AEE Electrical Engineering Core · Chapter 1

Circuit Laws, Network Theorems and Resonance

What to remember

Ohm's law, KCL and KVL solve every linear circuit. Network theorems (Thevenin, Norton, superposition, maximum power transfer) shorten the work.
Maximum power goes to the load when the load resistance equals the source (Thevenin) resistance, and then only 50% of the power is delivered to the load.
At series resonance the current is maximum and the impedance is purely resistive; at parallel resonance the impedance is maximum and the current is minimum.

Basic laws and units

Ohm's law: V = I R. Resistance R in ohm, conductance G = 1/R in siemens. Resistivity: R = ρ l / A.
Power and energy: P = V I = I² R = V² / R (watt); energy W = P t (joule; 1 kWh = 3.6 x 10⁶ J).
Kirchhoff's current law (KCL): the algebraic sum of currents at a node is zero (charge conservation).
Kirchhoff's voltage law (KVL): the algebraic sum of voltages round any closed loop is zero (energy conservation).
Series: same current, voltages add, R = R1 + R2 + ... Parallel: same voltage, currents add, 1/R = 1/R1 + 1/R2 + ...
Voltage divider: V1 = V R1 / (R1 + R2). Current divider: I1 = I R2 / (R1 + R2).
Capacitors and inductors: capacitors in series add like parallel resistors; in parallel they add directly. Inductors (no mutual coupling) add like resistors. Energy: ½ C V² in a capacitor and ½ L I² in an inductor.
Ideal sources: an ideal voltage source has zero internal resistance; an ideal current source has infinite internal resistance. Source conversion: a voltage source V in series with R equals a current source I = V/R in parallel with R.

Star-delta transformation

Delta to star	Star to delta
R1 = Rab Rca / (Rab + Rbc + Rca), and so on	Rab = (R1R2 + R2R3 + R3R1) / R3, and so on
Each star arm = product of the two adjacent delta arms / sum of delta arms	Each delta arm = sum of products of star arms / opposite star arm

If all three resistors are equal, R(delta) = 3 R(star) and R(star) = R(delta) / 3.

Worked example: a delta with three 30 ohm arms equals a star with three 10 ohm arms.

Network analysis methods

Mesh (loop) analysis uses KVL and loop currents. Best when there are fewer meshes than nodes.
Nodal analysis uses KCL and node voltages. Best when there are fewer nodes than meshes. Number of equations = nodes minus 1.
Supernode and supermesh are used when a voltage source lies between two nodes or a current source is shared between two meshes.
A network with b branches and n nodes has independent loops l = b - n + 1.

Network theorems

Theorem	Statement	Use
Superposition	In a linear bilateral network the response is the sum of the responses due to each independent source acting alone; others are replaced (voltage source by short, current source by open)	Multi-source circuits; not for power
Thevenin	Any linear two-terminal network equals a voltage source Vth in series with Rth	Load analysis
Norton	Equals a current source In in parallel with Rn; Rn = Rth, In = Vth / Rth	Parallel-form load analysis
Maximum power transfer	Load gets maximum power when RL = Rth (DC), or ZL = conjugate of Zth (AC)	Matching
Reciprocity	In a linear bilateral network a source in one branch gives the same current in another branch if source and meter positions are swapped	Passive networks
Millman	Many parallel voltage sources with series resistances combine into one: V = sum(V/R) / sum(1/R)	Parallel sources
Substitution	A branch with known voltage and current can be replaced by an equivalent source	Analysis
Tellegen	The sum of power over all branches is zero	Any lumped network
Compensation	A change in a branch resistance acts like a series source of the same change times the current	Sensitivity

To find Vth: remove the load, find the open-circuit voltage. To find Rth: deactivate all independent sources and find the resistance from the terminals. For circuits with dependent sources, find Rth as Voc / Isc.

Worked example (maximum power): Vth = 20 V and Rth = 5 ohm. For maximum power RL = 5 ohm. Current = 20 / 10 = 2 A. Pmax = Vth² / (4 Rth) = 400 / 20 = 20 W. Efficiency = 50%.

