Transmission Line Characteristic Impedance

If L is the inductance per unit length and C is the capacitance per unit length of a transmission line, then the characteristic impedance is given by

Zo = √(L/C) -----[9.3]

We will briefly present a derivation of this formula. Let us assume that a step voltage V starts propagating a transmission line at t = 0. In a time period of t, it traverses a distance l. If it causes a current I to flow in the transmission line, then its characteristic impedance is given by

Zo = V/I

Now, I = Q / t

Where Q is the charge accumulated in time t. The charge is given by capacitance times voltage across it. The capacitance for length l is Cl, where C is the capacitance per unit length. This gives

Q = C l V

In the previous section we have already seen that the propagation delay is given by , where L and C are inductance and Capacitance per unit length respectively. Therefore t, the delay for a length l is given by

t = l [L/C] ½

Substituting the values of Q and t, in the expression for I gives,

I = C lV / l [L/C] ½ = V [C/L] ½

Zo = V/I

Zo = √(L/C) -----[9.3]

Characteristic impedance is one of the most important formula a PCB designer will be using in the design of a high speed PCB.

A uniform transmission lines characteristic impedance depend only upon the physical property of the transmission line. It does not depend upon the frequency of the impressed signal.

As PCB designers we will often be required to maintain a given characteristic impedance for some traces. The next two sections will show formulas for calculating characteristic impedances of two most common configurations - microstrip and stripline.

If we increase the trace width, the capacitance of the trace will increase. This will lower the characteristic impedance. If we decrease the separation between the trace and the power or ground plane, the capacitance will increase and therefore the impedance will increase. By varying the trace width and the separation between the trace and the power plane we achieve the required characteristics impedance. The third factor that it depends upon is the relative permittivity, which depends upon the dielectric material and is usually between 3.5 to 4.7.

As PCB designer you will be required to create the stack up. You can use formula based method as a first approximation to calculate the impedance of common structures like microstrip and stripline.



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