Lesson Explainer: Impedance of Alternating Current Circuits Physics

In this explainer, we will learn how to calculate the impedance of simple resistive-capacitive-inductive circuits, using capacitive and inductive reactances.

Reactance has a specific, precisely defined meaning and is defined for circuits that are connected to sources of alternating potential difference.

Before considering the exact meaning of reactance, we can first say that it is a property that can be compared to resistance, but with significant differences from it.

The way in which reactance is like resistance is that they both act to reduce current in a circuit.

An inductor with inductance ๐ฟ reacts to a change in current ฮ”๐ผ in a time interval ฮ”๐‘ก by generating an opposing emf, ๐œ€, where ๐œ€=โˆ’๐ฟฮ”๐ผฮ”๐‘ก.

The induced opposing emf acts to reduce the instantaneous current through the inductor. The reduced instantaneous current produces a smaller opposing emf and so a smaller opposing current. The opposing current decreases with time.

To show how inductance relates to reactance, we must consider a circuit containing an inductor that is connected to a continuously varying source of potential difference, such as a generator that supplies an alternating emf at some frequency ๐‘“.

Whether or not such a circuit has inductance, the current in the circuit varies sinusoidally.

For such a circuit, we can define its inductive reactance, ๐‘‹๏Œซ. We can call the maximum value of current in the circuit ๐ผmax.

Recall that ๐œ€=โˆ’๐ฟฮ”๐ผฮ”๐‘ก.

It is the case that ฮ”๐ผฮ”๐‘กโˆ๐ผ.max

We can see this proportionality demonstrated in the following graph of two sinusoidal currents with the same frequency.

The greater the inductance of an inductor is, the more slowly the current in the inductor can change.

We see then that the greater the value of ๐ฟ, the less the value of ฮ”๐ผฮ”๐‘ก.

It then follows that for a given value of ๐‘“, the greater the value of ๐ฟ is, the less the maximum value of ๐ผ must be.

It is also important to note that the frequency of a sinusoidally alternating current must determine the rate of change of that current.

It must also then be the case that ฮ”๐ผฮ”๐‘กโˆ๐‘“.

The following graph shows three sinusoidal currents.

We see that the maximum value of current ๐ผ๏Šง is the same as the maximum value of current ๐ผ๏Šฉ.

We see also that the frequency of ๐ผ๏Šง is greater than the frequency of ๐ผ๏Šฉ; hence, the value of ฮ”๐ผฮ”๐‘ก of ๐ผ๏Šง at ๐‘ก=0 is greater than the value of ฮ”๐ผฮ”๐‘ก of ๐ผ๏Šฉ at ๐‘ก=0.

Current ๐ผ๏Šจ has the same frequency as current ๐ผ๏Šง but a lower maximum value.

At ๐‘ก=0, current ๐ผ๏Šจ has the same value of ฮ”๐ผฮ”๐‘ก as current ๐ผ๏Šฉ.

This inequality of the maximum values of ๐ผ๏Šง and ๐ผ๏Šจ is necessary for ๐ผ๏Šจ to have the same value of ฮ”๐ผฮ”๐‘ก as ๐ผ๏Šฉ at ๐‘ก=0 and to also have the same frequency as ๐ผ๏Šง.

We see then that the greater the value of ๐‘“, the less the maximum value of ๐ผ.

Considering the dependence of ๐œ€ on ๐ผmax, ๐ฟ, and ๐‘“, we see that ๐œ€โˆโˆ’๐ฟ๐ผ๐‘“.max

The value of ๐‘‹๏Œซ is defined as the ratio of ๐œ€ to ๐ผmax.

We see from this that ๐œ€๐ผโˆโˆ’๐ฟ๐‘“๐‘‹โˆโˆ’๐ฟ๐‘“.max๏Œซ

Reactance does not have a direction, and so the negative sign is irrelevant, and hence it can just as well be said that ๐‘‹โˆ๐ฟ๐‘“.๏Œซ

A formula for inductive reactance can be derived from these relationships.

Formula: Inductive Reactance

The inductive reactance, ๐‘‹๏Œซ, of a circuit with inductance ๐ฟ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=2๐œ‹๐‘“๐ฟ.๏Œซ

The unit of inductive reactance is the ohm. This is shown as follows: HzVsAVAVAAฮฉAฮฉร—โ‹…==ร—=.

We see that inductive reactance has the same unit as resistance. Inductive reactance is effectively a resistance that only applies to alternating current circuits containing inductors.

