kΩ: Vo = −(1 + 2 − 0.5) = −2.5 V.

Because the inverting node is a virtual ground, the inputs do not interact with each other. Uses: audio mixers, digital-to-analog converters (weighted resistors), and adding a DC offset to a signal. A second inverting amplifier of gain −1 can be added to get a non-inverted sum.

Barkhausen criteria

A feedback amplifier (gain A, feedback factor β) produces sustained oscillations when:

1. |Aβ| = 1 (loop gain magnitude is unity), and
2. the total phase shift around the loop is 0° or 360°.

Wien bridge oscillator

It is an RC sinusoidal oscillator using a non-inverting op-amp amplifier and a lead-lag RC network (series RC + parallel RC) for positive feedback. The four arms (series RC, parallel RC, R1, Rf) form a Wien bridge.

 Vo o---[R]---| C |---+ V+
         Z1 (series)  |
                      +---[R || C]--- GND
                      |     Z2
                      |   |\
                      +---|+\
                          |  >------+--- Vo
                  +-------|-/       |
                  |   V−  |/        |
                  +-----[Rf]--------+
                  |
                 [R1]
                  |
                 GND

(The left end of Z1 is joined to the output Vo. Z1 = R in series with C, from Vo to V+; Z2 = R parallel with C, from V+ to ground; Rf and R1 set the amplifier gain.)

Derivation of frequency

Z1 = R + 1/(jωC) = (1 + jωRC)/(jωC)
Z2 = R || (1/jωC) = R/(1 + jωRC)

β = V+/Vo = Z2/(Z1 + Z2)
  = [R/(1 + jωRC)] / [(1 + jωRC)/(jωC) + R/(1 + jωRC)]
  = jωRC / [(1 + jωRC)² + jωRC]
  = jωRC / [1 − ω²R²C² + j3ωRC]

Divide numerator and denominator by jωRC:

β = 1 / [3 + j(ωRC − 1/(ωRC))]

The amplifier is non-inverting (0° shift), so for the loop phase to be 0°, β must be real: the imaginary part must be zero.

ωRC − 1/(ωRC) = 0
ω² = 1/(R²C²)
ω  = 1/(RC)
f  = 1/(2πRC)

At this frequency β = 1/3. From |Aβ| = 1, the amplifier gain must be A = 3:

A = 1 + Rf/R1 = 3  →  Rf = 2R1

Result: f₀ = 1/(2πRC), with Rf ≥ 2R1 (slightly more than 2R1 to start oscillation).

Example: R = 10 kΩ, C = 10 nF gives f₀ = 1/(2π × 10⁴ × 10⁻⁸) ≈ 1.59 kHz.

Wien bridge oscillators give a low-distortion sine wave in the audio range (about 10 Hz to 1 MHz) and are tuned easily by ganged capacitors or resistors.