Worked example (Norton): a source of 24 V with 6 ohm in series. Norton current = 24 / 6 = 4 A, Rn = 6 ohm.

AC circuit basics

RMS value of a sine wave = peak / √2; average (half-cycle) = 2 x peak / π. Form factor = RMS / average = 1.11; peak factor = peak / RMS = 1.414.
Reactances: XL = ωL = 2πfL; XC = 1 / (ωC) = 1 / (2πfC). Impedance Z = √(R² + X²).
In a resistor V and I are in phase; in an inductor current lags voltage by 90°; in a capacitor current leads voltage by 90° ("ELI the ICE man").
Power: real power P = V I cosφ (watt); reactive power Q = V I sinφ (VAR); apparent power S = V I (VA); S² = P² + Q². Power factor = cosφ = P / S.
Complex impedance Z = R + jX. Admittance Y = 1 / Z = G + jB.
Power factor improvement: a capacitor in parallel supplies reactive power. Required kVAR = P (tanφ1 - tanφ2).

Worked example: a load of 10 kW at 0.8 lagging pf is raised to 1.0. tanφ1 = 0.75, tanφ2 = 0. Capacitor kVAR = 10 x 0.75 = 7.5 kVAR.

Resonance

Series RLC resonance. Occurs when XL = XC.

Resonant frequency: f0 = 1 / (2π √(LC)), ω0 = 1 / √(LC).
Impedance is minimum (= R); current is maximum (= V/R); power factor is unity.
Quality factor: Q = ω0 L / R = 1 / (ω0 C R) = (1/R) √(L/C). At resonance the voltages across L and C are each Q times the supply voltage (voltage magnification).
Bandwidth: BW = f2 - f1 = f0 / Q = R / (2πL). Half-power frequencies f1 and f2 are where the current is 1/√2 of its maximum; f0 = √(f1 f2).
Higher Q means a sharper (more selective) response and narrower bandwidth.

Worked example: R = 10 ohm, L = 0.1 H, C = 10 microfarad. ω0 = 1 / √(0.1 x 10 x 10⁻⁶) = 1 / √(10⁻⁶) = 1000 rad/s (f0 = about 159 Hz). Q = ω0 L / R = 1000 x 0.1 / 10 = 10. BW = R / L = 100 rad/s.

Parallel resonance (ideal inductor in parallel with capacitor, or a coil with resistance in parallel with a capacitor):

The impedance is maximum and the line current is minimum; it is also called a current-magnification or "rejector" circuit.
For a practical coil (R, L) in parallel with C: ω0 ≈ 1 / √(LC) when Q is high (Q much greater than 1). Dynamic impedance Zd = L / (C R).
Q for a parallel RLC with resistor R across it: Q = R / (ω0 L) = ω0 C R; the branch currents are Q times the supply current.

Feature	Series resonance	Parallel resonance
Impedance	Minimum	Maximum
Current	Maximum	Minimum
Magnification	Voltage	Current
Use	Tuner, filter	Tank circuit, rejector
Q formula	ω0 L / R	R / (ω0 L) for parallel R

Exam traps

In superposition a voltage source is replaced by a short, a current source by an open circuit.
Superposition does not apply to power, because power is a square-law quantity.
Maximum power transfer gives 50% efficiency, not maximum efficiency.
Thevenin resistance is found with sources deactivated, while Norton current is the short-circuit current.
Series resonance gives minimum impedance; parallel resonance gives maximum impedance.
Delta to star divides by the sum of delta arms; star to delta divides by the opposite arm.
Reciprocity fails for networks with dependent sources or non-linear elements.
High Q means narrow bandwidth, not wide.

One-liners

1. KCL follows conservation of charge; KVL follows conservation of energy.
2. Pmax = Vth² / (4 Rth).
3. A delta of equal resistors R equals a star of R / 3.
4. f0 = 1 / (2π √(LC)).
5. Series Q = ω0 L / R.
6. Bandwidth = f0 / Q.
7. Form factor of a sine wave is 1.11.
8. Current in an inductor lags the voltage by 90 degrees.
9. Reactive power is measured in VAR.
10. Parallel resonance dynamic impedance is L / (C R).
11. Millman's theorem combines parallel voltage sources.
12. Tellegen's theorem says total power in a network is zero.

