math4b-16 - Math 4B Lecture 16 Doug Moore Electrical...

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Math 4B Lecture 16 May 22, 2013 Doug Moore
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Electrical circuits give rise to complicated systems of differen- tial equations. Circuits which include linear resistors, inductors, and capacitors give rise to linear systems of equations. Other circuit elements, such as transistors or vacuum tubes, are often nonlinear, and lead to nonlinear systems. We might be interested in designing a radio from various circuit elements. To design such a circuit, we would divide the circuit into various stages, oscillators, tuners, amplifiers and so forth.
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Here we have pictured a relatively simple circuit. To find the equations for the currents flowing through an electric circuit such as this, we need to know the voltage drops associated with the various linear circuit elements.
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The voltage drops are: A current x ( t ) flowing through a resistor of resistance R causes a voltage drop E R = Rx ( t ). Here the resistance R is measured in ohms, the current x in amperes, and the voltage drop E R in volts. A current y ( t ) flowing through an inductor of inductance L causes a voltage drop E L = L ( dy/dt )( t ), where the inductance L is measuure in henrys. A charge q ( t ) accumulated on a capacitor of capacitance C causes a voltage drop E C = (1 /C ) q ( t ), where the capacitance C is measured in farads. Thus if z ( t ) is the current flowing through the capacitor, z = dq dt q ( t ) = Z t 0 z ( τ ) + constant E C = (1 /C ) Z t 0 z ( τ ) + constant .
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To determine the equations for the currents, we then apply Kirchhoff’s laws: Kirchhoff’s voltage law: The sum of voltage drops around any loop in the circuit must equal the voltage which is applied to that loop. Kirchhoff’s current law: The sum of currents flowing into a node must equal the the sum of currents flowing out of the node.
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According to Kirchhoff’s current law, if x ( t ) is the current flowing up through the left hand wire, and y ( t ) is the current flowing down through the middle wire, then the current flowing down through the right hand wire must be ( x - y )( t ). Hence, applying Kirchhoff’s voltage law to the left hand loop, we
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