NDSU

RC Filters (real poles and zeros)

ECE 311

RC Filters Objective Sketch the frequency response of an RC low pass filter Sketch the frequency response of an RC high pass filter

Single Stage RC Low Pass Filter: One of the simplest filters is a voltage divider with a single resistor and capacitor. If you have a single-stage RC filter (delete the red and blue stages):

R1

X

+ -

Single Stage RC Filgter

R3

R2

Y1

Y2

Y3

C1

C2

2-Stage RC Filter

3-Stage RC Filter

C3

By voltage division: Z Y 1 = ⎛⎝ Z 1 +Z2 2 ⎞⎠ X

Assuming a sinusoidal input, using phasor notation

⎛ jωC1 1 ⎞ Y1 = ⎜ 1 ⎟X ⎝ R 1 + jωC 1 ⎠ Y 1 = ⎛⎝ jωR 11C 1 +1 ⎞⎠ X Note that The gain is one at DC The corner frequency is ω = The gain drops off as

1 ω,

1 RC

or -20dB/decade.

Multi-Stage RC Low Pass Filter Suppose you cascaded several RC low pass filters (include the red and blue sections above). To prevent loading, make the impedance of each following stage 10x the previous stage.

R 2 = 10R 1 JSG

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NDSU C2 =

RC Filters (real poles and zeros)

ECE 311

1 C 10 1

R1C1 = R2C2 As a rough approximation, this allows you to ignore the effect of the subsequent stages. The gain of the overall filter is then

Y = ⎛⎝ jωR 11C 1 +1 ⎞⎠ ⎛⎝ jωR 21C 2 +1 ⎞⎠ ⎛⎝ jωR 31C 3 +1 ⎞⎠ X If RC is the same for all three stages, then the gain is 3

Y = ⎛⎝ jωR 11C 1 +1 ⎞⎠ X (You need to use voltage nodes or current loops to get the actual transfer function. It will be different since there is some loading - but it will be closed to this.) Note that The gain is one at DC The corner frequency is ω = The gain drops off as ⎛⎝ ω1 ⎞⎠

1 RC

3

, or -60 dB/decade.

The gain vs. frequency for 1 to 5 stages is show below on cartesian paper >w = [-1:0.01:1]'; >w = 10.^w;; >j = sqrt(-1); >s = j*w; >G1 = 1 ./ (j*w+1); >G2 = G1.^2; >G3 = G1.^3; >G4 = G1.^4; >G5 = G1.^5; >plot(w,abs(G1),w,abs(G2),w,abs(G3),w,abs(G4),w,abs(G5)) >xlabel('rad/sec'); >ylabel('Gain');

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NDSU

RC Filters (real poles and zeros)

ECE 311

On log-log paper, you can see the corner and the roll-off: at -20NdB/decade where N is the number of stages

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NDSU

RC Filters (real poles and zeros)

ECE 311

Low pass filters are useful for reducing the noise in a signal. White noise is equal amplitude at all frequencies. Most data is low frequency (such as temperature, volume, speed, flow, etc.) If there is noise on a signal, you can 'clean up' the signal by applying a filter which passes the data but rejects the noise. For example, suppose you have a 1Hz sine wave with white (Gaussian) noise with a standard deviation of 0.2.

Add an RC low pass filter to pass the 1Hz signal but remove the high frequency noise. 1Hz = 6.28 rad/sec, so let's pick 1/RC=20. The result of this data going through a single stage RC filtre is better:

A 3-stage filter is a little better still:

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NDSU

RC Filters (real poles and zeros)

ECE 311

The amplitude is no longer 1.00: each RC filter attenuates the signa a little. Each stage also adds some phase shift: 3

20 ⎞ Y f = ⎛⎝ jω+20 ⎠ ω=6.28 Y

Yf = RC High Pass Filter: A slight variation of the RC filter is to swap R and C so that you are measuring the voltage across the resistor. In this case, using phasor notation, the gain is

⎛ ⎞ Y = ⎜ 1R ⎟ X +R ⎝ jωC ⎠ or

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NDSU

RC Filters (real poles and zeros)

C3

C2 C1

X

+ -

Y1

Y2

R1

Single Stage RC Filgter

ECE 311

Y3

R3

R2

2-Stage RC Filter

3-Stage RC Filter

jωRC Y = ⎛⎝ jωRC+1 ⎞⎠ X

Like before, you can cascade RC high pass filters for an N-stage filter if you increase the impedance of each following stage by 10-ish while keepint RC constant:

R 2 = 10R 1 C2 =

1 C 10 1

R2C2 = R1C1 The frequency response on cartesian coordinates are: >G1 = (j*w) ./ (j*w+1); >G2 = G1.^2; >G3 = G1.^3; >G4 = G1.^4; >plot(w,abs(G1),w,abs(G2),w,abs(G3),w,abs(G4))

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NDSU

RC Filters (real poles and zeros)

ECE 311

On log-log paper, you can see the asymptotes and the corner. The low-frequency gain increases at +20N dB/decade for an N-stage RC high-pass filter. The corner is at 1 rad/sec in all cases.

RC high pass filters are used when you want to eliminate a low-frequency signal - usually at DC. This is what you do in ECE321 with transistor amplifiers.

Vcc

Rc

R1

vc vb

ve R2

Re

In ECE 321, you did (or will) analyze this circuit at DC. The resistors are selected to place the transistor in the active region. This creates a DC offset for all of the voltages (the Q-point). JSG

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NDSU

RC Filters (real poles and zeros)

ECE 311

Capitors are placed in series with this circuit. The capacitors act as high-pass filters. They block the DC signal outside this circuit so that you don't change the Q-point. They pass the AC signal that you're trying to aplifyl. One of the three capacitors is connected to the input, one is the output. The third is grounded. Which signal you choose for the input, output, and ground determines if you're using the transistor as a common-emitter, common-collector, or common base amplifier.

Comments RC filters are simple, easy to get working, and sometimes good enough. The are not great filters, though, and are constrained to have real poles and zeros at the origin. The gain in the pass-band region is less than one - much less than one for multi-stage filters. The gain in the reject region also isn't zero. If you could build a filter with complex poles and zeros, you could do better.

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