AC Circuits with Capacitors and Inductors 1.7 1.8

Impedance and Reactance

• Circuits with capacitors and inductors are more complicated than resistive circuits because their behavior depends on frequency.
• A "voltage divider" containing a capacitor or inductor will have a frequency-dependent division ratio.
• Circuits containing these components "corrupt" input waveforms such as square waves.
• Capacitors and inductors are linear devices, meaning the amplitude of the output waveform increases exactly in proportion to the input waveform's amplitude.
• The output of a linear circuit driven with a sinewave at some frequency f is itself a sinewave at the same frequency (with, at most, changed amplitude and phase).

Frequency Response

• The output of a linear circuit can be analyzed by asking how the output voltage (amplitude and phase) depends on the input voltage for sinewave input at a single frequency.
• A graph of the resulting frequency response is useful for thinking about many kinds of waveforms.
• A frequency response graph shows the ratio of output to input for each sinewave frequency.

Impedance

• Impedance (Z) is the "generalized resistance" that describes any circuit containing linear passive devices (resistors, capacitors, and inductors).
• Inductors and capacitors have reactance (X) and are reactive, with voltage and current always 90° out of phase.
• Resistors have resistance (R) and are resistive, with voltage and current always in phase.
• In a circuit that combines resistive and reactive components, the voltage and current have a complex impedance: impedance = resistance + reactance, or Z = R + jX.
• Impedance can be used to describe the behavior of a capacitor or inductor at a specific frequency.

Frequency Analysis of Reactive Circuits

• A capacitor driven by a sinusoidal voltage source V(t) = V0 sin(ωt) has a current I(t) = C dV/dt = ωCV0 cos(ωt), with a phase lead of 90°.
• The ratio of voltage to current amplitude is I = V / (ωC), similar to a frequency-dependent resistance R = 1 / (ωC).
• The reactance of a capacitor, XC, is defined as XC = 1 / (ωC).

RC Lowpass Filter

• A lowpass filter passes low frequencies and blocks high frequencies.
• The circuit has a capacitor and a resistor, with Vout/Vin = 1 / (1 + (1/ωRC)).
• The crossover frequency ω0 is defined as ω0 = 1/RC.
• At frequencies well below the crossover frequency, the output decreases inversely with increasing frequency.

RC Highpass Filter

• A highpass filter passes high frequencies and blocks low frequencies.
• The circuit has a capacitor and a resistor, with Vout/Vin = ωRC / (1 + ωRC).
• The crossover frequency ω0 is defined as ω0 = 1/RC.
• At frequencies well above the crossover frequency, the output increases directly with increasing frequency.

Blocking Capacitor

• A blocking capacitor is used to block DC voltage and pass AC signals.
• The capacitor is used in a highpass filter configuration to block DC and allow AC signals to pass through.
• The crossover frequency ω0 is defined as ω0 = 1/RC, and the product RC is chosen to be greater than 1/ωmin, where ωmin is the minimum frequency of interest.

Driving and Loading RC Filters

• When driving an RC filter, the output impedance of the signal source should be small compared to the input impedance of the filter.
• When loading an RC filter, the input impedance of the load should be large compared to the output impedance of the filter.
• The worst-case impedance of an RC filter is R, making it easy to design filters with predictable behavior.

Reactance of Inductors

• The reactance of an inductor (XL) is frequency-dependent and increases with increasing frequency.
• XL = ωL, where ω is the angular frequency and L is the inductance.
• Inductors can be used to pass DC and low frequencies while blocking high frequencies.

Voltages and Currents as Complex Numbers

• Voltages and currents can be represented using complex numbers.
• A single number is not enough to specify the current at a point in the circuit, as both magnitude and phase shift need to be considered.
• Complex numbers can be used to add or subtract voltages and currents, making calculations simpler.
• A rule is developed to convert between actual quantities and their complex number representations.

Converting Between Actual and Complex Representations

• Voltages and currents are represented by complex quantities V and I.
• The voltage V0 cos(ωt + φ) is represented by the complex number V0e^(jφ).
• Actual voltages and currents are obtained by multiplying the complex number representations by e^(jωt) and taking the real part.

