# Passive Filters¶

## Impedance Model¶

- Impedance Z is defined as Z=R+jX, where R is the resistance of the circuit element and X is the reactance of the circuit element
- Extending Ohm's Law to complex numbers, we get the relationship

where V is the potential difference between any two points in the circuit, I is the current through the conductor and Z is the impedance of the circuit element.

Tip

Impedance of a resistor: Z_{R} = R

Impedance of a capacitor: Z_{C} = \frac{1}{j\omega C}

Impedance of an inductor: Z_{L} = j\omega L

### Impedance Across A Resistor¶

Let's say we have a time-varying potential difference across a resistor v(t) and a time-varying current i(t) flowing through the resistor.

We know that there is no phase difference between the voltage and current waveforms.

For simplicity's sake, we let

Therefore, using Ohm's Law the impedance of the resistor is simply

### Impedance Across A Capacitor¶

Let's say we have a time-varying potential difference across a capacitor v(t) and a time-varying current i(t) flowing through the capacitor.

Capacitors have a unique property - they can store electric charges. The amount of electric charge Q that a capacitor stores can be calculated using

where C is the capacitance of the capacitor and V is the potential difference across the capacitor.

To find out the rate of change of electric charge in a capacitor, we can take the derivative of the equation:

We now have the relationship between i(t) and v(t). If we let

Using Ohm's Law we can find the impedance of the capacitor:

### Impedance Across An Inductor¶

Let's say we have a time-varying potential difference across an inductor v(t) and a time-varying current i(t) flowing through the inductor.

From Lenz's Law we know that

where \Phi_{B} is the magnetic flux and \mathcal{E} is the induced electromotive force.

Inductors have a quantifiable property known as inductance L, which is simply the ratio of the magnetic flux linkage \Phi_{B} to its time-varying current i(t) flowing through the inductor.

By separation of variables we can combine the equations together:

The induced electromotive force \mathcal{E} varies with time so we replace it with v(t). \frac{\partial \Phi_{B}}{dt} is simply L and can be replaced.

Thus the magnitude of the time-varying potential difference across an inductor v(t) is related to the current i(t) passing through it by the following relationship:

We now have the relationship between i(t) and v(t). If we let

Using Ohm's Law we can find the impedance of the capacitor: