1P AC Circuits 2 - Free Electrical Engineering Board Exam Reviewer

LessonAbout 1.5 h2 of 2 in 1P AC Circuits

AC Circuits 2

Alternating current circuit is an electric circuit energized by a time-varying sinusoidal or non-sinusoidal voltage source.

1. Parallel AC Circuits

Parallel AC circuit consists of two or more circuit branches connected across a common alternating voltage supply.

Voltage across each branch is identical
Total line current is the phasor sum of branch currents
Branch currents are determined by branch impedances

Equivalent Parallel Impedance

\[ \dfrac{1}{Z_t} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + \dots + \dfrac{1}{Z_n} \]

Where: \(Z_t\) = total impedance (\(\Omega\)), \(Z_1, Z_2, Z_n\) = branch impedances (\(\Omega\))

EE Board Exam October 1981

Example. A circuit consists of \(X_L = j5\ \Omega\), \(X_C = -j5\ \Omega\), and \(R = 5\ \Omega\) all connected in parallel. Find the equivalent impedance.

Solution.

\[ \dfrac{1}{Z_t} = \dfrac{1}{j5} + \dfrac{1}{-j5} + \dfrac{1}{5} = \dfrac{1}{5}\ \Omega^{-1} \]

\[ Z_t = \dfrac{1}{0.2} = 5.0\ \Omega \]

Answer: \(5.0\ \Omega\)

2. Admittance

Admittance is defined as the reciprocal of impedance.

Unit of admittance is siemens (S, formerly mho)
Named after German engineer Ernst Werner von Siemens
Admittance is a complex quantity with real and imaginary parts

Admittance

\[ Y = \dfrac{1}{Z} \]

\[ Y = G \pm j\beta \]

\[ Y = \sqrt{G^2 + \beta^2} \]

\[ I = E Y \]

Where: \(Y\) = admittance (siemens), \(Z\) = impedance (ohm), \(G\) = conductance (siemens), \(\beta\) = susceptance (siemens), \(I\) = current (ampere), \(E\) = supply voltage (volt)

EE Board Exam October 1986 / EE Board Exam April 1993

Example. A parallel resistor having a conductance of \(4\ \text{mhos}\), an inductive reactor having a susceptance of \(8\ \text{mhos}\), and a capacitive reactor having a susceptance of \(5\ \text{mhos}\) are connected in parallel. Find the equivalent impedance.

Solution.

\[ Y_t = G - j\beta_L + j\beta_C = 4 - j8 + j5 = 4 - j3 = 5\angle -36.869^\circ\ \text{mho} \]

\[ Z_t = \dfrac{1}{Y_t} = \dfrac{1}{5\angle -36.869^\circ} = 0.2\angle 36.869^\circ = 0.16 + j0.12\ \Omega \]

Answer: \(0.16 + j0.12\ \Omega\)

3. Conductance

Conductance is the real component of admittance that represents the ability to conduct alternating current.

Unit of conductance is siemens (mho)
In series circuit, conductance depends on resistance and reactance
Total conductance of parallel branches equals sum of branch conductances

Conductance

\[ G = \dfrac{R}{R^2 + X^2} \]

\[ G_t = G_1 + G_2 + \dots + G_n \]

Where: \(G\) = conductance (siemens), \(R\) = resistance (ohm), \(X\) = reactance (ohm), \(G_t\) = total conductance (siemens)

Example. Calculate the conductance of a branch containing \(50\ \Omega\) resistance and \(70\ \Omega\) inductive reactance in series.

Solution.

\[ G = \dfrac{R_L}{R_L^2 + X_L^2} = \dfrac{50}{50^2 + 70^2} = 0.00675\ \text{mho} \]

Answer: \(0.00675\ \text{mho}\)

4. Susceptance

Susceptance is the imaginary component of admittance representing the reactive power capability of a circuit.

Unit of susceptance is siemens (mho)
Inductive susceptance is negative imaginary quantity \(-j\beta_L\)
Capacitive susceptance is positive imaginary quantity \(+j\beta_C\)

Susceptance

\[ \beta = \dfrac{X}{R^2 + X^2} \]

\[ \beta_L = \dfrac{X_L}{R_L^2 + X_L^2} \qquad\qquad \beta_C = \dfrac{X_C}{R_C^2 + X_C^2} \]

\[ \beta_t = \beta_C - \beta_L \]

Where: \(\beta\) = susceptance (siemens), \(X\) = reactance (ohm), \(R\) = resistance (ohm), \(\beta_L\) = inductive susceptance (siemens), \(\beta_C\) = capacitive susceptance (siemens), \(\beta_t\) = net susceptance (siemens)

Example. Determine the capacitive susceptance of a \(50\ \mu\text{F}\) capacitor operating on a \(60\ \text{Hz}\) AC system.

