AC Circuit Notes
AC Circuits
AC Sources
Alternating Current (AC) circuits are commonly used in many electrical appliances.
An AC circuit consists of:
A voltage source providing an alternating voltage, \Delta v. where \Delta v = \Delta V_{max} \sin \omega t
Circuit elements: resistors, inductors, and capacitors.
Key questions to consider:
What are the amplitude and time characteristics of the alternating current?
What is the power consumed in AC circuits?
Transformers
AC Voltage
The output of an AC power source is sinusoidal and varies with time:
\Delta v = \Delta V_{max} \sin \omega t
\Delta v: instantaneous voltage
\Delta V_{max}: maximum output voltage (voltage amplitude)
\omega: angular frequency of the AC voltage
\omega = 2 \pi f = \frac{2 \pi}{T}
f: frequency of the source
T: period of the source
The voltage is positive during one half of the cycle and negative during the other half.
Resistors in an AC Circuit
Consider a circuit with an AC source and a resistor.
The instantaneous voltage across the resistor is: \Delta vR = \Delta V{max} \sin \omega t
The instantaneous current in the resistor is: iR = \frac{\Delta vR}{R} = \frac{\Delta V{max}}{R} \sin \omega t = I{max} \sin \omega t
Where: I{max} = \frac{\Delta V{max}}{R}
Ohm's law applies instantaneously: \Delta vR = iR R = I_{max} R \sin \omega t
Phase: The current and voltage reach their maximum values at the same time (in phase).
For a sinusoidal applied voltage, the current in a resistor is always in phase with the voltage across the resistor.
The direction of the current has no effect on the behavior of the resistor.
Resistors behave the same way in both DC and AC circuits.
Phasor Diagrams
A graphical tool to simplify AC circuit analysis.
A phasor is a vector:
Length is proportional to the maximum value of the variable it represents.
Rotates counterclockwise with angular speed \omega (angular frequency).
The projection of the phasor onto the vertical axis represents the instantaneous value.
Representations:
Rectangular coordinates: Voltage on the vertical axis, time on the horizontal axis.
Polar coordinates (phase space):
Radial coordinate: amplitude of the voltage.
Angular coordinate: phase angle.
Vertical axis coordinate of phasor tip: instantaneous voltage.
Horizontal coordinate: meaningless.
RMS (Root Mean Square) Values
The average current in one cycle is zero.
Resistors experience a temperature increase depending on the magnitude of the current, not the direction.
Power is related to the square of the current.
The rms current is the important average quantity in an AC circuit.
Alternating voltages can also be discussed in terms of rms values.
I{rms} = \frac{I{max}}{\sqrt{2}} = 0.707 I_{max}
\Delta V{rms} = \frac{\Delta V{max}}{\sqrt{2}} = 0.707 \Delta V_{max}
Power
The rate at which electrical energy is delivered to a resistor:
P = i^2 R where i is the instantaneous current.
The heating effect of an AC current with maximum value I_{max} is not the same as that of a DC current of the same value because the maximum current occurs for a small amount of time.
The average power delivered to a resistor carrying an alternating current:
P{avg} = I{rms}^2 R
AC ammeters and voltmeters are designed to read rms values.
Inductors in an AC Circuit
Applying Kirchhoff’s loop rule gives: \Delta v - \Delta v_L = 0 or \Delta v = L \frac{di}{dt}
Given the AC voltage \Delta v = \Delta V_{max} \sin \omega t,
The current in the inductor is: iL = \frac{\Delta V{max}}{L} \int \sin \omega t \ dt = -\frac{\Delta V_{max}}{\omega L} \cos \omega t
Where the max. current (Amplitude) is taken as: I{max} = \frac{\Delta V{max}}{\omega L}
Phase: The instantaneous current iL in the inductor and the instantaneous voltage \Delta vL across the inductor are out of phase by \frac{\pi}{2} rad = 90°.
iL(t) = I{max} \sin(\omega t - \frac{\pi}{2})
The current is a maximum when the voltage across the inductor is zero.
For a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90° (\frac{\pi}{2}).
Inductive Reactance
The factor \omega L has the same units as resistance (Ω) and is related to current and voltage, it is frequency-dependent.