Examples

• A voltage with a complex representation of V = 5j corresponds to a real voltage versus time of V(t) = -5 sin(ωt) volts.

Reactance of Capacitors and Inductors

• To apply complex Ohm's law to circuits with capacitors and inductors, we need to know their reactance.
• For a capacitor, the impedance (ZC) is given by -j/ωC, where ω is the frequency and C is the capacitance.
• The reactance (XC) of a capacitor is 1/ωC, and it is a real number.
• The impedance of a capacitor is purely imaginary, indicating a 90° phase shift between voltage and current.
• For a 1 μF capacitor, the impedance at 60 Hz is -2653j Ω and at 1 MHz is -0.16j Ω, with corresponding reactances of 2653 Ω and 0.16 Ω.
• At dc, the reactance (and impedance) of a capacitor is infinite.

Reactance of Inductors

• For an inductor, the impedance (ZL) is given by jωL, where ω is the frequency and L is the inductance.
• The reactance (XL) of an inductor is ωL, and it is a real number.

General Properties of Capacitors and Inductors

• A circuit containing only capacitors and inductors has a purely imaginary impedance, indicating a 90° phase shift between voltage and current.
• When resistors are added to the circuit, the impedance has a real part.
• In this case, the reactance refers to the imaginary part of the impedance only.

Generalized Ohm's Law

• Ohm's law in complex form: I = V/Z, V = IZ, where V is the voltage across a circuit with impedance Z, giving a current I.

Series and Parallel Impedance

• The complex impedance of devices in series: Z = Z1 + Z2 + Z3 + ...
• The complex impedance of devices in parallel: 1/Z = 1/Z1 + 1/Z2 + 1/Z3 + ...

Impedance of Devices

• Impedance of a resistor: ZR = R
• Impedance of a capacitor: ZC = -j/ωC = 1/jωC
• Impedance of an inductor: ZL = jωL

Analysis of AC Circuits

• AC circuits can be analyzed using series and parallel formulas and Ohm's law.
• Multiply-connected networks may require Kirchhoff's laws.

Example: Capacitor Circuit

• A 1μF capacitor is connected across a 115V 60Hz powerline.
• The impedance of the capacitor: Z = -j/ωC
• The current: I = V/Z = jωCA ≈ -0.061 sin ωt

Power in Reactive Circuits

• The phase angle between current and voltage in a two-terminal RLC circuit is equal to the angle of the complex impedance of that circuit.
• No power is dissipated by the capacitor in the example.

Power in Reactive Circuits

• Instantaneous power delivered to a circuit element is P = V I.
• In reactive circuits, V and I are not simply proportional, so multiplying their amplitudes together does not work.

Power in Capacitors

• During certain time intervals, power is delivered to the capacitor, causing it to charge up.
• During other intervals, the power delivered to the capacitor is negative, causing it to discharge.
• The average power over a whole cycle for a capacitor is zero.

Average Power

• Average power consumed by an arbitrary circuit can be found by adding up little pieces of V I product and dividing by the elapsed time: P = 1/T ∫ T 0 V(t)I(t) dt.
• Alternatively, the average power is given by P = Re(VI*), where V and I are complex rms amplitudes.

Example Circuit

• Consider a circuit with a 1 volt (rms) sinewave driving a capacitor: P = Re(VI*) = Re(-j/ωC) = 0, so the average power is zero.

Power in Series RC Circuit

• In a series RC circuit, the impedance is Z = R - j/ωC.
• The current is I = V/Z = V0 / (R - j/ωC).
• The average power is P = Re(VI*) = V²₀ R / (R² + (1/ω²C²)).

Power Factor

• The power factor is the ratio of the average power to the product of the magnitudes of V and I: power factor = P / (|V| |I|).
• It ranges from 0 (purely reactive circuit) to 1 (purely resistive).
• A power factor of less than 1 indicates some component of reactive current.