Solution.

\[ X_C = \dfrac{1}{2\pi f C} = \dfrac{1}{2\pi(60)(50 \times 10^{-6})} = 53.05\ \Omega \]

\[ \beta_C = \dfrac{1}{X_C} = \dfrac{1}{53.05} = 0.0188\ \text{mho} \]

Answer: \(0.0188\ \text{mho}\)

5. Series Resonance

Series resonance is a circuit condition in a series RLC circuit where inductive reactance equals capacitive reactance.

Power factor of circuit becomes unity (\(pf = 1\))
Total circuit impedance is minimum (\(Z = R\))
Total circuit current is maximum (\(I = E / R\))

Series Resonance

\[ X_L = X_C \]

\[ f_r = \dfrac{1}{2\pi\sqrt{LC}} \]

\[ Q = \dfrac{1}{R}\sqrt{\dfrac{L}{C}} = \dfrac{X_L}{R} \]

Where: \(f_r\) = resonant frequency (hertz), \(L\) = inductance (henry), \(C\) = capacitance (farad), \(Q\) = quality factor, \(R\) = resistance (ohm), \(X_L\) = inductive reactance (ohm), \(X_C\) = capacitive reactance (ohm)

EE Board Exam April 1988

Example. A loudspeaker with an inductance of \(1.15\ \text{H}\) is coupled through a condenser of \(2\ \mu\text{F}\). Find the resonant frequency.

Solution.

\[ f_r = \dfrac{1}{2\pi\sqrt{LC}} = \dfrac{1}{2\pi\sqrt{(1.15)(2 \times 10^{-6})}} = 105\ \text{Hz} \]

Answer: \(105\ \text{Hz}\)

6. Parallel Resonance

Parallel resonance is a circuit condition in parallel branches where total net susceptance is zero.

Also called anti-resonance
Total circuit impedance is maximum
Total supply line current is minimum
Circuit operating power factor is unity (\(pf = 1\))

Parallel Resonance

\[ \beta_L = \beta_C \]

\[ \dfrac{X_L}{R_L^2 + X_L^2} = \dfrac{X_C}{R_C^2 + X_C^2} \]

\[ f_r = \dfrac{1}{2\pi\sqrt{LC}}\sqrt{\dfrac{R_L^2 C - L}{R_C^2 C - L}} \]

\[ Z_r = \dfrac{L}{C R_L} \]

Where: \(\beta_L\) = inductive susceptance (siemens), \(\beta_C\) = capacitive susceptance (siemens), \(f_r\) = resonant frequency (hertz), \(L\) = inductance (henry), \(C\) = capacitance (farad), \(R_L\) = coil resistance (ohm), \(R_C\) = capacitor branch resistance (ohm), \(Z_r\) = resonant impedance (ohm)

EE Board Exam April 1989

Example. A coil of \(50\ \Omega\) resistance and \(100\ \Omega\) reactance is shunted by a lossless capacitor to achieve parallel resonance. Calculate the impedance of the combination at resonance.

Solution.

\[ Y_t = G_t = \dfrac{R_L}{R_L^2 + X_L^2} = \dfrac{50}{50^2 + 100^2} = 0.004\ \text{mho} \]

\[ Z_t = \dfrac{1}{G_t} = \dfrac{1}{0.004} = 250\ \Omega \]

Answer: \(250\ \Omega\)

7. Non-Sinusoidal Wave

Non-sinusoidal wave is a distorted periodic waveform consisting of a fundamental component and higher harmonic frequencies.

