The likewise equivalent resistance of dc circuit, we also need equivalent impedance for ac circuit. From its name, equivalent impedance means a single impedance of multiple elements without affecting other elements such as voltage and current source. Equivalent impedance is essential for analyzing the ac circuit.

*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 isFigure 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 the 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 bebalancedif 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.

## 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.**

Let

**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*in the circuit of Figure.(7)_{o}(t)
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**= Impedance of the parallel combination of the 10 mF capacitor and 5 H inductor.

_{2}**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

Figure 10 |

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*Reference: Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku*

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