Importance of Power Factor

• Power factor is a serious matter in large-scale electrical power distribution.
• Reactive currents don't result in useful power being delivered to the load, but cost the power company in terms of I²R heating in generators, transformers, and wiring.
• Power companies charge industrial users according to the power factor.

Generalized Voltage Dividers

• A generalized voltage divider is a circuit where either or both resistors are replaced with a capacitor or inductor (or a more complicated network made from R, L, and C).
• The division ratio Vout/Vin of such a divider depends on frequency.
• The analysis of the circuit is straightforward, with the current I equal to Vin divided by the total impedance Ztotal.
• Ztotal is the sum of impedances Z1 and Z2.

Characteristics of Generalized Voltage Dividers

• The output voltage Vout is equal to IZ2, which can also be expressed as Vin times Z2 divided by the sum of Z1 and Z2.
• In nonlinear circuits, the current waveform is not proportional to the voltage waveform, which will be discussed further in section 9.7.1.

Examples of Generalized Voltage Dividers

• RC highpass and lowpass filters are simple but important examples of generalized voltage dividers.
• These filters were approximated earlier in section 1.7.1.

RC Highpass Filters

• A highpass filter is a circuit that passes high-frequency signals while rejecting low-frequency signals.
• The complex Ohm's law (or the complex voltage-divider equation) gives the output voltage of a highpass filter as:
  • Vout = Vin * R / (R + j/ωC)
  • Vout = Vin * R / √(R2 + (1/ω2C2))
• The impedance of the series RC combination is:
  • Ztotal = √(R2 + 1/ω2C2)
  • φ = tan^(-1)(-1/ωC/R)
• The frequency response of the highpass filter is:
  • Vout ∝ ω
  • Vout = R / √(R2 + (1/ω2C2)) * Vin
  • The -3 dB breakpoint is given by:
    • ω3dB = 1/RC
    • f3dB = 1/(2πRC)

Impedance and Reactance

• The impedance of a capacitor is given by:
  • Z = 1/jωC
  • |Z| = 1/ωC
• The graph in Figure 1.100 provides a useful reference for the impedance of capacitors and inductors versus frequency.
• The impedance of a load driven by a highpass filter should be much larger than the output impedance of the filter to prevent circuit loading effects.
• The driving source should be able to drive a load with an impedance similar to the output impedance of the filter to prevent circuit loading effects on the signal source.

Example of a Highpass Filter

• The filter shown in Figure 1.101 has a 3 dB point at 15.9 kHz.
• The impedance of the load driven by this filter should be much larger than 1.0k in order to prevent circuit loading effects on the filter's output.

Lowpass Filters

• A lowpass filter can be created by interchanging R and C in an RC circuit.
• The output of a lowpass filter can be described by the equation: Vout = 1 / (1 + ω2R2C2)1/2 Vin.
• The 3 dB point of a lowpass filter occurs at a frequency of f = 1/2πRC.
• Lowpass filters are useful in real life for eliminating interference from nearby radio and television stations.
• The frequency response of a lowpass filter can be viewed as a signal source with output impedance that drops to zero at high frequencies.
• The signal driving the filter sees a load of R plus the load resistance at low frequencies, dropping to just R at high frequencies.

Frequency Response

• The frequency response of a lowpass filter can be plotted on logarithmic axes, showing the phase shift and amplitude response.
• The phase shift is -45° at the -3 dB point and is within 6° of its asymptotic value for a decade of frequency change.
• The amplitude response of a lowpass filter becomes a straight line at large attenuations, with a slope of -20 dB/decade.
• A rule of thumb for single-section RC filters is that the phase shift is ≈ 6° from its asymptotic value at 0.1 f3 dB and at 10 f3 dB.

Filter Design

• It is not possible to create a filter with arbitrary specified amplitude and phase responses due to the demands of causality.
• The relationship between phase and amplitude response of realizable analog filters is known as the Kramers-Kronig relation.

RC Differentiators and Integrators in the Frequency Domain

• An RC differentiator is equivalent to a high-pass filter, and its operation can be viewed in either the time domain or frequency domain.
• For an RC differentiator to operate properly, the signal frequency must be well below the 3 dB point.
• The condition for proper operation can be expressed as ωRC ≪ 1, where ω is the signal frequency.
• If the input signal contains multiple frequencies, the condition must be met for the highest frequency present.

