## 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 Vn = 0,  for each node    ∑n In = 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) = ZC = 1/(iωC),
Z(inductance) = iωL,
Z(resistance) = ZR = R.
Any impedance may be written as Z = R + iX.

Power:
Pavg = ½Re(VI*) = I2rmsR.  The maximum power is delivered to a load when Zload = Zeff*. (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 Zeff.

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

• The Thevenin voltage εeff is equal to the open circuit voltage at the terminals.
• The Thevenin impedance Zeff 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 Ieff in parallel with an impedance Zeff.

• Ieff is found by determining the open circuit voltage εeff at the terminals and dividing it by Zeff.