doPhysics · Electromagnetism 2
Alternating Current and the LCR Circuit
Sinusoidal voltages and currents, their mean and rms values, and phasors. The resistor, inductor and capacitor on AC, the series LCR circuit with its impedance and phase angle, resonance and the quality factor, and power, the power factor and the choke coil.
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Why this chapter
Why this chapter exists
The electricity in every Indian home arrives as an alternating current, reversing a hundred times a second. A heater, a fan and a radio each answer to it differently. This chapter measures an alternating current by its heating, the rms value, and pictures it as a turning arrow, a phasor. With phasors, a resistor, a coil and a capacitor in series become one right triangle. The same triangle explains resonance, which tunes a radio. It also explains why a tube light carries a choke and why the power company watches the power factor.
You will learn
By the end of this chapter you will
- Write an alternating voltage as \(v_0\sin\omega t\), and find its mean value and its rms value, for a sine and for other waveforms.
- Draw phasors, read instantaneous values from them, and add two sinusoids of one frequency as vectors.
- State and explain the phase of the current in a resistor, an inductor and a capacitor, and use the reactances \(\omega L\) and \(1/\omega C\).
- Draw the phasor diagram of a series LCR circuit, and find its impedance, its current and its phase angle.
- Find the resonant frequency, the quality factor and the bandwidth of a series LCR circuit, and explain radio tuning.
- Find the average power and the power factor of a circuit, and explain the wattless current and the use of a choke.
Before you begin
Before you begin
You should know the sine and cosine as functions of time, and their derivatives. You should know the phase of simple harmonic motion from the chapter Kinematics of SHM. The forced oscillator and its resonance, from the chapter Superposition, Damped and Forced Oscillations, is the mechanical twin of this chapter. From earlier chapters of this unit you need Ohm's law, Joule heating and Kirchhoff's loop rule. You also need the charge \(q = Cv\) on a capacitor and the emf \(-L\,dI/dt\) of a coil of inductance \(L\).
Lessons
1
Alternating Current and Its RMS Value
The mains voltage is a sine wave, \(v = v_0\sin\omega t\), that reverses a hundred times a second. Its mean over a cycle is zero, so it is measured by its heating instead. The rms value is the steady current that heats a resistor equally, \(i_0/\sqrt{2}\) for a sine.
Sinusoidal voltage and currentMean value of a sinusoidRms value and heatingRms of other waveforms
2
Phasors, and the Resistor
A sinusoidal voltage is the projection of an arrow turning at \(\omega\), a phasor. The angle between two phasors is their phase difference, and phasors add like vectors. In a resistor the current is in phase with the voltage, and the power it takes never goes negative.
The phasorThe resistor in an ac circuitAdding two sinusoids with phasorsPower in a resistor
3
The Inductor and the Capacitor
In a pure inductor the current lags the voltage by a quarter cycle, and in a capacitor it leads by a quarter cycle. Their opposition, the reactances \(\omega L\) and \(1/\omega C\), depends on the frequency. Neither takes any average power.
The inductor current lags by a quarter cycleThe capacitor current leads by a quarter cycleReactance against frequencyNo average power in an inductor or a capacitor
4
The Series LCR Circuit
In a series circuit one current flows through \(R\), \(L\) and \(C\). Their voltages are a quarter cycle apart, so they add as phasors, not as numbers. The result is the impedance \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) and the phase angle \(\tan\varphi = (X_L - X_C)/R\).
The phasor diagram of a series circuitImpedance and the phase angleInductive or capacitiveWhy the voltages do not add up
5
Resonance
At one frequency, \(\omega_0 = 1/\sqrt{LC}\), the two reactances cancel. The impedance falls to \(R\), the current peaks and the phase angle is zero, while the inductor and capacitor voltages can far exceed the supply. The quality factor sets how sharp the peak is, which lets a radio pick one station.
The resonance conditionVoltages at resonanceQ factor and bandwidthTuning a radio
6
Power, the Power Factor and the Choke
The average power in an AC circuit is \(V I\cos\varphi\), and only the resistance takes it. The part of the current a quarter cycle from the voltage carries no power but still heats the lines. A choke, a coil of large inductance and small resistance, holds a current down without wasting power.
