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Energy Storage Elements Capacitance/Inductance and RC Op Amp Circuits Kevin D. Donohue, University of Kentucky 1 Energy Storage in Electric Circuits Analogous to a spring storing the energy used to compress it or a flywheel storing the energy used rotate it … a capacitor can store energy in an electric field from the voltage used to move charge into it. i(t) t q Cv , i C dv 1 , v(t ) v(t0 ) dt C i ( x)dx , p(t ) Cv dv 1 , w(t ) Cv 2 dt 2 C - t0 + v(t) An inductor can store the energy in a magnetic field from the current used to create lines of flux around it. i(t) t di 1 Li , v L , i (t ) i (t0 ) dt L di 1 v( x)dx , p(t ) Li , w(t ) Li 2 dt 2 t0 Kevin D. Donohue, University of Kentucky L + v(t) 2 Examples Solve for voltages, currents, charge, power, and energy in simple circuits containing inductors and capacitors. Kevin D. Donohue, University of Kentucky 3 Ideal and Practical Models What happens if current changes instantaneously in an ideal inductor? What happens if voltage changes instantaneously in a ideal capacitor? What would be an equivalent model for an ideal inductor in a DC circuit? What would be an equivalent model for an ideal capacitor in a DC circuit? Kevin D. Donohue, University of Kentucky 4 Ideal and Practical Models A small amount of current leaks through the dielectric in an actual capacitor. A practical model can be constructed from 2 ideal lumped-parameter models The coils used to construct an inductor may have a significant resistance component. A practical model can be constructed from 2 ideal lumpedparameter models Kevin D. Donohue, University of Kentucky C Rleak L Rleak 5 Capacitor Combinations Series capacitors can be combined according to the following formula: C1 C2 … CN Ceq Ceq 1 N k 1 1 Ck Parallel capacitors can be combined according to the following formula: N … C1 C2 CN Ceq Ceq Kevin D. Donohue, University of Kentucky C k k 1 6 Inductor Combinations Series inductors can be combined according to the following formula: N … L1 L2 LN Leq Leq L k k 1 Parallel inductors can be combined according to the following formula: 1 Leq … L1 L2 LN Leq N k 1 Kevin D. Donohue, University of Kentucky 1 Lk 7 Examples Simplify circuits with series and parallel combinations of inductor and capacitors. Kevin D. Donohue, University of Kentucky 8 Application - Integrator Circuit Show that this circuit integrates the input signal vs(t) according to the equation below for time greater than 0: t C 1 vo (t ) vo (0) RC R + vs(t) vs ( x)dx 0 vo(t) - Kevin D. Donohue, University of Kentucky 9 Application - Differentiator Circuit Show that this circuit differentiates the input signal vs(t) according to the equation below: R dvs (t ) vo (t ) RC dt C + vs(t) vo(t) - Kevin D. Donohue, University of Kentucky 10