Alternating Current (A.C.) Circuits
An A.C. circuit is one in which the magnitude of the current changes periodically with time. A.C. is produced by an alternating voltage supply. The pattern of the A.C. voltage follows a sinusoidal wave, meaning it varies like a sine curve with constant amplitude and frequency.
\( V_0 \) is the maximum or peak voltage, which represents the maximum displacement (amplitude).
\( V \) is the instantaneous voltage, representing the displacement at any given time.
The relationship between peak voltage and instantaneous voltage is given by:
\[ \frac{V}{V_0} = \sin \theta \] \[ V = V_0 \sin 2\pi f t = V_0 \sin \omega t \]Root Mean Square (RMS) Voltage
The RMS voltage is defined as the steady voltage that would produce the same heating effect per second in a given resistor as the alternating voltage.
Similarly, the RMS value of current is the steady current that would dissipate power at the same rate in a given resistor.
\[ V_0 = \sqrt{2} V_{\text{rms}} \]\[V_{\text{rms}} = \frac{V_0}{\sqrt{2}}\]
\[ V = \sqrt{2} V_{\text{rms}} \sin \omega t \] \[ V = \sqrt{2} V_{\text{rms}} \sin (2\pi f t) \]where \( \omega = 2\pi f \).
Alternating Current
\[ I = I_0 \sin \theta = \sqrt{2} I_{\text{rms}} \sin \omega t \]\[I_{\text{rms}} = \frac{I_0}{\sqrt{2}}\]
\[ I = \sqrt{2} I_{\text{rms}} \sin (2\pi f t) \] \[ I_0 = I_{\text{rms}} \sqrt{2} \]Definitions
- \( V \) : Instantaneous voltage
- \( V_0 \) : Peak voltage
- \( V_{\text{rms}} \) : Root mean square (RMS) voltage
- \( I \) : Instantaneous current
- \( I_0 \) : Peak current
- \( I_{\text{rms}} \) : Root mean square (RMS) current
- \( \theta \) : Phase angle between voltage and current
- \( \omega \) : Angular velocity
A.C. in Resistors
The ability of a resistor to restrict the flow of current in an A.C. circuit is called its resistance \( R \).
When an A.C. voltage is applied to a resistor, both the current and the voltage reach their maximum and minimum values at the same time. This means they are in phase.
According to Ohm’s law:
\[ R = \frac{V}{I} \] \[ I_0 = \frac{V_0}{R} \] \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{R} \]
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Capacitive Reactance
The ability of a capacitor to resist the flow of current in an A.C. circuit is called its Capacitive Reactance, denoted as \( X_c \).
The reactance of a capacitor is given by:
\[ X_c = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]where:
- \( X_c \) = Capacitive Reactance (in ohms, \( \Omega \))
- \( \omega \) = Angular frequency (\( \omega = 2\pi f \))
- \( f \) = Frequency of the A.C. supply (Hz)
- \( C \) = Capacitance (Farads)
The RMS current is given by:
\[ I_{\text{rms}} = \frac{V_{\text{rms}}}{X_c} \]Similarly, the peak current is:
\[ I_0 = \frac{V_0}{X_c} \]Phase Relationship in a Capacitor
In an A.C. circuit containing a capacitor, the current \( I_c \) leads the voltage \( V_c \) by:
\[ \frac{\pi}{2} \text{ radians} \quad \text{or} \quad 90^\circ \quad \text{or by} \quad \frac{1}{4} \text{ cycle} \]Inductance in A.C. Circuits
The voltage across an inductor, \( V_L \), leads the current, \( I_L \), by \( \frac{\pi}{2} \) radians or \( 90^\circ \). The induced electromotive force (e.m.f.) in the inductor \( L \) opposes changes in current.
As a result, the current lags behind the voltage in the circuit by \( \frac{\pi}{2} \) radians or \( 90^\circ \), which is equivalent to a quarter of a cycle.
The phase difference between \( I \) and \( V \) is:
\[ V = V_0 \sin(\omega t) \] \[ I = I_0 \sin(\omega t - \frac{\pi}{2}) \]Inductive Reactance
Like resistors and capacitors, an inductor \( L \) opposes the flow of current, producing an impedance effect known as inductive reactance, \( X_L \).
\[ V = I X_L \]The unit of \( X_L \) is ohms (\(\Omega\)).
\[ X_L = 2\pi f L \]Where:
- \( L \) is inductance in Henry (H)
- \( f \) is frequency in hertz (Hz)
- \( X_L \) is inductive reactance in ohms (\(\Omega\))
Reactance
Reactance is the opposition to the flow of alternating current (A.C.) offered by a capacitor, an inductor, or both.
Inductor and Resistor in series (RL circuit)
When an inductor is connected in series with a resistor, the voltage leads the current by \( 90^\circ \).