Practice questions

Kirchhoff's current law is based on the conservation of
1. Energy
2. Power
3. Charge
4. Momentum
Answer
C. Charge
KCL says charge entering a node equals charge leaving it.
Kirchhoff's voltage law is based on the conservation of
1. Mass
2. Current
3. Charge
4. Energy
Answer
D. Energy
The algebraic sum of voltages around a closed loop is zero.
In superposition, an unused ideal voltage source is replaced by
1. An open circuit
2. A short circuit
3. A capacitor
4. A resistor of 1 ohm
Answer
B. A short circuit
Voltage sources are shorted and current sources opened.
In superposition, an unused ideal current source is replaced by
1. An open circuit
2. An inductor
3. A battery
4. A short circuit
Answer
A. An open circuit
A current source is deactivated by opening its terminals.
Maximum power is delivered to a DC load when
1. Load resistance is zero
2. Load resistance equals Thevenin resistance
3. Load resistance is infinite
4. Load resistance is twice Rth
Answer
B. Load resistance equals Thevenin resistance
Matching RL = Rth gives maximum power.
At series resonance the circuit impedance is
1. Minimum and equal to R
2. Purely inductive
3. Maximum
4. Purely capacitive
Answer
A. Minimum and equal to R
XL = XC cancel, leaving only R.
At parallel resonance of an ideal LC circuit the line current is
1. Maximum
2. Infinite in the source
3. Equal to the supply voltage
4. Minimum
Answer
D. Minimum
The parallel tank has maximum impedance and minimum current.
The resonant frequency of a series RLC circuit is
1. 1 / (2 pi LC)
2. 2 pi sqrt(LC)
3. 1 / (2 pi sqrt(LC))
4. sqrt(L / C)
Answer
C. 1 / (2 pi sqrt(LC))
f0 = 1 / (2 pi sqrt(LC)).
The form factor of a sinusoidal wave is
1. 0.637
2. 1.414
3. 1.11
4. 0.707
Answer
C. 1.11
Form factor = RMS / average = 1.11.
A delta network of three 30 ohm resistors is equivalent to a star of
1. 15 ohm each
2. 10 ohm each
3. 30 ohm each
4. 90 ohm each
Answer
B. 10 ohm each
For equal arms R(star) = R(delta) / 3 = 10 ohm.
Parallel resistors of 6 ohm and 12 ohm have an equivalent resistance of
1. 2 ohm
2. 9 ohm
3. 18 ohm
4. 4 ohm
Answer
D. 4 ohm
(6 x 12) / (6 + 12) = 4 ohm.
A 12 V supply is connected across 2 ohm and 4 ohm in series. The voltage across the 4 ohm resistor is
1. 8 V
2. 4 V
3. 12 V
4. 6 V
Answer
A. 8 V
Divider: 12 x 4 / (2 + 4) = 8 V.
A 6 A current enters a parallel combination of 2 ohm and 4 ohm. The current in the 2 ohm branch is
1. 6 A
2. 2 A
3. 4 A
4. 3 A
Answer
C. 4 A
Divider: 6 x 4 / (2 + 4) = 4 A.
A Thevenin source has Vth = 20 V and Rth = 5 ohm. The maximum power to a load is
1. 20 W
2. 10 W
3. 40 W
4. 80 W
Answer
A. 20 W
Pmax = Vth squared / (4 Rth) = 400 / 20 = 20 W.
A series RLC circuit has R = 10 ohm, L = 0.1 H and C = 10 microfarad. Its resonant angular frequency is
1. 10000 rad/s
2. 500 rad/s
3. 100 rad/s
4. 1000 rad/s
Answer
D. 1000 rad/s
w0 = 1 / sqrt(0.1 x 10 x 10^-6) = 1000 rad/s.
For the same circuit (R = 10 ohm, L = 0.1 H, C = 10 microfarad) the quality factor is
1. 100
2. 1
3. 0.1
4. 10
Answer
D. 10
Q = w0 L / R = 1000 x 0.1 / 10 = 10.
A series resonant circuit has f0 = 1000 Hz and Q = 50. Its bandwidth is
1. 200 Hz
2. 20 Hz
3. 50 Hz
4. 5000 Hz
Answer
B. 20 Hz
BW = f0 / Q = 1000 / 50 = 20 Hz.
A load of 10 kW at 0.8 lagging power factor is corrected to unity. The capacitor rating needed is
1. 10 kVAR
2. 7.5 kVAR
3. 6 kVAR
4. 12.5 kVAR
Answer
B. 7.5 kVAR
tan(phi1) = 0.75, so Q = 10 x 0.75 = 7.5 kVAR.
A source of 24 V in series with 6 ohm is converted to a Norton source. The Norton current is