Inductive Reactance: X_L = \omega L
The current can be expressed in terms of the inductive reactance:
I{max} = \frac{\Delta V{max}}{X_L}
I{rms} = \frac{\Delta V{rms}}{X_L}
Voltage Across the Inductor
\Delta vL = -L \frac{di}{dt} = - \Delta V{max} \sin \omega t = -I{max} XL \sin \omega t
As frequency increases, inductive reactance increases.
Inductors in an AC Circuit – Phasor Diagram
The phasors are at 90° with respect to each other, representing the phase difference between current and voltage.
In an inductor, the current lags behind the voltage by 90°.
Capacitors in AC Circuits
The circuit contains a capacitor and an AC source.
The potential difference of the source is the same as that of the capacitor: \Delta v = \Delta vC = \Delta V{max} \sin \omega t
The charge on the capacitor is: q(t) = C \Delta V_{max} \sin \omega t
The instantaneous current is given by:
i = \frac{dq}{dt} = \omega C \Delta V{max} \cos \omega t = \omega C \Delta V{max} \sin(\omega t + \frac{\pi}{2})
The current is \frac{\pi}{2} rad = 90° out of phase with the voltage.
The current reaches its maximum value one-quarter of a cycle before the voltage reaches its maximum value.
Capacitive Reactance
The maximum current in the circuit occurs at \cos \omega t = 1 which gives
I{max} = \omega C \Delta V{max} = \frac{\Delta V_{max}}{1/\omega C}
The unit of \frac{1}{\omega C} is Ohm (Ω).
Capacitive Reactance: X_C \equiv \frac{1}{\omega C}
Voltage across a capacitor
The instantaneous voltage across the capacitor:
\Delta vC = \Delta V{max} \sin \omega t = I{max} XC \sin \omega t
As the frequency of the voltage source increases, the capacitive reactance decreases, and the maximum current increases.
As the frequency decreases approaching zero, X_C approaches infinity, and the current approaches zero (acts like an open circuit in DC).
Capacitors in an AC Circuit – Phasor Diagram
The current always leads the voltage across a capacitor by 90°.
RLC Series Circuit
Combining a resistor, inductor, and capacitor in an AC circuit.
The instantaneous voltage applied from the source: \Delta v = \Delta V_{max} \sin \omega t
The instantaneous current : I = I_{max} \sin (\omega t - \varphi)
\varphi is the phase angle between the current and the applied voltage.
The current is the same at all points in the circuit (same amplitude and phase).
I and V Relationship
\Delta VR is the maximum voltage across the resistor and \Delta VR = I_{max}R
\Delta VL is the maximum voltage across the inductor and \Delta VL = I{max}XL
\Delta VC is the maximum voltage across the capacitor and \Delta VC = I{max}XC
The sum of these voltages must equal the voltage from the AC source.
They cannot be added directly but must be added as vectors due to different phase relationships.
Vector Addition of \Delta V
Total Voltage in RLC Circuits
From the vector diagram: \Delta V{max} = \sqrt{\Delta VR^2 + (\Delta VL - \Delta VC)^2}
\Delta V{max} = \sqrt{(I{max}R)^2 + (I{max}XL - I{max}XC)^2} = I{max} \sqrt{R^2 + (XL - X_C)^2}
Impedance
The current in an RLC circuit: I{max} = \frac{\Delta V{max}}{Z}
Z: impedance of the circuit (plays the role of resistance in the circuit) and it is measured in Ohms.
Z = \sqrt{R^2 + (XL - XC)^2}
Phase Angle
From the vector diagram, the phase angle \varphi for the current with respect to the applied voltage:
\tan \varphi = \frac{XL - XC}{R}
The phase angle can be positive or negative and determines the nature of the circuit.
Circuit Classification
If \varphi is positive:
XL > XC (high frequencies).
The current lags the applied voltage by phase angle \varphi.
The circuit is more inductive than capacitive.
If \varphi is negative:
XL < XC (low frequencies).
The current leads the applied voltage by phase angle \varphi.
The circuit is more capacitive than inductive.
If \varphi is zero:
XL = XC
The circuit is purely resistive.
