# Phase Shifter circuit with Op-Amp All Pass Filter

Phase shifters are important circuits in audio processing and communication systems, used for various purposes including audio equalization, signal mixing such as in SSB modulation, AM modulation and image processing. An all-pass filter is a special type of filter that allows all frequencies to pass through without attenuation, but alters the phase response of the signal. The phase shifter with op-amp described earlier is an example of all pass filter that produces phase shift from 0 to 180 degree without any degradation of the amplitude. Here we will explore how an all-pass filter can be used to create a phase shifter circuit that can produce larger phase shifts.

What is an all-pass filter?

An all-pass filter is a filter that has a constant magnitude response but introduces a phase shift that varies with frequency. The output of an all-pass filter is a delayed version of the input signal, with the amount of delay varying with frequency. All-pass filters are useful in audio processing applications where phase relationships between different frequencies are important, such as in equalization or reverb effects. All pass filter is also called phase filter or time delay filter.

How does an all-pass filter work?

An all-pass filter is designed to have a frequency response that is the inverse of a complementary filter. The complementary filter is designed to have a frequency response that is the inverse of the all-pass filter. When the output of an all-pass filter is combined with the output of a complementary filter, the resulting signal has a constant magnitude response and a phase shift that depends on the frequency.

All pass filter can be implemented with passive component like resistor, capacitor and inductor or active devices like operational amplifiers. Here all pass filter designed with op-amp are explained.

### First order All pass filter

The following shows circuit diagram of lagging first order all pass filter.

The filter is called first order because it has one capacitor C1. This first order filter is called lagging because the phase shift from 0 to -180 and the input signal lags the output signal. The filter design equations for this filter are as follows,

(1) Voltage gain, $$A_v = 1$$

(2) Cutoff  frequency of R1C1 network, $$f_c = \frac{1}{2 \pi R_1 C_1}$$

(3) Phase difference, $$tan(\frac{\phi}{2}) = \frac{f_i}{f_c}$$

where $$f_i$$ is the input signal frequency

The following shows how the lagging first order all pass filter works and how the phase shift happens.

As one can see the phase difference difference between the input and output changes from 0 to 180 with output signal lagging the input signal. The amplitude and frequency of the output signal is same as the input signal.

If the position of the resistor R1 and capacitor C1 is interchanged we get leading first order all pass filter whose circuit diagram is shown below.

This lead all pass filter produces phase shift from 0 to 180 degrees. That is the output signal can lead the input signal in phase by up to 180 degree. See the following leading first order all pass filter circuit animation to see the phase shifting.

### Second order All pass filter

The circuit diagram of second order all pass filter is shown below.

The second order all pass filter has two capacitor and four resistors. The design equation for the op-amp based 2nd order all pass filter are as follows.

$$R_1 = \frac{1}{2Qw_cC}$$    ------>(1)

where $$C=C_1=C_2$$ and $$w_c=2\pi f_c$$ and $$f_c$$ is the cutoff frequency of the filter.

$$R_2 = \frac{2Q}{w_cC}$$    ------->(2)

$$\frac{R_2R_3}{R_1R_4}=4$$   ------->(3)

Let $$R_3=R_1$$ then from (3)

$$R_4=\frac{R_2}{4}$$   ------->(4)

The voltage gain is,

$$A_v = \frac{R_4}{R_3+R_4}$$ ------->(5)

The Q and Av are related by the following equation,

$$A_v=\frac{Q^2}{1+Q^2}$$ ------->(6)