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.


Circuits without transients

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

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

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


AC circuits

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*.


ladder networkLadder networks:

Consider an infinite ladder network of 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.  Here 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.