RC Integrators

• An RC integrator is equivalent to a low-pass filter.
• The criterion for a good integrator is that the lowest signal frequencies must be well above the 3 dB point.

Inductors versus Capacitors

• Inductors can be used with resistors to make low-pass or high-pass filters, but they are rarely used due to their bulkiness, expense, and departure from ideal behavior.
• Capacitors are generally preferred over inductors for filtering applications.
• An important exception is the use of ferrite beads and chokes in high-frequency circuits, which add inductance to prevent oscillations without introducing significant series resistance.
• Inductors are essential components in LC tuned circuits and switch-mode power converters.

Phasor Diagrams

• Phasor diagrams are a graphical method to understand reactive circuits, especially RC filters.
• The diagram represents the impedance of a circuit, with real (resistive) and imaginary (reactive) components.
• In a series circuit, the axes also represent the complex voltage, as the current is the same everywhere.

RC Filters

• At the frequency f = 1/2πRC, an RC filter attenuates by 3 dB.
• The phasor diagram shows that the ratio of output voltage to input voltage is 1/√2, or -3 dB.
• The angle between the vectors gives the phase shift from input to output, which is 45° at the 3 dB point.

Poles and Decibels per Octave

• An RC lowpass filter has a 6 dB/octave falloff beyond the "knee" frequency.
• A filter with multiple RC sections has a steeper falloff, e.g., 12 dB/octave for two sections, 18 dB/octave for three sections.
• A "pole" refers to an RC section, so a "three-pole filter" is a filter with three RC sections.

Multistage Filters

• Cascading identical filter sections does not produce a frequency response that is the concatenation of individual responses.
• Each stage loads the previous one, changing the overall response.
• Solutions include using higher impedance for each successive section or active circuits with interstage "buffers" or active filters.

Resonant Circuits

• Resonant circuits are a type of circuit that combines capacitors and inductors to produce a sharp frequency response.
• They are used in various audio and RF devices.

LC Circuits

• LC circuits consist of a combination of inductors (L) and capacitors (C) in a circuit.
• The impedance of the LC combination at frequency f is given by the formula:

ZLC = j(1/ωL - ωC)

• When an LC circuit is combined with a resistor (R), it forms a voltage divider.

Parallel LC Resonant Circuit

• The impedance of the parallel LC goes to infinity at the resonant frequency (f0) given by the formula:

f0 = 1/2π√LC

• This results in a peak in the response at the resonant frequency.
• The quality factor (Q) of a parallel LC circuit is a measure of the sharpness of the peak and is given by:

Q = ω0RC

Series LC Resonant Circuit

• The impedance of the series LC goes to zero at the resonant frequency (f0) given by the formula:

f0 = 1/2π√LC

• This circuit is known as a "notch" filter or "trap" because it shorts signals at or near the resonant frequency to ground.
• The quality factor (Q) of a series RLC circuit is given by:

Q = ω0L/R

Time Domain Response

• In the time domain, the response of an LC circuit to a pulse or step input is characterized by a damped oscillation ("ringing") waveform.
• The signal voltage falls to 1/e (37%) in Q/π cycles, and the stored energy falls to 1/e (61% in amplitude) in Q/2π cycles.

Alternative Resonant Circuits

• Other alternatives to LC resonant circuits include quartz-crystal, ceramic, and surface acoustic-wave (SAW) resonators; transmission lines; and resonant cavities.

Resonant Circuits

• Resonant circuits are a type of circuit that combines capacitors and inductors to produce a sharp frequency response.
• They are used in various audio and RF devices.

LC Circuits

• LC circuits consist of a combination of inductors (L) and capacitors (C) in a circuit.
• The impedance of the LC combination at frequency f is given by the formula:

ZLC = j(1/ωL - ωC)

• When an LC circuit is combined with a resistor (R), it forms a voltage divider.

