AC and transient circuits

Circuits with transients

Passive circuit elements:
Resistor:  V = IR
Capacitor:  V = Q/C
Inductor:  V = LdI/dt

Kirchhoff's rules for filamentary circuits:
For each loop   ∑_n V_n = 0,  for each node    ∑_n I_n = 0.

AC circuits without transients

Assume V(t), I(t), ε(t) are all proportional to exp(iωt).
Assume idealized circuit elements.  Define the impedance Z = V/I.  Then
Z(capacitance) = Z_C = 1/(iωC), 
Z(inductance) = iωL,
Z(resistance) = Z_R = R.
Any impedance may be written as Z = R + iX.

Power:
P_avg = ½Re(VI*) = I²_rmsR.  The maximum power is delivered to a load when Z_load = Z_eff^*. (See below, equivalent circuits.)

Circuits with or without transients

AC and DC circuits
Any two-terminal network of passive elements is equivalent to an effective impedance Z_eff.

Thevenin equivalent circuits: Any two terminal network can be replaced by a generator ε_eff in series with an impedance Z_eff.

• The Thevenin voltage ε_eff is equal to the open circuit voltage at the terminals.
• The Thevenin impedance Z_eff is the impedance measured at terminals with all voltage sources replaced by short circuits and all current sources replaced by open circuits.

Norton equivalent circuits: Any two terminal network can be replaced by a current source I_eff in parallel with an impedance Z_eff.

• I_eff is found by determining the open circuit voltage ε_eff at the terminals and dividing it by Z_eff.

Ladder networks:

Consider an infinite ladder network of reactive elements, for example the network shown on.
The network has some impedance Z₀.  Adding another element to the front of the infinite ladder network does not change Z₀. The network with impedance Z₀ is therefore equivalent to the network shown below, terminated by Z₀.
We can calculate Z₀.  For the example shown, Z₀ = (Z₁²/4 + Z₁Z₂)^½.
Z₀ = R + iX.  If the ladder contains no resistors, then Z₀ is either real or imaginary.  If Z₀ is real, the circuit absorbs energy, if Z₀ is imaginary, it does not absorb energy.  If Z₀ is real signals can pass to the load, if Z₀ is imaginary, signals cannot pass to the load, we have a filter.