We see that increasing the inductance of a circuit increases its inductive reactance. We see also that increasing the frequency of an alternating current increases the inductive reactance of the circuit.

Let us look at an example involving inductive reactance.

Example 1: Determining an Inductive Reactance

An alternating voltage source with a frequency of 75 Hz is connected to a 35 mH inductor. What is the reactance of the circuit? Give your answer to two decimal places.

Answer

The inductive reactance, ๐‘‹๏Œซ, of a circuit with inductance ๐ฟ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=2๐œ‹๐‘“๐ฟ.๏Œซ

Substituting the values given in the question, we obtain ๐‘‹=2๐œ‹ร—75ร—0.035.๏ŒซHzH

To two decimal places, ๐‘‹๏Œซ is 16.49 ฮฉ.

As well as resistors and inductors, a circuit can also contain capacitors. Capacitors also have reactance, called capacitive reactance, ๐‘‹๏Œข.

Before defining capacitive reactance, let us consider how a capacitance acts to reduce the average current in a circuit.

Let us consider a circuit containing a resistor and a capacitor in series.

A capacitor can store charge that is deposited on the capacitor by the current in a circuit that contains the capacitor. The maximum charge, ๐‘„, that can be stored by a capacitor with capacitance ๐ถ that is connected to a potential difference ๐‘‰ is given by ๐‘„=๐ถ๐‘‰.

A capacitor initially stores zero charge and hence has zero potential difference across it. The more the charge stored by a capacitor, the greater the potential difference across it.

The more charge that is stored by the capacitor, the more the current in the circuit is reduced. When a capacitor is storing the maximum charge that it can store, the current in the circuit containing the capacitor must be zero.

The relationship between the potential difference across a capacitor and the current in a circuit containing a capacitor can be expressed as ฮ”๐‘„ฮ”๐‘ก=๐ผ=๐ถฮ”๐‘‰ฮ”๐‘ก.

If a capacitor has zero capacitance, when connected to a circuit, it is initially storing its maximum charge, and so the current in the circuit must be zero. The greater the capacitance of the capacitor is, the more the charge that can be transported by the current in the circuit before the current becomes zero.

We can see from this that when a circuit containing a capacitor is connected to a potential difference source over a certain time interval, the current in the circuit will be reduced less if a capacitor of smaller capacitance is used.

What we see from this is that a capacitor can also act to reduce the current in a circuit, similarly to the way that an inductor can. However, the greater the capacitance of the capacitor, the less the current is reduced.

Capacitive reactance has a specific, precisely defined meaning.

To show how capacitance relates to capacitive reactance, we must consider a circuit containing a resistor and a capacitor in series that is connected to a continuously varying source of potential difference, such as a generator that supplies an alternating emf at some frequency ๐‘“.

Whether or not such a circuit has capacitance, the current in the circuit varies sinusoidally.

For such a circuit, we can define its capacitive reactance, ๐‘‹๏Œข. We can call the maximum value of the potential difference across the capacitor ๐‘‰max.

Recall that ฮ”๐‘„ฮ”๐‘ก=๐ผ=๐ถฮ”๐‘‰ฮ”๐‘ก.

It is the case that ฮ”๐‘‰ฮ”๐‘กโˆ๐‘‰.max

It is also the case that ฮ”๐‘‰ฮ”๐‘กโˆ๐‘“.

We see then that ๐ผโˆ๐ถ๐‘‰๐‘“.max

The value of ๐‘‹๏Œข is defined as the ratio of ๐‘‰max to ๐ผ.

We see from this that ๐ผ๐‘‰โˆ๐ถ๐‘“๐‘‰๐ผโˆ1๐ถ๐‘“๐‘‹โˆ1๐ถ๐‘“.maxmax๏Œข

A formula for capacitive reactance can be derived from these relationships.

Formula: Capacitive Reactance

The capacitive reactance, ๐‘‹๏Œข, of a circuit with capacitance ๐ถ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=12๐œ‹๐‘“๐ถ.๏Œข

The unit of capacitive reactance is the ohm. This is shown as follows: 1ร—===ร—ร—ร—ร—=ร—ร—=.HzFsFsFsssAฮฉCsAฮฉCsฮฉCฮฉCVCVCs

We see that capacitive reactance has the same unit as resistance. Capacitive reactance is effectively a resistance that only applies to alternating current circuits containing capacitors.

We see that increasing the capacitance of a circuit decreases its capacitive reactance. We see also that increasing the frequency of an alternating current decreases the capacitive reactance of the circuit.

It is useful to note that the relationship of capacitance and of frequency to capacitive reactance is the inverse of the relationship of inductance and of frequency to inductive reactance.