1. 0.25 A
2. 6 A
3. 4 A
4. 144 A
Answer
C. 4 A
In = V / R = 24 / 6 = 4 A.
Two sources, 10 V with 2 ohm and 20 V with 2 ohm, are connected in parallel. By Millman's theorem the common voltage is
1. 15 V
2. 5 V
3. 10 V
4. 30 V
Answer
A. 15 V
V = (10/2 + 20/2) / (1/2 + 1/2) = 15 / 1 = 15 V.
A network has 6 branches and 4 nodes. The number of independent loops is
1. 3
2. 2
3. 4
4. 6
Answer
A. 3
l = b - n + 1 = 6 - 4 + 1 = 3.
An inductor of 0.1 / pi henry is connected to a 50 Hz supply. Its reactance is
1. 5 ohm
2. 20 ohm
3. 50 ohm
4. 10 ohm
Answer
D. 10 ohm
XL = 2 pi f L = 2 x pi x 50 x 0.1 / pi = 10 ohm.
A circuit has R = 3 ohm and XL = 4 ohm in series. The impedance magnitude is
1. 1 ohm
2. 7 ohm
3. 5 ohm
4. 12 ohm
Answer
C. 5 ohm
Z = sqrt(9 + 16) = 5 ohm.
A load draws 5 kVA at 4 kW. The reactive power is
1. 1 kVAR
2. 3 kVAR
3. 9 kVAR
4. 4 kVAR
Answer
B. 3 kVAR
Q = sqrt(S squared - P squared) = sqrt(25 - 16) = 3 kVAR.
A coil in parallel with a capacitor has L = 0.1 H, C = 10 microfarad and coil resistance 10 ohm. The dynamic impedance at resonance is
1. 10000 ohm
2. 10 ohm
3. 100 ohm
4. 1000 ohm
Answer
D. 1000 ohm
Zd = L / (C R) = 0.1 / (10 x 10^-5) = 1000 ohm.
A peak value of 100 V in a sine wave gives an RMS value of about
1. 100 V
2. 70.7 V
3. 63.7 V
4. 141 V
Answer
B. 70.7 V
RMS = peak / sqrt(2) = 70.7 V.
In a series resonant circuit the voltage across the inductor at resonance is
1. Zero
2. 1 / Q times the supply voltage
3. Q times the supply voltage
4. Equal to the supply voltage always
Answer
C. Q times the supply voltage
Voltage magnification equals Q.
Which theorem fails for power calculations?
1. Superposition
2. Thevenin
3. Norton
4. Reciprocity
Answer
A. Superposition
Power is a square-law quantity, so it cannot be added source by source.
The efficiency at maximum power transfer in a DC circuit is
1. 75 percent
2. 50 percent
3. 100 percent
4. 25 percent
Answer
B. 50 percent
Equal source and load resistance share the power equally.
Which theorem says that the sum of power over all branches of a lumped network is zero?
1. Reciprocity theorem
2. Millman's theorem
3. Tellegen's theorem
4. Compensation theorem
Answer
C. Tellegen's theorem
Tellegen's theorem expresses conservation of power.
A higher quality factor in a series RLC circuit means
1. A wider bandwidth
2. A lower resonant frequency
3. Greater resistance
4. A narrower bandwidth
Answer
D. A narrower bandwidth
BW = f0 / Q, so BW falls as Q rises.
To find the Thevenin resistance, the independent sources are
1. Deactivated and the resistance seen at the terminals found
2. Replaced by capacitors
3. Doubled
4. Left on and the load removed
Answer
A. Deactivated and the resistance seen at the terminals found
Voltage sources are shorted, current sources opened.
In a purely capacitive circuit the current
1. Lags the voltage by 90 degrees
2. Is in phase
3. Leads by 45 degrees
4. Leads the voltage by 90 degrees
Answer
D. Leads the voltage by 90 degrees
In a capacitor current leads the voltage by 90 degrees.
Consider: 1. In a delta to star conversion each star arm equals product of two adjacent delta arms divided by the sum of the three. 2. For equal arms R(delta) = 3 R(star).
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both statements are correct.
Consider: 1. Series resonance has maximum impedance. 2. Parallel resonance has maximum impedance.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Series resonance has minimum impedance.
