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Chapter 12: Direct Current Circuit Theory (Time-Domain)

Recommended Article : Circuit Theory Circuit Theory Table of Contents


1. Series-Parallel Composition

2. Kirchhoff’s Laws

3. Superposition Principle

4. Equivalent Circuit

5. Transient Phenomena



1. Series-Parallel Composition(series-parallel simplification)

⑴ Series-Parallel Composition of Power Sources

① The composite voltage due to series connection of voltage sources is equal to the sum of the voltages of each voltage source.

② Parallel connection of voltage sources producing different voltages can lead to explosions.

③ Composite voltage of parallel connection of voltage sources producing the same voltage v is v (increases lifespan).

④ Series connection of current sources producing different currents can lead to explosions.

⑤ Composite current of series connection of current sources producing the same current i is i (increases lifespan).

⑥ The composite current due to parallel connection of current sources is equal to the sum of the currents of each current source.

⑵ Series-Parallel Composition of Load : Several components can be treated as one component

① Series connection of Resistors

Figure. 1. Series connection of Resistors

② Parallel connection of Resistors

Figure. 2. Parallel connection of Resistors

③ Series-Parallel Composition of Capacitors

④ Series-Parallel Composition of Coils

Application : The sum of powers of each component is equal to the power of the composite component.

○ All power is supplied from the power source.

○ Current flowing through the power source is the same when calculated for each component or the composite component.

⑶ Millman’s Theorem

① In the case of two resistances

Figure. 3. Millman’s Theorem

② Can be extended similarly for R1, …, Rn



2. Kirchhoff’s Laws

⑴ Kirchhoff’s Current Law (KCL, Kirchhoff current law) : Conservation of charge. Node analysis.

Figure. 4. Kirchhoff’s Current Law

① Kirchhoff’s law applies regardless of linearity/non-linearity, lumped/distributed parameters, and time-varying/time-invariant systems.

② Whether current is flowing out or in, it must be unified : Negative values are also considered.

○ Electromagnetic perspective : Outgoing current from the system is considered positive, similar to heat physics. The sum of outgoing currents from any node is zero.

○ Electrical engineering perspective : Incoming current to the system is considered positive, similar to heat chemistry. The sum of incoming currents to any node is zero.

③ Also known as node voltage analysis : Node voltages are defined, and KCL is applied.

④ Super-node : Composed of two or more nodes connected to a power source. Reduces the number of unknowns.

Figure. 5. Super-node

Tip: Node voltage analysis is recommended over mesh current analysis.

Example 1:

Figure. 6. Example 1

Example 2:

Figure. 7. Example 2

⑵ Kirchhoff’s Voltage Law (KVL, Kirchhoff voltage law) : Mesh or loop analysis.

① Types of paths

○ Mesh : The most basic circuit

○ Loop : A circuit with no open branches. Closed curve, also known as a closed loop.

○ Examples

Figure. 8. Types of paths [Footnote: 2]

○ ⒜ : This is not a path. KVL does not hold.

○ ⒝ : This is not a path. It passes the same node twice.

○ ⒞ : This is not a mesh since it encloses another loop.

○ ⒟ : Similar to ⒞, this is not a mesh due to the same reason.

○ ⒠, ⒡ : These are loops and also meshes.

○ Mesh determination is not essential : But using mesh helps in creating a system of linear equations strategically.

② The sum of voltage drops around any closed path is zero.

Figure. 9. Kirchhoff’s Voltage Law

③ Also known as mesh current analysis : Mesh currents are defined, and KVL is applied.

○ Mesh currents are assigned to each mesh.

○ Current flowing through each component is expressed as the sum or difference of mesh currents.

○ Apply KVL around each mesh.

Tip: You can also use variables for unknown points without specifying mesh currents.

Proof 1:

○ Fundamentally, the electric field E is a conservative field. Any closed path X(t), a ≤ t ≤ b can be parameterized (where X(a) = X(b)).

○ Mathematical development : For grad V = E

○ (Note) This proof is also used when deriving the law of conservation of energy for conservative fields.

○ (Note) Dot product of E and d l signifies voltage drop.

○ (Note) Combining the law of conservation of energy and voltage drop analysis: Work done on a unit charge is equal to the decrease in potential.

Proof 2:

○ Maxwell’s Third Law : Faraday’s Law. Change in an external magnetic field induces an electric field.

○ When there is no change in the external magnetic field, curl E vector = 0 : Using Green’s theorem.

⑥ Limitation : Some problems can only be solved with KCL (e.g., Op Amp).

Example 1:

Figure. 10. Example 1

⑶ All circuits can be solved using KCL and KVL.

① Therefore, the voltage and current at any point can be determined.

② (Note) Existence of equivalent circuits can be proven through induction.



3. Superposition Principle

⑴ Definition : The sum of effects caused by each independent power source in a circuit with multiple independent power sources is equal to the actual effect.

