Unity Gain Frequency (fu)

CIRCUIT

Schematic: Op_Amp_fu.asc
Op Amp Symbol: Opamp_1.asy
Op Amp Shematic: Opamp_1.asc
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SPEC IT

Intro	In sensor and signal acquisition systems, it's critical to respond to a voltage change quickly. A fast response requires a HIGH bandwidth (fc) and a SHORT settling time (tr). But beware, both an op amp's open-loop unity-gain freq (fu) and closed-loop signal gain (Kcl) can limit the circuit's overall bandwidth. A simple insight can ensure your design meets spec!
Definition	fu  - frequency where open-loop gain crosses unity. Aol(fu) = 1V/V (0dB)
Design Goal	Closed-loop Bandwidth:  fc = 1MHz
Design Spec	fc > 1.0 MHz tr_63% < 0.159 us

DESIGN IT

• Circuit Design
  •  Kcl = (R2+R1)/R1 = 4V/V
  • Choose: R1=10k
  • Calc: R2=30k
• Op Model Param
  • 1MHz device spec
  • Aol=100000 fu=1e6 Ro=1
• Circuit Test
  • Input: .param Vstep=2.5V
  • Output:  Vo = Vstep*Kcl = 2.5V*4 = 10V
  • Expected 63% Risetime: tr = 0.159us (fc = 1MHz)

TEST IT

• Run a Transient simulation. (.TRAN)
• Plot v(vo) and measure the risetime (tr) to 63.7% or 6.37V
• Did it meet spec?
  • If tr < 0.159us, then PASS, else FAIL
•  DESIGN ISSUE: The bandwidth fails to meet spec! 

SOLVE IT

• Incrementally increase fu (by factors of 2).
• Rerun the SPICE sim.
• What's the minimum fu needed for tr < 0.159us?

THEORY REFRESH

• The closed-loop bandwidth (fc) is the unity-gain freq (fu) scaled by the feedback gain (B).

• where B is the gain from vo to vfb

• You can also think of fc as fu divided by Noise Gain Kn

• What is Kn? Simply the gain from the vin+ input to the output vo.

• Note these two handy insights for the non-inverting amp!
  • The Noise gain equals Signal gain:
       Kn = Kcl
  • Noise Gain and Feedback gain are related by
       Kn = 1/B
• What happened to our original design?
  • B = R1/(R1+R2) = 0.25
  • Kn = 1/B = (R1+R2)/R1 = 4
  • fc = fu/Kn = 1MHz/4 = 250kHz.
  • Yikes! The bandwidth is 4 times lower than the spec of 1MHz.
    (Consequently, the rise time will be 4 times longer!)

• Let's solve it
  • What fu is required for a fc = 1MHz and Kcl = Kn = 4?
  • Solve fc = fu / Kn for fu.
  • Then calculate fu = fc * Kn
• Retest the circuit with the new fu. Did your circuit meet spec?

Way to go! Select an op amp that meets or exceeds your required fu.

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