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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| KVL | sum of all voltages around a closed loop = 0 | V_k: voltage across each element in the loop; sign convention: voltage rise is positive, voltage drop is negative |
| KCL | sum of all currents entering a node = 0 | I_k: current at each branch connected to the node; currents entering are positive, currents leaving are negative |
| Thevenin's theorem | V_th = open-circuit voltage at terminals; R_th = resistance seen at terminals with sources deactivated | V_th: Thevenin voltage; R_th: Thevenin resistance; equivalent circuit is V_th in series with R_th |
| Norton's theorem | I_N = short-circuit current; R_N = R_th | I_N: Norton current; R_N: Norton resistance equals Thevenin resistance; V_th = I_N * R_N |
| Maximum power transfer | P_max = V_th^2 / (4 R_th) | P_max: maximum power delivered to load; condition: R_L = R_th; V_th: Thevenin voltage; R_th: source resistance |
| Superposition | V_total = sum of V_k due to each independent source acting alone | V_k: contribution from k-th independent source; all other independent voltage sources replaced by short circuits, current sources by open circuits |
Key concepts
Mesh analysis
Assign a mesh current to each independent loop. Write KVL for each mesh. The voltage across a shared branch uses both mesh currents. Solve the resulting system of equations. Number of equations equals the number of mesh currents.
Nodal analysis
Assign node voltages at every node except a reference (ground). Write KCL at each non-reference node expressing branch currents in terms of node voltages and element values. This method is systematic for any planar or non-planar network.
Z-parameters (open-circuit)
V1 = Z11 I1 + Z12 I2; V2 = Z21 I1 + Z22 I2. Z11 is input impedance with port 2 open. Z12 is reverse transfer impedance. Reciprocal networks have Z12 = Z21.
Y-parameters (short-circuit)
I1 = Y11 V1 + Y12 V2; I2 = Y21 V1 + Y22 V2. Y11 is input admittance with port 2 short-circuited. For reciprocal networks Y12 = Y21. Y matrix is the inverse of Z matrix.
Source transformation
A voltage source V_s in series with resistance R can be replaced by a current source I_s = V_s/R in parallel with R. The transformation is reversible. Dependent sources cannot be transformed independently.
Reciprocity theorem
In a linear bilateral network with no dependent sources, if a voltage source V placed in branch A produces current I in branch B, then placing V in branch B produces the same current I in branch A. This is used to simplify analysis by choosing convenient source placement.
Tables
Two-Port Parameter Relationships
| Parameter | Equations | Measured at |
|---|---|---|
| Z (impedance) | V = Z I | Open-circuit ports |
| Y (admittance) | I = Y V | Short-circuit ports |
| h (hybrid) | V1=h11 I1+h12 V2 | Mixed open/short |
| ABCD (chain) | V1=AV2-BI2 | Output side vars |
Impedance of Basic Elements
| Element | Impedance Z(s) | At DC (s=0) | At high freq |
|---|---|---|---|
| Resistor R | R | R | R |
| Inductor L | sL | 0 (short) | inf (open) |
| Capacitor C | 1/(sC) | inf (open) | 0 (short) |
| RL series | R + sL | R | inf |
| RC series | R + 1/(sC) | inf | R |
| LC parallel | sL/(1+s^2LC) | 0 | 0 |
Network Theorem Applicability
| Theorem | Applies to | Key condition |
|---|---|---|
| Superposition | Linear networks | Each source alone |
| Thevenin | Any two-terminal | Linear network |
| Norton | Any two-terminal | Linear network |
| Max power transfer | Source to load | R_L = R_th |
| Reciprocity | Linear bilateral | No dependent sources |
| Millman's | Parallel branches | Same terminal pair |
Quick facts
- For maximum power transfer the load resistance must equal the Thevenin resistance. The efficiency at maximum transfer is only 50%.
- In a delta-to-star conversion: R_star = product of two adjacent delta resistors / sum of all three delta resistors.
- The number of independent mesh equations for a planar network is B - N + 1, where B is branches and N is nodes.
- The number of independent nodal equations is N - 1, where N is the total number of nodes including the reference.
- For a purely resistive network, the Thevenin resistance equals V_oc / I_sc, the ratio of open-circuit voltage to short-circuit current.
- In an AC circuit, maximum power transfer requires Z_L = Z_th* (complex conjugate of Thevenin impedance).
- Millman's theorem: voltage across a set of parallel branches = sum(V_k/R_k) / sum(1/R_k).
- The h-parameter h21 is the forward current gain, equal to beta (hFE) for a BJT in common-emitter configuration.
Exam shortcuts
- Thevenin resistance shortcut when there are no dependent sources: deactivate all independent sources (short voltage sources, open current sources) and calculate resistance at the terminals using series-parallel combinations. No need to solve KVL/KCL equations.
- Superposition shortcut: when the circuit has only one type of source (all voltage or all current), superposition offers no simplification over direct analysis. Use it only when two or more independent sources of different types are present.
- Norton to Thevenin conversion: V_th = I_N * R_N and R_th = R_N. Use whichever form is easier to compute first, then convert. Short-circuit current is easier to find when the load path has simple series elements.
- Mesh vs nodal selection rule: use nodal analysis when there are more meshes than nodes minus one. Use mesh analysis when there are more nodes. This minimizes the number of simultaneous equations to solve.
- Star-delta conversion memory aid: for delta to star, the star resistor connected to a node equals the product of the two delta resistors at that node divided by the sum of all three. For star to delta, multiply each pair of star resistors and sum all three products, then divide by the star resistor opposite to the delta branch needed.