Bias current effect; closed-loop output impedance

Effect of input bias current

The input transistors of an op-amp draw small dc base currents IB1 (inverting input) and IB2 (non-inverting input). These flow through external resistors and produce an output voltage even with zero input.

                 Rf
          +----/\/\/----+
          |             |
 Vi --R1--+--> IB1 -(-) |
                     >--+---- Vo
     +-----> IB2 ---(+)
     |
   Rcomp
     |
    GND

With Vi = 0 and Rcomp = 0, the inverting input is at virtual ground, so all of IB1 flows through Rf:

Vo = IB1·Rf     (e.g. 500 nA × 1 MΩ = 0.5 V)

Adding Rcomp = R1‖Rf at the non-inverting input makes V− = −IB2·Rcomp, and

Vo = IB1·Rf - IB2·Rcomp(1 + Rf/R1) = Rf(IB1 - IB2) = Rf·Ios

so only the much smaller offset current causes error.

Closed-loop output impedance of the inverting amplifier

Model the op-amp by its open-loop gain A, input resistance Ri (very large, ignored) and output resistance Ro. To find the closed-loop output resistance, set the input source to zero, apply a test voltage vx at the output and find the current ix drawn.

                    Rf
      +-----------/\/\/-----------+
      |                           |
 v- --+-----(-)                   |
      |         >--(A·vd)---Ro----+----o vx (test source,
      R1   +---(+)                         ix flows in)
      |    |
     GND  GND

Feedback voltage at inverting input:
  v- = vx·R1/(R1 + Rf) = β·vx
  vd = v+ - v- = -β·vx
Current through Ro:
  io = (vx - A·vd)/Ro = vx(1 + Aβ)/Ro
Current through R1 + Rf:
  iF = vx/(R1 + Rf)   (very small, neglected)
  ix ≈ io = vx(1 + Aβ)/Ro
Rof = vx/ix = Ro / (1 + Aβ),   β = R1/(R1 + Rf)

Because the output voltage is sampled and fed back (voltage feedback), the output resistance is divided by (1 + Aβ). Example: 741 with Ro = 75 Ω, A = 2 × 10⁵ and β = 0.1 gives Rof = 75/(1 + 2 × 10⁴) ≈ 3.75 mΩ, practically zero. The same result holds for the inverting and non-inverting amplifiers, since with the input grounded both have the same feedback network.