Impedance in Series and in Parallel
Series Combination: Z{eff} = Z1 + Z2 + Z3
Parallel Combination: \frac{1}{Z{eff}} = \frac{1}{Z1} + \frac{1}{Z2} + \frac{1}{Z3}
Power in AC Circuits
The average power delivered by the AC source is converted to thermal energy in the resistor (ideal capacitor and inductor).
The instantaneous power in the circuit is:
P = I \Delta V = (I{max} \sin(\omega t - \varphi))(\Delta V{max} \sin(\omega t))
The average power over one cycle:
P{avg} = \frac{1}{T} \int0^T P \ dt
The average power is then: P{avg} = I{rms} \Delta V_{rms} \cos \varphi
Also: P{avg} = I{rms}^2 R
Resonance in AC Circuits
Resonance occurs when inductive reactance equals capacitive reactance
XL = XC
\omega0 L = \frac{1}{\omega0 C}
\omega_0 = \frac{1}{\sqrt{LC}}
Resonance happens at \omega_0 independent of R.
As R decreases, the curve becomes narrower and taller; if R = 0, the current would be infinite.
Power as a function of frequency
The average power consumed in an RLC circuit:
P{av} = \frac{\Delta V{rms}^2 R \omega^2}{ R^2 + (\omega L - \frac{1}{\omega C})^2 }
At resonance, the average power is a maximum.
Quality Factor in AC Circuits
Describes the quality of the circuit in terms of energy: the ratio between the maximum energy stored to the energy dissipated.
Describes the sharpness of the resonance curve.
Q = \omega \frac{Energy \ stored}{Energy \ dissipated} = \frac{I^2 X_L}{I^2 R} = \frac{\omega L}{R}
A high-Q circuit responds to a narrow range of frequencies, while a low-Q circuit responds to a broader range.
Transformers
Two coils of wire wound around a core of iron.
Primary coil: connected to the input AC voltage source, with N_1 turns.
Secondary coil: connected to a load resistor, with N_2 turns.
The core increases magnetic flux into the secondary coil.
Eddy-current losses are minimized by using a laminated core (90% to 99% power efficiency).
Voltages are related by
\frac{\Delta v2}{\Delta v1} = \frac{N2}{N1}
Transformers - Classification
Step-up transformer: N2 > N1
Step-down transformer: N2 < N1
The power input into the primary equals the power output at the secondary
I1 \Delta V1 = I2 \Delta V2
The equivalent resistance of the load resistance when viewed from the primary:
R{eq} = RL (\frac{N1}{N2})^2
Problems - Ch 33 (AC circuits)
P1
An inductor (L = 400 mH), a capacitor (C = 4.43 μF), and a resistor (R = 500 Ω) are connected in series. A 50.0-Hz AC source produces a peak current of 250 mA in the circuit.(a) Calculate the required peak voltage ∆Vmax. (b) Determine the phase angle by which the current leads or lags the applied voltage. (c) Write an expression for such current as a function of time.
P2
A series AC circuit contains a resistor, an inductor of 150 mH, a capacitor of 50 μF , and a source with ∆Vmax = 240 V operating at 50.0 Hz. The maximum current in the circuit is 100 mA. Calculate (a) the inductive reactance, (b) the capacitive reactance, (c) the impedance, (d) the resistance in the circuit, and (e) the phase angle between the current and the source voltage.
P3
In a certain series RLC circuit, Irms= 9.00 A, ΔVrms = 180 V, and the current leads the voltage by 37°. (a) What is the total resistance of the circuit? (b) Calculate the reactance of the circuit (XL - XC).
P4
A sinusoidal voltage ΔV= 40.0 sin 100πt, where ΔV is in volts and t is in seconds, is applied to a series RLC circuit with L = 160 mH, C = 99.0 μF, and R = 68.0 Ω. (a) What is the impedance of the circuit? (b) What is the maximum current? (c) Determine the numerical values for ω and in the equation I = Imax sin (ωt -). (d) Is the circuit inductive or capacitive? (e) Calculate the power factor of the circuit and the average power consumption in the circuit.
LC Circuits
Magnetic and Electric Energy