① Linear Components : Components where the superposition principle can be applied, such as resistors (R), capacitors (C), inductors (L), linear power sources, etc.

② Superposition principle holds due to linearity of electric fields.

③ Power is not considered.

④ Superposition principle can also be applied in small-signal analysis.

⑵ Superposition Principle for Average Power : For periodic power sources

① AC power indicates average power.

Figure. 11. Superposition Principle for Average Power

② Superposition principle for average power is not applicable in most cases.

③ Applicable when composed of AC power sources with different frequencies.

⑶ Example

Figure. 12. Superposition Principle Example

① Removing the power source

Figure. 13. Removing the power source

② 1st. Removing independent current source : Setting the current to 0 ⇔ Open circuit

Figure. 14. Removing independent current source

③ 2nd. Removing independent voltage source : Setting the voltage to 0 ⇔ Short circuit

Figure. 15. Removing independent voltage source

④ 3rd. Load voltage is linear with respect to two independent power sources

Figure. 16. Load voltage

Figure. 17. Load voltage



4. Equivalent Circuit: Circuit with identical load effects

⑴ Loading Effect

① Open-Circuit Voltage

Figure. 18. Open-Circuit Voltage

② Load Voltage

Figure. 19. Load Voltage

Thevenin Equivalent Circuit

① Representation

Figure. 20. Thevenin Equivalent Circuit Representation

○ If Rth is negative, dependent power source exists.

○ If only independent power sources exist, Thevenin equivalent resistance Rth is always positive.

② Experimental Method

Figure. 21. Experimental Method

○ Calculate Vth and Rth from the Itest - Vtest graph.

○ Measure open-circuit voltage : Open-circuit voltage measured using voltmeter with internal resistance R1, R2 is defined as V1, V2.

③ Thevenin Equivalent Circuit Example

Figure. 22. Thevenin Equivalent Circuit Example

Norton Equivalent Circuit: Dual to Thevenin equivalent circuit

① Representation

Figure. 23. Norton Equivalent Circuit Representation

⑷ Thevenin - Norton Reciprocity (Source Transformation)

① Formulation

② Thevenin - Norton Reciprocity : Resistance

Figure. 24. Thevenin - Norton Reciprocity of Resistance

③ Thevenin - Norton Reciprocity : Capacitor

④ Thevenin - Norton Reciprocity : Inductor

⑤ Proof of Thevenin - Norton Reciprocity

Figure. 25. Proof of Thevenin - Norton Reciprocity

Lemma 1: All circuits can be equivalenced (existence)

① Not all circuits can be equivalenced.

○ The load part can have any component.

Case 1: Circuits entirely composed of resistors and power sources where equivalence is applicable

○ Strategy : Start with the closest component to a specific component in the circuit and draw Thevenin equivalent circuits.

○ When encountering a resistor in between : Use Thevenin - Norton reciprocity (source transformation).

○ When encountering a voltage source in between : Introduce a change to vth in vtest = itestRth + vth.

○ When encountering a current source in between : Convert to a Norton equivalent circuit and apply KCL.

○ In a circuit with only independent power sources, Rth can never be negative.

Case 2: All power sources in the circuit have the same frequency: R, L, C can be considered as resistors.

Lemma 2: Every circuit has a unique Thevenin equivalent circuit (uniqueness)

① Assume a circuit has Thevenin equivalent circuits A and B.

② In this case, the open-circuit voltage at two terminals in the circuit must be the same ⇒ Vth of A and B are the same.

③ The short-circuit current at two terminals in the circuit must be the same ⇒ Vth ÷ ith = Rth of A and B are the same.

④ Conclusion : A and B are identical.

Lemma 3: Power consumption of any circuit and its equivalent circuit may be different

Figure. 26. Example showing that power consumption of equivalent circuits is not always the same [Footnote: 4]

① In case ⒜ : Power consumption in resistance within N is 10 W.

② In case ⒝ : Power consumption in resistance within Nth is 1 W.

③ Since ⒜ and ⒝ have the same loading effect for both terminals, the load power is always the same.

⑻ Maximum power that a system can deliver to a load resistance

① Conditions in signal domain

② Conditions in power domain

③ Conditions for resistance R in signal domain



5. Transient Phenomena

⑴ Definition : Using differential equations for analysis of R, L, C components

⑵ General Solution, Complete Solution = Homogeneous Solution + Particular Solution

⑶ Homogeneous Solution, Transient Solution

① Definition : Solutions determined by coefficients of differential equations

⑷ Particular Solution, Steady-State Solution

① Definition : Solutions determined by form of the driving term

② Driving Term : Typically, left-hand side is a linear differential equation, right-hand side is left as a specific function, referred to as that specific function.


Input: 2016.01.05 19:49

Modified: 2018.12.11 23:52

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