The *RC*, *RL*, and *RLC* circuits have several ac applications; such as: coupling circuits, phase shifter circuits, filters, resonant circuits, ac bridge circuits, and transformers.

**Contents**show

*RC*phase-shifting circuits and ac bridge circuits. Phase shift operation principle is based on phasor.

Make sure to read what is ac circuit first.

Make sure to read:

- What is phasor
- Impedance and admittance
- Kirchhoff’s laws for ac circuit
- Power calculation in ac circuit
- Three phase circuit

And its applications:

Now we will learn about phase-shifting circuits.

## Phase Shifter

A phase-shifting circuit is often used to correct an undesirable phase shift which presents in a circuit or to produced special needed effect.

An *RC* circuit is capable for this purpose because the capacitor causes the circuit current to lead the applied voltage.

The two common examples can be seen in Figure.(1). (*RL* circuits or any reactive circuits could also serve the same purpose.)

Figure 1. Series RC shift circuits : (a) leading output, (b) lagging output |

In Figure.(1a), the circuit current **I** leads the applied voltage **V**_{i} by some phase angle θ, where 0 < θ < 90^{o}, depending on the values of *R* and *C*. If X_{C} = -1/ωC, then the total impedance is **Z** = R + jX_{C}, and the phase shift is given by

(1) |

Notice that the phase shift amount is depended on the values of R, C, and the operating frequency.

Since the output voltage **V**_{o} across the resistor is in phase with the current, **V**_{o} leads (positive phase shift) **V**_{i} as shown in Figure.(2a).

Figure 2. Phase shift in RC circuits : (a) leading output, (b) lagging output |

In Figure.(1b), the output is taken across the capacitor.

The current **I** leads the input voltage **V**_{i} by θ, but the output voltage *v _{o}(t)* across the capacitor lags (negative phase shift) the input voltage

*v*as drawn in Figure.(2b).

_{i}(t)We have to keep in mind that the simple RC circuits in Figure.(1) also act as voltage divider.

Therefore, as the phase shift θ approaches 90^{o}, the output voltage **V**_{o} approaches zero.

For this reason, these simple RC circuits are used only when small amounts phase shift are required.

If it is desired to have phase shifts greater than 60^{o}, simple RC networks are cascaded, thereby providing a total phase shift equal to the sum of the individual phase shifts.

In practice, the phase shifts due to the stages are not equal, because the succeeding stages load down the earlier stages unless op-amps are used to separate the stages.

## Phase Shifter Examples

For better understanding let us review examples below :

**1. Please make a circuit design to provide a 90 ^{o }phase leading.**

*Solution :*

If we select circuit components of equal ohmic value, say R = |X_{C}| = 20 Ω, at a particular frequency, according to Equation.(1), the phase shift is exactly 45^{o}.

By cascading two similar RC circuits in Figure.(1a), we obtain the circuit in Figure.(3),

Figure 3 |

providing a positive or leading phase shift of 90^{o}, as we shall soon show. Using the series-parallel combination technique, **Z** in Figure.(3) is obtained as

**2. For the RL circuit shown in Figure.(4a), calculate the amount of phase shift produced at 2 kHz.**

At 2 kHz, we transform the 10 mH and 5 mH inductances to the corresponding impedances.

Consider the circuit in Figure.(4b). The impedance **Z** is the parallel combination of j125.7 Ω and 100 + 62.83 Ω. Thus,

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