Let us look at an example involving capacitive reactance.

Example 2: Determining Capacitive Reactance

An alternating voltage source with a frequency of 50 Hz is connected to a 125 ยตF capacitor. What is the reactance of the capacitor? Give your answer to the nearest ohm.

Answer

The capacitive reactance, ๐‘‹๏Œข, of a circuit with capacitance ๐ถ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=12๐œ‹๐‘“๐ถ.๏Œข

Substituting the values given in the question, we obtain ๐‘‹=12๐œ‹ร—50ร—1.25ร—10.๏Œข๏Šฑ๏ŠชHzF

To the nearest ohm, ๐‘‹๏Œข is 25 ฮฉ.

The current in an alternating current circuit at a given instant depends on the combined effect of the resistance and the reactance of the circuit. This is called the impedance, ๐‘, of the circuit.

A circuit containing a resistor (R), inductor (L), and capacitor (C) connected to an alternating potential difference source is shown in the following figure.

The resistor provides the circuit with resistance. The inductor and capacitor provide the circuit with reactance.

For any alternating current circuit, the potential difference, ๐‘‰, across the circuit varies with time. We can denote by ๐‘‰๏ the potential difference across the circuit at the instant ๐‘ก.

If such a circuit has resistance, ๐‘…, but no reactance, the current at a given instant, ๐ผ๏, is determined as it would be in a direct current circuit, using Ohmโ€™s law. We can express this as ๐ผ=๐‘‰๐‘….๏๏

However, if an alternating current circuit has a reactance, the instantaneous current in the circuit is affected by the reactance of the circuit. The instantaneous current in such a circuit is given by ๐ผ=๐‘‰๐‘.๏๏

The capacitive and inductive reactances are not simply summed to determine their contribution to the impedance.

The instantaneous current passed by a capacitor is not in phase with the instantaneous potential difference across the capacitor. The current leads the potential difference by a phase angle of 90โˆ˜.

The instantaneous current passed by an inductor is not in phase with the instantaneous potential difference across the inductor. The current lags the potential difference by a phase angle of 90โˆ˜.

These phase relationships are shown in the following figure.

The phase difference between current and potential difference is 90โˆ˜ for both capacitors and inductors, but in opposite directions.

We can see then that the phase difference between the current and the potential difference for a capacitor compared to an inductor is 90+90โˆ˜โˆ˜, which is 180โˆ˜.

This means that the effect on the instantaneous current of a capacitor and that of an inductor are opposite to each other.

From this, we can see that if the capacitive and inductive reactances of a circuit are equal, the effect on the current due to the total reactance is zero. In this case, the impedance of the circuit is just the resistance of the circuit.

From these relationships, a formula for impedance can be obtained.

Formula: Impedance

The impedance, ๐‘, of a circuit is given by ๐‘=๏€บ๐‘…+(๐‘‹โˆ’๐‘‹)๏†,๏Šจ๏Œซ๏Œข๏Šจ๏Ž ๏Žก where ๐‘… is the resistance of the circuit, ๐‘‹๏Œซ is the inductive reactance of the circuit, and ๐‘‹๏Œข is the capacitive reactance of the circuit.

Resistance and reactance both have the unit ohm, so the unit of impedance is the ohm.

It is important to notice that a circuit can have nonzero impedance despite having zero resistance. For an alternating current circuit, the reactance of the circuit alone can constrain the value of current. This is not the case for circuits supplied by a constant potential difference.

Let us look at an example involving impedance.

Example 3: Determining Impedance

An alternating current circuit contains a resistor with a resistance of 125 ฮฉ, an inductor with a 450 ฮฉ inductive reactance, and a capacitor with a 28 ฮฉ capacitive reactance. What is the impedance of the circuit? Give your answer to the nearest ohm.

Answer

The impedance, ๐‘, of a circuit is given by ๐‘=๏€บ๐‘…+(๐‘‹โˆ’๐‘‹)๏†,๏Šจ๏Œซ๏Œข๏Šจ๏Ž ๏Žก where ๐‘… is the resistance of the circuit, ๐‘‹๏Œซ is the inductive reactance of the circuit, and ๐‘‹๏Œข is the capacitive reactance of the circuit.

Substituting the values given in the question, we obtain ๐‘=๏€บ(125)+(450โˆ’28)๏†.ฮฉฮฉฮฉ๏Šจ๏Šจ๏Ž ๏Žก

To the nearest ohm, ๐‘ is 440 ฮฉ.

Let us look at another such example.