Consider: 1. Norton resistance equals Thevenin resistance. 2. Norton current equals Vth / Rth.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both relations are correct for the same network.
Consider: 1. Superposition applies to linear bilateral networks. 2. Superposition can be used to add powers directly.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
It applies to voltage and current, not to power.
Consider: 1. Nodal analysis uses KCL. 2. Mesh analysis uses KVL.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are correct.
Consider: 1. Reciprocity holds for linear bilateral passive networks. 2. It holds for circuits with dependent sources.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Dependent sources can break reciprocity.
Consider: 1. At resonance of a series RLC circuit the power factor is unity. 2. Q is the ratio of reactance to resistance at resonance.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both statements are correct.
Consider: 1. A current source has zero internal resistance. 2. A voltage source has zero internal resistance.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
An ideal current source has infinite internal resistance.
Match in order: Thevenin, Norton, Millman, Tellegen
1. Current source with Rn, voltage source with Rth, sum of power is zero, parallel voltage sources
2. Parallel voltage sources, sum of power is zero, voltage source with Rth, current source with Rn
3. Sum of power is zero, parallel voltage sources, current source with Rn, voltage source with Rth
4. Voltage source with Rth, current source with Rn, parallel voltage sources, sum of power is zero
Answer
D. Voltage source with Rth, current source with Rn, parallel voltage sources, sum of power is zero
Each theorem is matched to its content.
Match in order: Series resonance, Parallel resonance, Form factor, Peak factor
1. 1.414, 1.11, maximum impedance, minimum impedance
2. 1.11, 1.414, minimum impedance, maximum impedance
3. Maximum impedance, minimum impedance, 1.414, 1.11
4. Minimum impedance, maximum impedance, 1.11, 1.414
Answer
D. Minimum impedance, maximum impedance, 1.11, 1.414
These are the standard values for a sine wave and the two resonance cases.
Two capacitors of 6 microfarad and 3 microfarad are connected in series. The equivalent capacitance is
1. 2 microfarad
2. 4.5 microfarad
3. 9 microfarad
4. 18 microfarad
Answer
A. 2 microfarad
(6 x 3) / (6 + 3) = 2 microfarad.
A 230 V supply is applied across a 46 ohm resistor. The power consumed is
1. 230 W
2. 1150 W
3. 5 W
4. 10580 W
Answer
B. 1150 W
P = V squared / R = 52900 / 46 = 1150 W.

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1.The charge flowing through a circuit in 10 minutes is 300 C. Find the current drawn by the circuit.
2.The fuse wire in electric circuits works on the principle of _________.
3.The electric ______ between two points in an electric circuit carrying some current is the work done to move a unit charge from one point to the other.
4.The rate at which electrical energy is dissipated in an electric circuit is called ___________.
5.The resistors of values 3 Ω, 6 Ω and 15 Ω are connected in parallel. What will be equivalent resistance in the circuit?
6.The Work-Energy theorem states that the total work done on an object is equal to:
7.The term for maximum value of electric current in a circuit is-
8.The main purpose of the earth wire in a domestic circuit is:
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