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Peter's Index Physics Home Lecture 12a Course Index Lecture 14

A Semester of First Year Physics with Peter Eyland

Lecture 13 (Pure R,C and L in AC)

In this lecture the following are introduced:

• The average or D.C value

• A pure resistor in an A.C. circuit

• The power dissipated in a resistor

• The effective or r.m.s. value

• A pure capacitor in an A.C. circuit

• The power in a capacitor

• A pure inductor in an A.C. circuit

• The power in an inductor

The Average or D.C value

The average or D.C. value of a periodically varying function is given by the sum over one period divided by the time for one period, as written mathematically: |

Example

For the half-wave rectified sine potential shown. Find the average value of the potential |

The angular frequency, ω, is related to the period, T, by ω = 2π/T, where π is the Greek letter pi.

Example

For the full-wave rectified sine potential shown. Find the average value of the potential. |

The half and full-wave rectified e.m.f.s shown above, are D.C., because the average value is not zero and the current always flows in the same direction. An A.C. has to change direction, as shown in the diagram. E.m.f.s may be represented as the superposition of a steady D.C. component (the average value) and an A.C. component which changes direction. |

A Resistor in an AC circuit

The complex applied emf is jωt]. |
The complex potential difference across the resistor is equal to the complex applied emf. |

The *complex impedance* of a circuit element is the ratio of the potential difference across the element
to the current flowing through the element.
For a pure resistor this is purely real, i.e. there is no imaginary part.

The complex impedance of a pure resistor is:

The physical potential and current are the imaginary parts of the complex values: |
Because there is no initial phase in either quantity, the emf is in phase with the current, as shown in the graph. |

The Power dissipated in a Resistor

The instantaneous power is: |
By inspection, the average of sinθ is zero and the average of sin |

Taking the average over one period:

Effective or r.m.s. values

From above |
Here
ean of the mquare of the current,
or s current.r.m.s. |

The r.m.s. value calculation above does not give a universal result,
as it will depend on the waveshape supplied by the source.
For example, with the *sinusoidal* potential difference and resistor above:

Since most A.C. meters measure the r.m.s. values of current and potential, unless stated otherwise,
* all values shown on circuit diagrams will be r.m.s. values*.
For example, the 240 V domestic supply has a nominal maximum (or

Example

The 240 V, 50 Hz mains electricity supply is connected in series with a 1 kW heater.
Find the rms and instantaneous values of

(a) the p.d. across the heater, and

(b) the current in the circuit, and

(c) the power dissipated in the heater.

In the following, upper case letters denote r.m.s. or steady D.C. values and lower case letters denote instantaneous values.

A Pure Capacitor in an AC circuit

The complex applied emf is jωt]. |
The complex potential difference across the capacitor is equal to the complex applied emf. |

The instantaneous current is:
Multiplying by j is a rotation of 90 |
The complex impedance The complex impedance |

The physical potential and current are:

In brief:

• The current is 90^{0} ahead of the applied potential difference, as shown in the graphs.

• The current is limited by the capacitance and the angular frequency.

Where X_{C} is called the *capacitative reactance*.
It is not a *resistance* but it has units of Ohm because it is a ratio of potential to current and limits the current.

The Power in a capacitor

The instantaneous power in a capacitor is given by: |
The instantaneous power changes at |

Example

The 240 V, 50 Hz mains electricity supply is connected in series with a 5μF capacitor.
Find the rms and instantaneous values of

(a) the current in the circuit, and

(b) the potential difference across the capacitor.

A Pure Inductor in an AC circuit

The complex applied emf is jωt]. |
The complex potential difference across the inductor is equal to the complex applied emf. |

The integration constant is a possible steady D.C. current.
It is set equal to zero here because there is no steady D.C. potential.
Dividing by j is equivalent to multiplying by -j ( since 1/j×j/j = -j ).
Multiplying by -j is a clockwise rotation of 90^{0} in phase, so the current will lag the emf by 90^{0}.

The current in circuit is given by: |
The complex impedance |

The physical potential and current are:

In brief:

• The current is 90^{0} behind the applied potential difference.

• The current is limited by the inductance and the angular frequency.

,
where is called
the *inductive reactance*.
As before, it is not a *resistance* because inductors do not dissipate heat.
Neverthless it has units of Ohm because it is a ratio of potential to current and limits current.

Example

The 240 V, 50 Hz mains electricity supply is connected in series with a 200mH inductor.
Find the rms and instantaneous values of

(a) the current in the circuit, and

(b) the p.d. across the inductor.

*Summarising:*

The average or D.C value:

The potential difference across a resistor is in phase with the current through it.

The average power dissipated in a resistor:

The effective or r.m.s. value of a sinusoidal waveform:

The complex impedance of a pure capacitor: Z_{C} = X_{C} = -j(1/ωC).
The potential difference across a capacitor is 90^{0} behind the phase of the current through it.
The average power in a capacitor is zero

The complex impedance of a pure inductor: Z_{L} = X_{L} = j·ωC.
The potential difference across an inductor is 90^{0} ahead of the phase of the current through it.
The average power in an inductor is zero

Current reference mnemonic: |
"level in the Room, down to the Cellar, up to the Loft". |

Acknowledgement: These notes are based in part on "Alternating Current Circuit Theory" by G.J.Russell and K.Mann NSWUP 1969.

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