---
title: "Transfer function of op-amp phase-shift circuit"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:tf-opamp-allpass-r1-c-r2"
source: https://www.bctengineer.com/q/semester_3/control_systems/tf-opamp-allpass-r1-c-r2
site: BCT Engineering
---

# Transfer function of op-amp phase-shift circuit

Control Systems · Chapter 2: Component Modeling

- Asked in: 2075 Asoj, 4 marks; 2068 Chaitra, 6 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-opamp-allpass-r1-c-r2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

Find the transfer function of given circuit. [Figure: op-amp circuit; input Ei is applied through resistor R1 to the inverting (−) input and through capacitor C to the non-inverting (+) input; R2 connects the non-inverting input to ground; feedback resistor R1 from output to the inverting input; output Eo]

## Short answer

The transfer function is Eo(s)/Ei(s) = (sR2C - 1)/(sR2C + 1). It is an all-pass filter with unity gain and phase shift from 180° to 0°.

## Answer

**Assumption:** ideal op-amp (infinite gain and input impedance), so $V_- = V_+$ and no current enters the inputs.

```text
                R1 (feedback)
            +----/\/\/----+
            |             |
Ei --/\/\/--+---(-)|\     |
       R1          | >----+---- Eo
Ei ---||----+---(+)|/
       C    |
           [R2]
            |
           GND
```

#### Non-inverting input
$C$ and $R_2$ form a voltage divider:

$$
V_+ = E_i\frac{R_2}{R_2 + \dfrac{1}{sC}} = E_i\frac{sCR_2}{1 + sCR_2}
$$

#### Inverting input (KCL)
The current through the input $R_1$ equals the current through the feedback $R_1$, with $V_- = V_+$:

$$
\frac{E_i - V_+}{R_1} = \frac{V_+ - E_o}{R_1} \;\Rightarrow\; E_o = 2V_+ - E_i
$$

#### Transfer function
$$
\begin{aligned}
E_o &= E_i\left(\frac{2sCR_2}{1+sCR_2} - 1\right) = E_i\,\frac{sCR_2 - 1}{sCR_2 + 1} \\
\frac{E_o(s)}{E_i(s)} &= \frac{sR_2C - 1}{sR_2C + 1} = -\frac{1 - sR_2C}{1 + sR_2C}
\end{aligned}
$$

This is an **all-pass (phase-shift) network**: $|E_o/E_i| = 1$ at all frequencies, and the phase is $\phi = 180^\circ - 2\tan^{-1}(\omega R_2C)$, going from $180^\circ$ at $\omega = 0$ to $0^\circ$ at high frequency ($90^\circ$ at $\omega = 1/R_2C$). It has a zero at $s = +1/R_2C$ (non-minimum phase) and a pole at $s = -1/R_2C$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. V+ at the non-inverting input is
   A. Ei/(1 + sCR2)
   B. Ei sCR2/(1 + sCR2)
   C. Ei R2/(R1+R2)
   D. Ei sCR2
   Answer: B. Divider of C and R2: R2/(R2+1/sC). (Key point: Non-inverting voltage)

2. With equal R1 resistors and V-=V+, Eo =
   A. V+ - 2Ei
   B. Ei - V+
   C. 2V+ - Ei
   D. 2Ei - V+
   Answer: C. KCL: (Ei-V+)/R1=(V+-Eo)/R1. (Key point: Output relation)

3. The transfer function and type are
   A. (sR2C + 1)/(sR2C - 1), a low-pass filter
   B. 1/(1+sR2C), a low-pass filter
   C. sR2C/(1+sR2C), a high-pass filter
   D. (sR2C - 1)/(sR2C + 1), an all-pass network
   Answer: D. The magnitude is 1 at all frequencies. (Key point: Result)

## Related in Control Systems

- [Transfer function of op-amp with Rs ∥ C feedback](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-rs-c-r1-r2) — Control Systems, chapter 2
- [Transfer function of op-amp integrator-type circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-r1-r2-c) — Control Systems, chapter 2

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