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
⑤ 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
⑤ Tip: Node voltage analysis is recommended over mesh current analysis.
⑥ Example 1:
Figure. 6. Example 1
⑦ 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.
④ 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.
⑤ 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).
⑦ 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
⑵ 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
⑶ 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
⑸ Lemma 1: All circuits can be equivalenced (existence)
① Not all circuits can be equivalenced.
○ The load part can have any component.
② 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.
③ Case 2: All power sources in the circuit have the same frequency: R, L, C can be considered as resistors.
⑹ 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.
⑺ 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