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Unformatted text preview: Project Idea # 5 - Electrical Circuit Theory MT-A355 Differential Equations February 22, 2003 Introduction: In this problem we will explore the utility of differential equations in describing electrical circuits. We will examine a four degree of freedom R-C circuit, as depicted in the accompanying diagram. There will be an input voltage V in and four nodal voltages, v j ( t ), 1 ≤ j ≤ 4. We will soon see that this circuit acts as a low-pass filter , i.e., if we put in an input voltage containing low-, mid-, and high-range frequencies (relative to some fixed reference frequency) then the voltage at each of the four nodes will reflect an attenuation of the various frequency components. The term low-pass indicates that low frequencies will be allowed to pass through the circuit with little attenuation. Thus we expect the attenuation to increase as frequency increases. One can obtain a high-pass filter by simply switching the locations of the resistors and capacitors in the circuit. Figure 1: Fourth order RC low-pass filter circuit. The uses of such filters are wide ranging, but one common application of a low-pass filter is the denoising of signals. Loosely, noise refers to any unwanted components of a signal. We will consider two types of noise in this problem. First, we will examine the effects of random noise like static over a phone or snow on a television. Second, we will include noise that results from interference, which stems from interaction with nearby electrical devices. Since most circuits carry periodic signals we expect noise from interference also to be periodic. In order to obtain a system of differential equations to describe the behavior of our low-pass filter circuit we will use Kirchoff’s Law for Electrical Current, which says that the sum of all in- coming and out-going currents to a node in the circuit must be zero. Our circuit has four nodes,coming and out-going currents to a node in the circuit must be zero....
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- Fall '09
- Electrical Circuits, Resistor, Electrical impedance, Voltage drop, Electrical Circuit Theory