Average power and the power factorWattless current and why the power company caresThe choke coilWhy a choke and not a resistor
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- alternating current
- A current that reverses its direction periodically. The mains current is a sine of time, \(i = i_0\sin\omega t\), at 50 Hz in India.
- angular frequency
- The rate \(\omega = 2\pi f\) at which the phase of a sinusoid grows, in rad/s. The mains has \(\omega = 314\,\text{rad/s}\).
- apparent power
- The product \(V_{\text{rms}} I_{\text{rms}}\) of a circuit, written in volt-amperes (VA). It equals the true power only when the power factor is 1.
- bandwidth
- The gap \(\Delta\omega = R/L\) between the two half-power points of a series LCR circuit. At those points the current is \(1/\sqrt{2}\) of its peak.
- capacitive
- Said of a circuit in which \(X_C\) is larger than \(X_L\), so the current leads the supply voltage.
- capacitive reactance
- The opposition \(X_C = 1/\omega C\) of a capacitor to an alternating current, in ohms. It falls as the frequency rises, and the current leads the voltage by 90°.
- choke coil
- A coil of large inductance and small resistance, often on a laminated iron core. It limits an alternating current with almost no loss of power.
- direct current
- A current that flows in one direction only and is steady in size, as from a cell.
- impedance
- The ratio \(Z = V/I\) of the rms voltage to the rms current of a circuit, in ohms. For a series LCR circuit \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
- impedance triangle
- A right triangle with \(R\) along the base, \(X_L - X_C\) as the height and \(Z\) as the hypotenuse. Its corner angle is the phase angle \(\varphi\).
- inductive
- Said of a circuit in which \(X_L\) is larger than \(X_C\), so the current lags the supply voltage.
- inductive reactance
- The opposition \(X_L = \omega L\) of an inductor to an alternating current, in ohms. It rises with frequency, and the current lags the voltage by 90°.
- instantaneous value
- The value of an alternating voltage or current at one instant, such as \(v = v_0\sin\omega t\) at time \(t\).
- mean value
- The steady current that would carry the same charge in the same time as an alternating current. For a sine it is zero over a cycle and \(2i_0/\pi\) over a half cycle.
- peak value
- The largest value, \(v_0\) or \(i_0\), reached by a sinusoidal voltage or current in each cycle. The mains peak is 311 V.
- phase angle
- The angle \(\varphi\) by which the supply voltage of a circuit leads its current, with \(\tan\varphi = (X_L - X_C)/R\) in a series circuit. It is negative when the current leads.
- phase difference
- The fixed angle between the phasors of two sinusoids of the same frequency. The one ahead, anticlockwise, reaches its peak first.
- phasor
- An arrow whose length is the peak value of a sinusoid and which turns anticlockwise at \(\omega\). Its projection on the vertical axis is the instantaneous value.
- power factor
- The factor \(\cos\varphi\) in the average power \(P = V_{\text{rms}} I_{\text{rms}}\cos\varphi\) of an AC circuit. It is \(R/Z\) for a series circuit, 1 for a resistor and 0 for a pure inductor or capacitor.
- quality factor
- The number \(Q = \omega_0L/R\) that measures the sharpness of resonance. It equals \(\omega_0\) divided by the bandwidth, and \(V_L/V\) at resonance.
- resonance
- The state of a series LCR circuit driven at the frequency where \(X_L = X_C\). The impedance falls to \(R\), the current is largest and it is in phase with the supply.
- resonant frequency
- The frequency \(\omega_0 = 1/\sqrt{LC}\) at which a series LCR circuit resonates, or \(f_0 = 1/(2\pi\sqrt{LC})\) in hertz.
- rms value
- Root mean square: the square root of the mean of the square of a current or voltage over a cycle. It is the steady value that gives the same heating in a resistor, \(i_0/\sqrt{2}\) for a sine.
- wattless current
- The part \(I\sin\varphi\) of an alternating current that is a quarter cycle out of phase with the voltage. It carries no average power but still heats the wires.