Expressed as a Fourier series expansion
First introduced by French mathematician Jean Baptiste Joseph Fourier
Consists of DC component, fundamental wave, and harmonic components

Fourier Series Waveform

\[ e(t) = E_{\text{dc}} + E_{m1}\sin(\omega t \pm \alpha_1) + E_{m2}\sin(2\omega t \pm \alpha_2) + \dots + E_{mn}\sin(n\omega t \pm \alpha_n) \]

\[ i(t) = I_{\text{dc}} + I_{m1}\sin(\omega t \pm \beta_1) + I_{m2}\sin(2\omega t \pm \beta_2) + \dots + I_{mn}\sin(n\omega t \pm \beta_n) \]

Where: \(e(t)\) = instantaneous voltage (volt), \(i(t)\) = instantaneous current (ampere), \(E_{\text{dc}}\) = DC voltage (volt), \(I_{\text{dc}}\) = DC current (ampere), \(E_{mn}\) = peak voltage of \(n\)-th harmonic (volt), \(I_{mn}\) = peak current of \(n\)-th harmonic (ampere), \(\alpha_n, \beta_n\) = phase angles (degree)

8. RMS Value of Non-Sinusoidal Wave

RMS value of non-sinusoidal wave is the effective value equal to the square root of the sum of the squares of DC and harmonic components.

Independent of phase angles of individual harmonics
Calculated using peak or effective values of each harmonic component
AC meters measure the combined effective value

Effective Value of Non-Sinusoidal Wave

\[ I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + \dfrac{1}{2}\left(I_{m1}^2 + I_{m2}^2 + \dots + I_{mn}^2\right)} \]

\[ I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_1^2 + I_2^2 + \dots + I_n^2} \]

\[ E_{\text{rms}} = \sqrt{E_{\text{dc}}^2 + E_1^2 + E_2^2 + \dots + E_n^2} \]

Where: \(I_{\text{rms}}\) = effective current (ampere), \(E_{\text{rms}}\) = effective voltage (volt), \(I_{\text{dc}}\) = DC current (ampere), \(E_{\text{dc}}\) = DC voltage (volt), \(I_{mn}\) = peak current of \(n\)-th harmonic (ampere), \(I_n\) = rms current of \(n\)-th harmonic (ampere)

EE Board Exam April 1994

Example. An alternating current with an effective value of \(5\ \text{A}\) and a direct current of \(10\ \text{A}\) flow simultaneously in a conductor. Find the reading of an AC ammeter in the circuit.

Solution.

\[ I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_{\text{ac}}^2} = \sqrt{10^2 + 5^2} = 11.18\ \text{A} \]

Answer: \(11.18\ \text{A}\)

9. Active Power Drawn by Non-Sinusoidal Circuit

Active power drawn by non-sinusoidal circuit is the total real power obtained by adding average power of DC component and each identical frequency harmonic component.

Harmonics of different frequencies produce zero net average power
Analyzed using the Superposition Theorem
Total power factor is total active power divided by total apparent power

Active Power in Non-Sinusoidal Circuit

\[ P = E_{\text{dc}} I_{\text{dc}} + \dfrac{E_{m1}I_{m1}}{2}\cos(\alpha_1 - \beta_1) + \dfrac{E_{m2}I_{m2}}{2}\cos(\alpha_2 - \beta_2) + \dots \]

\[ P = E_{\text{dc}} I_{\text{dc}} + E_1 I_1 \cos\theta_1 + E_2 I_2 \cos\theta_2 + \dots \]

\[ P = I_{\text{rms}}^2 R_t \]

\[ pf = \dfrac{P}{E_{\text{rms}} I_{\text{rms}}} \]

Where: \(P\) = active power (watt), \(E_{\text{dc}}, I_{\text{dc}}\) = DC voltage and current (volt, ampere), \(E_n, I_n\) = rms voltage and current of \(n\)-th harmonic (volt, ampere), \(\theta_n\) = phase difference of \(n\)-th harmonic (degree), \(R_t\) = circuit equivalent resistance (ohm), \(pf\) = overall power factor

Example. Find the average power dissipated in a \(10\ \Omega\) resistor if the current is \(i = 12\sin\omega t + 8\sin 3\omega t + 3\sin 5\omega t\ \text{A}\).

Solution.

\[ I_{\text{rms}} = \sqrt{\dfrac{1}{2}(12^2 + 8^2 + 3^2)} = 10.416\ \text{A} \]

\[ P = I_{\text{rms}}^2 R = (10.416)^2(10) = 1085\ \text{W} \]

Answer: \(1085\ \text{W}\)

