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What is the transfer function for the below circuit?

schematic

simulate this circuit – Schematic created using CircuitLab

Since the op-amp has unity gain, the transfer function should be the same as a passive high pass RC filter.

This can be found by calculating the voltage across \$R\$ using the potential divider rule, in the \$s\$ domain:

The impedance of a capacitor in the S domain is \$\frac{1}{sC}\$, so the transfer function is:

$$H(s) = \frac{R}{R+\frac{1}{sC}}$$

In standard form this is:

$$H(s) = \frac{RCs}{RCs+1}$$

$$H(s) = RC * s * \frac{1}{RCs+1}$$

DC Gain: \$20log(RC)\$ dB

Gain due to single zero at origin: \$20log(\omega)\$ dB ; Argument: \$90^o\$

Gain due to pole:

  • at high frequencies: \$-20log(RC\omega) = -20log(RC)-20log(\omega)\$ dB
  • at low frequencies: \$20log(1) = 0\$ dB
  • with corner frequency: \$\frac{1}{RC}\$

Argument (phase) due to pole: \$-tan^{-1}(RC\omega)\$

So the complete gain is:

  • \$for \:\omega >> \frac{1}{RC}\$

\$|H(j\omega)| = 0\$ dB

  • \$for \:\omega << \frac{1}{RC}\$

\$|H(j\omega)| = 20log(\omega) + 20log(RC)\$ dB

Complete phase response:

\$\angle H(j\omega) = 90-tan^{-1}(RC\omega)\$

However, in my lecture notes it says that the transfer function for the above circuit is:

$$H(s) = \frac{s}{1+sCR}$$

Without any derivation.

Which is correct?

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The last expression cannot be correct because the numerator has the dimension rad/s. The whole function H(s) must be dimensionless. (Perhaps the author assumes a factor T=1 second, however should be mentioned in this case). But your DC gain is not correct. Instead, we have H(s=0)=0. –  LvW Aug 14 at 12:47

1 Answer 1

up vote 3 down vote accepted

Your expression for \$H(s)\$ is correct:

$$H(s)=\frac{RCs}{1+RCs}$$

where \$\tau=RC\$ is the time constant of the high pass filter. The most important features of this transfer function are the location of the pole (\$s_{\infty}=-1/RC\$) and the location of the zero (\$s_0=0\$). Note that the location of the pole and the zero is of course the same for the transfer function in your lecture notes. However, the gain factor in your notes is wrong. The consequence of this is that for \$s\rightarrow\infty\$ (i.e. for very high frequencies) your transfer function converges to unity gain, whereas the one in your notes converges to \$1/RC\$.

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Thanks for this answer. In my notes there is also a transfer function for a similar circuit, but with the op-amp in a non inverting configuation, given as \$\frac{Ks}{1+sCR}\$, where \$K=1+\frac{R_1}{R_2}\$. Im guessing this is also wrong, and there should be an \$RC\$ in the numerator of this as well? –  Blue7 Aug 14 at 14:44
    
The circuit as shown is a non-inverting configuration - however, with a gain of +1. For resistive feedback with R1 and R2 the maximum gain (for very larger frequencies) is - as mentioned by you - 1+R2/R1. And - yes, this expression must appear as a factor in the numerator. –  LvW Aug 14 at 14:54

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