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Nonlinear elements - example Nonlinear resistor is given by: For I>0

Nonlinear circuits Analysing nonlinear circuits is often difficult. Only a few simple circuits are adequately described by equations that have a closed form solution. A trivial example is a circuit consisting of a current source and an exponential diode. Voltages and currents in circuits containing only a few nonlinear circuit elements may be found using graphical methods for solving nonlinear equations that describe the behaviour of the circuit.

Nonlinear elements - example Nonlinear resistor is given by: For I>0

Nonlinear element - methods Analytical Graphical using load line Graphical using graphical summing for Kirchoffs laws

Analytical method For the nodes A-B: For the voltage drops in left loop: For the node A: For nonlinear element we have: Then combining the above formulas (U1=I1R1) we can write: We know that: then

We can combine the equation to calculate the current: And for the given value of source E: Substituting the values: We take into account only positive value The characteristic of the nonlinear element was given for the I>0 For the current I=2 we calculate the voltage:

The voltage and current of the nonlinear element are called the operating poit of the circuit Finding a circuit's dc operating points is an essential step in its design and involves solving systems of nonlinear algebraic equations. Then the operating point for our exapmle: U=2V I=2A U=2V I=2A

Graphical method (using load line) First we find with Thevenin method the equivalent source and resistance of the circuit on the left from AB-nodes (we remove the nonlinear element, then the circuit that remain is linear and we can use the Thevenin’s or Nortons’s Method) 1.

Than we can redraw our circuit using the calculated source and resistance:

We plot the characteristic of the nonlinear element and the summation characteristic of the source with resistor. U AB =E T —I*R T The operating point is as we can see the same U=2V I=2A U=2V I=2A

Graphical using graphical summing for Kirchoffs laws 1.We draw the characteristic of the nonlinear element 2.We draw the characteristic of parallel connected resistor R2 3.We are adding this two characteristics to obtain the relation of voltage U from the sum of I and I2; U=f(I+I2), I+I2=I, then the red line is for U=f(I1) Black line Blue line Red line

4. We draw the characteristic of the R1 resistor 5. We adding it to the obtained earlier sum of characteristic (red line) Brown line Pink line

4. We draw the characteristic of the ideal source 5. We mark the point the ideal source line and our sum of characteristic crosses as P – this point gives us the value of the current I1 4. We can also subtract the pink line from the brown one and see when the substracted characteristic cross zero. Brown dashed line

Then we can calculate all the curents and voltages in the circuit. We do it in reverse order as we drew characteristics. I1 – point P. I1 =4A For I1 we can see the voltage drops of the elements though this current flews U1 – point P1 – crosses of I1=4 with R1 characteristics (brown dash-dot line) U1=8V U – point P2 – crosses of I1=4 with function U=f(I1)=f(I+I2) (red dotted line) U=2V

Then for given U (U=2V from last step) we can read the currents of the branches with U drop of voltage (the branches between the A and B node) I2 – point P3 – crosses of U=2 with characteristic of resistor R1 (blue dotted), then I2=2A I – point P4 – crosses of U=2V with characteristic of nonlinear resistor R N (black curve) I=2A. The voltage and current of the nonlinear element R N (point P4) is the operating point of the circuit. U=2V I=2A U=2V I=2A Comment: For given set of parameters, the points: P3 and P4 are in the same place. But it is only a coincidence.

Comments 1. The methods can be combined, for example using Thevenin’s method in analytical method or calculating the fragment of the circuit with Thevenin’s method and then summing the characteristics. 2. U-I characteristic of nonlinear element could be given by a formula, in a table or on the graph 3. There is possible, that the characteristic on the graph is gives as U=f(I) (horizontal U and vertical I) or I=f(U) (horizontal I and vertical U) In the first case the summation is done as in example, but in the second case summation of voltages will be done horizontally and summation of currents vertically 4. The graphical method of summation is adecuate to summing the two nonlinear elements.

Another example An expamle taken from: AnantAgarwaland Jeffrey Lang, course materials for Circuits and Electronics, Spring MIT OpenCourseWare( Massachusetts Institute of Technology.

Incremental Analysis An expamle taken from: AnantAgarwaland Jeffrey Lang, course materials for Circuits and Electronics, Spring MIT OpenCourseWare( Massachusetts Institute of Technology.

For linear element:

~linear:

The incremental method (small signal method)

Mathematical consideration For the nonlinear: We assume: We can use Taylor’s expansion of f(v D ) near v D =V D : We neglect the higher order terms because Δv D =0

Mathematical consideration - 2

Static and dynamic resistance Static resistance Dynamic resistance Static Resistance is the normal ohmic resistance in accordance with Ohm's Law. It is the ratio of voltage and current and is a constant at a given temperature. Dynamic Resistance is a concept of resistance used in nonlinear circuit. Dynamic resistance refers to the change in current in response to a change in voltage at a specific region of the VI curve. When the voltage is increased, the current may not increase proportionally. In some cases, the current may actually decrease Dynamic conductance

Negative resistance Negative resistance is the phenomenon in which the current through a device decreases as the voltage increases. This is in contrast to conventional logic in which current increases as the voltage increases. While negative resistance exists, there is no such thing as a negative resistor. Negative Resistance exists along some are of the V-I curve in certain electronic components such as the Gunn Diode used in microwave electronics. In certain regions of the V-I curve, the current drops as the voltage rises.

source: wikipedia Negative resistance examples

Example for small signals For the given nonlinear resistor described by Flows the current: Using small singal method, calculate the voltage drop. Given a=2[MΩ/A 2 ], b=40[kΩ/A], c=0.4[kΩ], I 0 =10mA, I m =0.1mA 1. DC analysis SUPERPOSITION Then operating point: U0=10V I0=10mA U0=10V I0=10mA

1. DC analysis –dynamic resistance 2. AC analysis The Ohms law ! 3. Superposition

Example Tuneling diode with characteristic given at the graph is working in circuit system as shown below.Find the voltage drop on resistor R. J 0 =4mA, J m = 0.1mA, ω=10 6 rad/s, R=1kΩ, L=0.5mH, C=2nF. Answer: Istnieją dwa rozwiązania stabilne: Trzecie rozwiązanie odpowiadające stałoprądowemu punktowi pracy (U D =4V, I D =4mA) jest niestabilne i należy je odrzucić.