## 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*) = I^{2}_{rms}R. 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 Z_{0}. Adding another element to
the front of the infinite ladder network does not change Z_{0}. The network
with impedance Z_{0} is therefore equivalent to the network shown below, terminated by Z_{0}.

We can calculate Z_{0}. For the example
shown, Z_{0} = (Z_{1}^{2}/4
+ Z_{1}Z_{2})^{½}.

Z_{0 }= R + iX.
If the ladder contains no resistors, then Z_{0} is either real or
imaginary. If Z_{0} is real, the circuit absorbs energy, if Z_{0}
is imaginary, it does not absorb energy. If Z_{0} is real
signals can pass to the load, if Z_{0} is imaginary, signals
cannot pass to the load, we have a **filter**.