The likewise equivalent resistance of dc circuit, we also need equivalent impedance for ac circuit. This is a crucial thing for analyzing electric circuit.

From its name, equivalent impedance means a single impedance of multiple elements without affecting other elements such as voltage and current source.

**Contents**show

*Make sure to read what is ac circuit first.*

## Equivalent Impedance for AC Circuits

Consider the N series-connected impedance shown in Figure.(1). The same current I flows through the impedances. Applying KVL around the loop gives

(1) |

Figure 1. N impedances in series |

The equivalent impedance at the input terminals is

(2) |

showing that the total or equivalent impedance of series-connected impedances is the sum of the individual impedances. This is similar to the series connection of resistances.

If *N* = 2, as shown in Figure.(2), the current through the impedances is

Figure 2. Voltage division |

(3) |

Since **V _{1}** =

**Z**and

_{1}I**V**=

_{2}**Z**, then

_{2}I(4) |

which is the *voltage-division* relationship.

In the same manner, we can get the equivalent impedance and admittance of the N parallel-connected impedances as can be seen in Figure.(3).

Figure 3. N impedances in parallel |

The voltage across each impedance is the same. Applying KCL at the top node,

(5) |

The equivalent impedance is

(6) |

and the equivalent admittance is

(7) |

This indicates that the equivalent admittance of a parallel connection of admittances is the sum of the individual admittances.

When *N* = 2, as drawn in Figure.(4),

Figure 4. Current division |

the equivalent impedance becomes

(8) |

Also, since **V** = **Z _{eq}I** =

**I**

_{1}**Z**=

_{1}**I**

_{2}**Z**

_{2}the currents in the impedances are

(9) |

which is the *current-division* principle.

The delta-to-wye and wye-to-delta transformations that we applied to resistive circuits are also valid for impedances. With reference to Figure.(5), the conversion formulas are as follows.

Figure 5. Superimposed Y and delta networks |

Wye-delta conversion :

A delta or wye circuit is said to be

balancedif it has equal impedances in all three branches.

When a delta-wye circuit is balanced, Equations.(10) and (11) become

As you can see in this post, the principles of voltage division, current division, circuit reduction, impedance equivalence, and wye-delta transformation all apply to ac circuits. In the next post, we will cover another technique for ac circuits such as :

- Superposition
- Nodal analysis
- Mesh analysis
- Source transformation
- Thevenin theorem
- Norton theorem.

The application of sinusoidal circuit and impedance formula results in phase shifter and ac bridge.

Read also : superposition theorem

## Equivalent Impedance for AC Circuits Examples

For better understanding let us review examples below :

**1. Find the input impedance of the circuit in Figure.(6). Assume that the circuit operates at ω = 50 rad/s.**

**Z**= Impedance of the 2 mF capacitor

_{1}**Z**= Impedance of the 3 Ω resistor in series with the 10 mF capacitor

_{2}**Z**= Impedance of the 0.2 H inductor in series with 8 Ω resistor

_{3}Then

**2. Determine v_{o}(t) in the circuit of Figure.(7)**

To do analysis in the frequency domain, we must first transform the domain circuit in Figure.(7) to the phasor domain equivalent in Figure.(8). The transformation produces

**Z**= Impedance of the 60 Ω resistor

_{1}**Z _{2}** = Impedance of the parallel combination of the 10 mF capacitor and 5 H inductor.

**3. Find current I in the circuit of Figure.(9).**

Figure 9 |

*Solution :*

The delta network connected to nodes *a*, *b*, and *c* can be converted to the Y network of Figure.(10). We obtain the Y impedances as follows using Equation.(11) :

The total impedance at the source terminals is

The desired current is