Formula Summary

Formula Name	Formula	Variables
Equivalent Parallel Impedance	\(\dfrac{1}{Z_t} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + \dots + \dfrac{1}{Z_n}\)	\(Z_t, Z_1, Z_2, Z_n\) = branch impedance (\(\Omega\))
Admittance	\(Y = \dfrac{1}{Z}\) \(Y = G \pm j\beta\) \(Y = \sqrt{G^2 + \beta^2}\) \(I = E Y\)	\(Y\) = admittance (S), \(Z\) = impedance (\(\Omega\)), \(G\) = conductance (S), \(\beta\) = susceptance (S), \(I\) = current (A), \(E\) = voltage (V)
Conductance	\(G = \dfrac{R}{R^2 + X^2}\) \(G_t = G_1 + G_2 + \dots + G_n\)	\(G, G_t, G_1, G_2\) = conductance (S), \(R\) = resistance (\(\Omega\)), \(X\) = reactance (\(\Omega\))
Susceptance	\(\beta = \dfrac{X}{R^2 + X^2}\) \(\beta_L = \dfrac{X_L}{R_L^2 + X_L^2}\) \(\beta_C = \dfrac{X_C}{R_C^2 + X_C^2}\) \(\beta_t = \beta_C - \beta_L\)	\(\beta, \beta_L, \beta_C, \beta_t\) = susceptance (S), \(X, X_L, X_C\) = reactance (\(\Omega\)), \(R, R_L, R_C\) = resistance (\(\Omega\))
Series Resonance	\(X_L = X_C\) \(f_r = \dfrac{1}{2\pi\sqrt{LC}}\) \(Q = \dfrac{1}{R}\sqrt{\dfrac{L}{C}} = \dfrac{X_L}{R}\)	\(f_r\) = resonant frequency (Hz), \(L\) = inductance (H), \(C\) = capacitance (F), \(Q\) = quality factor, \(R\) = resistance (\(\Omega\)), \(X_L, X_C\) = reactance (\(\Omega\))
Parallel Resonance	\(\beta_L = \beta_C\) \(\dfrac{X_L}{R_L^2 + X_L^2} = \dfrac{X_C}{R_C^2 + X_C^2}\) \(f_r = \dfrac{1}{2\pi\sqrt{LC}}\sqrt{\dfrac{R_L^2 C - L}{R_C^2 C - L}}\) \(Z_r = \dfrac{L}{C R_L}\)	\(\beta_L, \beta_C\) = susceptance (S), \(f_r\) = resonant frequency (Hz), \(L\) = inductance (H), \(C\) = capacitance (F), \(R_L, R_C\) = resistance (\(\Omega\)), \(Z_r\) = resonant impedance (\(\Omega\))
Fourier Series Waveform	\(e(t) = E_{\text{dc}} + E_{m1}\sin(\omega t \pm \alpha_1) + \dots + E_{mn}\sin(n\omega t \pm \alpha_n)\) \(i(t) = I_{\text{dc}} + I_{m1}\sin(\omega t \pm \beta_1) + \dots + I_{mn}\sin(n\omega t \pm \beta_n)\)	\(e(t)\) = voltage (V), \(i(t)\) = current (A), \(E_{\text{dc}}, I_{\text{dc}}\) = DC components (V, A), \(E_{mn}, I_{mn}\) = peak values (V, A), \(\alpha_n, \beta_n\) = phase angle (\(^\circ\))
Effective Value of Non-Sinusoidal Wave	\(I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + \dfrac{1}{2}\left(I_{m1}^2 + \dots + I_{mn}^2\right)}\) \(I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_1^2 + \dots + I_n^2}\) \(E_{\text{rms}} = \sqrt{E_{\text{dc}}^2 + E_1^2 + \dots + E_n^2}\)	\(I_{\text{rms}}\) = effective current (A), \(E_{\text{rms}}\) = effective voltage (V), \(I_{\text{dc}}, E_{\text{dc}}\) = DC components (A, V), \(I_{mn}\) = peak harmonic current (A), \(I_n\) = rms harmonic current (A)
Active Power in Non-Sinusoidal Circuit	\(P = E_{\text{dc}} I_{\text{dc}} + \dfrac{E_{m1}I_{m1}}{2}\cos(\alpha_1 - \beta_1) + \dots\) \(P = E_{\text{dc}} I_{\text{dc}} + E_1 I_1 \cos\theta_1 + \dots\) \(P = I_{\text{rms}}^2 R_t\) \(pf = \dfrac{P}{E_{\text{rms}} I_{\text{rms}}}\)	\(P\) = active power (W), \(E_{\text{dc}}, I_{\text{dc}}\) = DC voltage and current (V, A), \(E_n, I_n\) = rms harmonic components (V, A), \(\theta_n\) = phase difference (\(^\circ\)), \(R_t\) = resistance (\(\Omega\)), \(pf\) = power factor