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High-Pass Filters

The Circuits

High pass RC filter.

The basic first-order high-pass filters use the same components as the low-pass filters we just studied. However, their positions are swapped. Thus, the RC high-pass filter has the capacitor in series with the signal and the resistor across the output, as shown in the first diagram to the right. At high frequencies, C has very low impedance, and the signal passes through unhindered. As the frequency decreases, however, XC becomes significant, until at the cutoff frequency, XC = R, just as with the low-pass filter. At still lower frequencies, XC increases, and less of the signal reaches the output.

One useful feature of the RC high-pass filter is that the capacitor serves to block direct current between vIN and vOUT. Thus, two circuits that operate at different DC voltages can be connected by this type of high-pass filter without encountering any problems with dc component bias voltages as a consequence.

High pass RL filter.

The RL version of the high-pass filter uses a series resistor and a shunt inductor to accomplish its purpose. At high frequencies, XL is large, so the inductor is nearly an open circuit for such signals. At low frequencies, XL is very small, and effectively connects those signals directly to ground. As before, the cutoff frequency occurs where R = XL, so that L is just beginning to have a significant effect on the signal.



Frequency Response

Normalized frequency response of a high pass filter.

As shown to the right, the frequency response of a basic high-pass filter is actually a mirror image of its low-pass counterpart. At the cutoff frequency where R = XL or R = XC, the attenuation is only 3 db, so the signal voltage is still 70.7% of its higher-frequency value.

Below the cutoff frequency, attenuation increases at the rate of 20 db per decade, which is the same roll-off as for the low-pass filter. Above the cutoff frequency, attenuation rapidly decreases to nothing, and all higher frequencies pass with ease.

The green line in the graph is the straight line extension of the constant slopes of the actual frequency response. As with the low-pass filter, the intersection point is the cutoff frequency. This straight-line approximation of the real frequency response curve is very easy to draw, and is sufficiently accurate for some kinds of applications. Of course, the actual curve near the cutoff frequency is understood.



Phase Response

Voltage vectors for first-order high-pass filters.

As we have already seen, in a first-order low-pass filter, vOUT always lags vIN by some phase angle betweeen 0 and 90°. Exactly the reverse is true for a first-order high-pass filter: as shown in the vector diagrams to the right, vOUT is always taken from across the component whose voltage leads vIN by some phase angle, φ.

For the RL filter, vOUT is taken from across L, so its phase angle necessarily leads vIN as shown in the upper vector diagram. For the RC filter, vOUT is taken from across R, which again leads vIN as shown in the lower vector diagram.


Phase response of a first-order high-pass filter.

Of course, the actual phase angle by which vOUT leads vIN depends on the specific frequency of the signal, as compared to the cutoff frequency of the filter. As shown in the phase diagram to the right, signals more than 10 times the cutoff frequency show little or no appreciable phase shift, while signals less than 0.1 times the cutoff frequency are shifted close to 90°. Most of the change in phase occurs within a factor of 0.1 to 10 times ωCO.

As with the low-pass filter, non-sinusoidal signals with frequency components at or near ωCO will be distorted when passing through the high-pass filter.



The Equations

The equations for the high-pass filter are very similar to the ones for the low-pass filter, as you might expect. Certainly the basic comparisons are the same. Thus, for the RC circuit at the cutoff frequency,

R  =  XC
 
   =  1

ωCOC
 
 
ωCO  =  1

RC

This much is the same as for the RC low-pass filter. However, because the components have been swapped, the equation for attenuation over a frequency range has become:

vOUT  =  R    


vIN Z
 
 
   =  R  

(R² + XC²)½
 
 
   =  R

(R² + (1/ωC)²)½
 
 
   =  1

1/R × (R² + (1/ωC)²)½
 
 
   =  1

(1 + (1/RωC)²)½

In this equation, the higher the value of ω, the less effect it has on vOUT. This is exactly the reverse of the low-pass filter, where high values of ω would seriously reduce vOUT. This is in fact the essential difference between the low-pass filter and the high-pass filter.

The RL filter behaves the same way, except that at the cutoff frequency:

XL  =  R
 
ωCOL  =  R
 
ωCO  =  R

L

Then, over the frequency spectrum,

vOUT  =  XL    


vIN Z
 
 
   =  XL  

(R² + XL²)½
 
 
   =  ωL  

(R² + (ωL)²)½
 
 
   =  1

1/ωL × (R² + (ωL)²)½
 
 
   =  1

((R/ωL)² + 1)½

At high frequencies, the fraction R/ωL becomes very small and has negligible effect on the signal. At very low frequencies, this fraction becomes very large and blocks nearly all of the signal. And at cutoff, ω = ωCO = R/L, so that vOUT/vIN = 1/ = 0.707 as expected. Thus, the frequency response of the RL filter is exactly the same as the frequency response of the RC filter.

As with the low-pass filters, the phase response for both high-pass filters near ωco is exactly the same, and may be calculated as:

φ  =  arctan  XC  =  arctan  R


R XL
 
   =  arctan  1  =  arctan  R


ωRC ωL

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