Voltage Relationship
From the vector diagram, the total voltage is given by:
\[ V^2 = V_L^2 + V_R^2 \]Since \( V_L = I X_L \) and \( V_R = I R \), we can rewrite the equation as:
\[ V^2 = I^2 X_L^2 + I^2 R^2 \] \[ \frac{V^2}{I^2} = X_L^2 + R^2 \]Impedance (Z)
By definition, the impedance \( Z \) is:
\[ Z = \frac{V}{I} \]Squaring both sides:
\[ Z^2 = \frac{V^2}{I^2} \] \[ Z^2 = X_L^2 + R^2 \] \[ Z = \sqrt{X_L^2 + R^2} \]Phase Angle (θ)
The phase angle is given by:
\[ \tan \theta = \frac{V_L}{V_R} = \frac{I X_L}{I R} = \frac{X_L}{R} \]Capacitor and Resistor in Series (RC Circuit)
When a capacitor is connected in series with a resistor, the total opposition to the current flowing through the circuit is called Impedance (Z). In this case, the current leads the voltage by \( 90^\circ \).
Voltage Relationship
From the vector diagram, the total voltage is given by:
\[ V^2 = V_C^2 + V_R^2 \]Since \( V_C = I X_C \) and \( V_R = I R \), we can rewrite the equation as:
\[ V^2 = I^2 X_C^2 + I^2 R^2 \] \[ \frac{V^2}{I^2} = X_C^2 + R^2 \]Impedance (Z)
By definition, the impedance \( Z \) is:
\[ Z = \frac{V}{I} \]Squaring both sides:
\[ Z^2 = \frac{V^2}{I^2} \] \[ Z^2 = X_C^2 + R^2 \] \[ Z = \sqrt{X_C^2 + R^2} \]Phase Angle (θ)
The phase angle is given by:
\[ \tan \theta = \frac{V_C}{V_R} = \frac{I X_C}{I R} = \frac{X_C}{R} \]A.C Circuit Containing Only Inductor and Capacitor (L-C Circuit)
In an L-C circuit, the total opposition to current flow is given by the impedance \( Z_{LC} \), which depends on the inductive reactance \( X_L \) and the capacitive reactance \( X_C \).
Impedance in an L-C Circuit
\[ Z_{LC} = \sqrt {X_L - X_C} \]Current in an L-C Circuit
\[ I_0 = \frac{V_0}{\sqrt{X_L - X_C}} \]Phase Relationship
In an L-C circuit, the inductor voltage \( V_L \) and the capacitor voltage \( V_C \) are out of phase by \( 180^\circ \), meaning:
\[ V_L = -V_C \]Effective Voltage
The total effective voltage across the circuit is given by:
\[ V = V_L - V_C \]Series Circuit Containing Resistance (R), Inductance (L), and Capacitance (C)
If an alternating voltage is applied across the circuit:
\[ V = V_0 \sin(2\pi ft) \]A steady-state current flows through the circuit given by:
\[ I = I_0 \sin(2\pi ft) \]The maximum or peak value of the current is:
\[ I_0 = \frac{V_0}{\sqrt{R^2 + (X_L - X_C)^2}} \] \[ I_0 = \frac{V_0}{\sqrt{R^2 + X^2}}, \quad \text{where} \quad X = X_L - X_C \]Defining the impedance \( Z \) as:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]Then:
\[ I_0 = \frac{V_0}{Z} \] \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} \]Impedance (Z) is the overall opposition of a circuit containing a resistor, an inductor, and/or a capacitor. It is measured in ohms (\(\Omega\)).
Reactance Values
The capacitive reactance (\( X_C \)) is:
\[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]The inductive reactance (\( X_L \)) is:
\[ X_L = \omega L = 2\pi f L \]Thus, the impedance can also be written as:
\[ Z = \sqrt{R^2 + \left( \omega L - \frac{1}{\omega C} \right)^2} \] \[ Z = \sqrt{R^2 + \left( 2\pi f L - \frac{1}{2\pi f C} \right)^2} \]Summary of Voltages
- \( V = IR \)
- \( V_L = I X_L \)
- \( V_C = I X_C \)
- \( V = IZ \)
- \( V = I \sqrt{R^2 + (X_L - X_C)^2} \)
Resonance in A.C Circuit
Resonance is said to occur in an a.c series circuit when the maximum current is obtained from such a circuit. The frequency at which this resonance occur is called the resonance frequency (fo).
Resonance Condition
At resonance, the inductive reactance \( X_L \) and the capacitive reactance \( X_C \) are equal:
\[ X_L = X_C \]This implies that:
\[ V_L = V_C \]Resonance Frequency
The resonance frequency \( f_0 \) is given by:
\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]Maximum Current at Resonance
At resonance, the impedance of the circuit is minimized, and the maximum current is:
\[ I_{\text{max}} = \frac{V}{R} \]Power in an A.C Circuit
Average Power
The average power in an A.C circuit is given by:
\[ P = IV \cos \phi \]Where:
- \( I \) and \( V \) are the effective (RMS) values of current and voltage, respectively.
- \( \phi \) is the phase angle (angle of lag or lead between them).
Power Factor
The quantity \( \cos \phi \) is known as the power factor of the device.
The power factor can have any value between zero and unity for \( \phi \) varying from \( 90^\circ \) to \( 0^\circ \). For \( \phi = 90^\circ \) or \( \cos \phi = 0 \), the average power \( P \) is zero.
A power factor of zero means the device is either a pure reactance (inductance or capacitance). This implies that no power is dissipated in an inductor or capacitor.
Power in a Circuit with Resistance
If \( I \) is the RMS value of the current in a circuit containing a resistance \( R \), the power absorbed in the resistance is given by:
\[ P = I^2 R \]Instantaneous Power
For an A.C circuit, the instantaneous power is given by:
\[ P = IV \]