Example 4: Determining the Resistance Required for a Particular Impedance

An alternating current circuit has an impedance of 750 ฮฉ. The circuit contains a resistor, an inductor with a 250 ฮฉ inductive reactance, and a capacitor with a 45.0 ฮฉ capacitive reactance. What is the resistance of the resistor? Give your answer to the nearest ohm.

Answer

The impedance, ๐‘, of a circuit is given by ๐‘=๏€บ๐‘…+(๐‘‹โˆ’๐‘‹)๏†,๏Šจ๏Œซ๏Œข๏Šจ๏Ž ๏Žก where ๐‘… is the resistance of the circuit, ๐‘‹๏Œซ is the inductive reactance of the circuit, and ๐‘‹๏Œข is the capacitive reactance of the circuit.

Substituting the values given in the question, we obtain 750=๏€บ๐‘…+(250โˆ’45.0)๏†.ฮฉฮฉฮฉ๏Šจ๏Šจ๏Ž ๏Žก

This equation must be rearranged to make ๐‘… the subject, as follows: (750)=๐‘…+(250โˆ’45.0)(750)=๐‘…+(205)๐‘…=(750)โˆ’(205)โˆš๐‘…=๏„(750)โˆ’(205)๐‘…=๏„(750)โˆ’(205).ฮฉฮฉฮฉฮฉฮฉฮฉฮฉฮฉฮฉฮฉฮฉ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ

To the nearest ohm, ๐‘… is 721 ฮฉ.

Let us look at an example in which currents are determined.

Example 5: Determining Instantaneous Currents in a Circuit with Resistance, Capacitance, and Inductance

An alternating current circuit contains a resistor with a resistance of 25 ฮฉ, an inductor with a 32 ฮฉ inductive reactance, and a capacitor with a 12.8 ฮฉ capacitive reactance. The peak voltage produced by the alternating voltage source powering the circuit is 120 V.

  1. What is the peak current in the circuit? Give your answer to one decimal place.
  2. What is the root-mean-square current in the circuit? Give your answer to one decimal place.

Answer

Part 1

The peak current, ๐ผmax, is given by ๐ผ=๐‘‰๐‘.maxmax

The value of the peak voltage of the alternating power source for the circuit is 120 V, so we see that ๐ผ=120๐‘.maxV

The impedance, ๐‘, of a circuit is given by ๐‘=๏€บ๐‘…+(๐‘‹โˆ’๐‘‹)๏†,๏Šจ๏Œซ๏Œข๏Šจ๏Ž ๏Žก where ๐‘… is the resistance of the circuit, ๐‘‹๏Œซ is the inductive reactance of the circuit, and ๐‘‹๏Œข is the capacitive reactance of the circuit.

Substituting the values given in the question, we obtain ๐‘=๏€บ(25)+(32โˆ’12.8)๏†.ฮฉฮฉฮฉ๏Šจ๏Šจ๏Ž ๏Žก

To the nearest ohm, ๐‘ is approximately 31.522 ฮฉ.

We have then that ๐ผโ‰ˆ12031.522.maxV

To one decimal place, the peak current is 3.8 A.

Part 2

The root-mean-square current is given by ๐ผ=๐ผโˆš2.RMSmax

Using the value of ๐ผmax obtained, to one decimal place, ๐ผRMS is 2.7 A.

Let us now summarize what has been learned in this explainer.

Key Points

  • The inductive reactance, ๐‘‹๏Œซ, of a circuit with inductance ๐ฟ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=2๐œ‹๐‘“๐ฟ.๏Œซ The unit of inductive reactance is the ohm.
  • The capacitive reactance, ๐‘‹๏Œข, of a circuit with capacitance ๐ถ that is carrying an alternating current with a frequency ๐‘“ is given by ๐‘‹=12๐œ‹๐‘“๐ถ.๏Œข The unit of capacitive reactance is the ohm.
  • The impedance, ๐‘, of a circuit is given by ๐‘=๏€บ๐‘…+(๐‘‹โˆ’๐‘‹)๏†,๏Šจ๏Œซ๏Œข๏Šจ๏Ž ๏Žก where ๐‘… is the resistance of the circuit, ๐‘‹๏Œซ is the inductive reactance of the circuit, and ๐‘‹๏Œข is the capacitive reactance of the circuit.
    The unit of impedance is the ohm.
  • The instantaneous current, ๐ผ๏, in an alternating current circuit is given by ๐ผ=๐‘‰๐‘,๏๏ where ๐‘‰๏ is the instantaneous potential difference across the circuit.

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