Parallel LC Resonant Circuit

• The impedance of the parallel LC goes to infinity at the resonant frequency (f0) given by the formula:

f0 = 1/2π√LC

• This results in a peak in the response at the resonant frequency.
• The quality factor (Q) of a parallel LC circuit is a measure of the sharpness of the peak and is given by:

Q = ω0RC

Series LC Resonant Circuit

• The impedance of the series LC goes to zero at the resonant frequency (f0) given by the formula:

f0 = 1/2π√LC

• This circuit is known as a "notch" filter or "trap" because it shorts signals at or near the resonant frequency to ground.
• The quality factor (Q) of a series RLC circuit is given by:

Q = ω0L/R

Time Domain Response

• In the time domain, the response of an LC circuit to a pulse or step input is characterized by a damped oscillation ("ringing") waveform.
• The signal voltage falls to 1/e (37%) in Q/π cycles, and the stored energy falls to 1/e (61% in amplitude) in Q/2π cycles.

Alternative Resonant Circuits

• Other alternatives to LC resonant circuits include quartz-crystal, ceramic, and surface acoustic-wave (SAW) resonators; transmission lines; and resonant cavities.

LC Filters

• Combining inductors with capacitors produces filters with sharper behavior in frequency response than RC filters.
• Example: Figure 1.111 shows a mixer-digitizer circuit board with six LC lowpass filters designed to cut off at 1.0 MHz.
• LC lowpass filter outperforms an RC lowpass filter with the same 1 MHz cutoff frequency in a frequency sweep test.

Capacitor Applications

Bypassing

• Capacitors can bypass ac signals while allowing dc voltage to pass through.
• Capacitor impedance decreases with increasing frequency, making them useful for bypassing.
• Bypass capacitors are chosen to have low impedance at signal frequencies.

Power-Supply Filtering

• Capacitors are used to filter ripple from rectifier circuits in power supplies.
• These capacitors are often referred to as "filter capacitors," but they are actually storage capacitors.
• Storage capacitors are large and used for energy storage in power supplies.

Timing and Waveform Generation

• Capacitors charged with a constant current generate a ramp waveform.
• This principle is used in ramp and sawtooth generators, analog function generators, oscilloscope sweep circuits, and timing circuits.
• RC circuits are used for timing and form the basis of delay circuits (monostable multivibrators).

Thévenin's Theorem Generalized

• Thévenin's theorem can be applied to any two-terminal network of resistors, capacitors, inductors, and signal sources.
• The network can be equivalent to a single complex impedance in series with a single signal source.
• The complex impedance and signal source can be found from the open-circuit output voltage and the short-circuit output current.

AM Radio

• An AM radio signal consists of a radio frequency (RF) carrier whose amplitude is varied by the audio frequency signal.
• The audio frequency waveform (20Hz ~ 5kHz) is added to the RF carrier (~1MHz) to create a modulated signal.
• The AM signal is received and the task is to select the desired station and extract the modulating envelope, which is the desired audio signal.

Simple AM Radio Circuit

• The simplest AM radio consists of a parallel LC resonant circuit, tuned to the station's frequency by a variable capacitor.
• A diode acts as a half-wave rectifier, passing only the positive half-cycles of the modulated carrier.
• A small capacitor is added to prevent the output from following the fast half-cycles of the carrier.
• The time constant R1C2 is chosen to be long compared to a carrier period, but short compared to the period of the highest audio frequency.

AM Radio Signal Processing

• The bare antenna receives plenty of low-frequency pickup (mostly 60 Hz ac powerline) and a tiny bit of signal from all the AM stations at once.
• When the signal is connected to the LC resonant circuit, the low-frequency noise disappears and the selected AM station is amplified.
• The resonant circuit's high Q stores energy from multiple cycles of the signal, increasing the amplitude of the selected station.

Observations and Experiments

• The observed waveforms at point "X" show the low-frequency noise disappearing and the radio signal getting larger when the LC resonant circuit is connected.
• The observed waveforms at point "Y" show the rectified wave with and without the smoothing capacitor C2.
• Probing point "X" with a length of BNC cable can change the resonant frequency and tune to a different station, even changing languages.