L and C connected together. To be able: To analyse some basic circuits.

Transcription

1 circuits: Sinusoidal Voltages and urrents Aims: To appreciate: Similarities between oscillation in circuit and mechanical pendulum. Role of energy loss mechanisms in damping. Why we study sinusoidal signals RMS urrent and Voltage To be able: To analyse some basic circuits. and connected together The capacitor is charged up to a voltage V What happens when the switch is closed?. discharges through. urrent in decays and charges with the reverse e.m.f. (in the reverse polarity) 3. discharges through and so on.

2 oscillation Energy V +V Pendulum h Energy mgh I +I v mv V -V h mgh I -I mv v 3 Form of oscillation The voltage across the inductor is but for the capacitor, di V = dt dv I = dt so This is the differential equation describing simple harmonic motion The solution is The details depend on the initial conditions. ω is the ANGUAR frequency (radians per second) The true frequency (oscillations per second or HERTZ) is ω f = = π π Hz 4

3 Form of oscillation The voltage across the inductor is but for the capacitor, so di V = dt dv I = dt di I = dt This is the differential equation describing simple harmonic motion The solution is I = I sinωt 0 V V cosωt 0 ω is the ANGUAR frequency (radians per second) The true frequency (oscillations per second or HERTZ) is The details depend on the initial = conditions. ω = 5 rad s - ω f = = π π Hz Relationship between peak current and voltage onservation of energy: High capacitance high current High inductance high voltage 6 3

4 Relationship between peak current and voltage onservation of energy: V = I I V = High capacitance high current High inductance high voltage 7 Sinusoidal oscillations V P Voltage urrent phase difference φ I P time Period τ Frequency = /τ Sine waves are fundamental to electronic systems The natural form of oscillations in circuits (and sound and radio waves) The form of voltages generated by rotating dynamos Any complex waveform can be built up from superpositions of fundamental sinusoidal waves (Fourier Series) 8 4

5 Damping With ideal components, the oscillation will continue indefinitely (no energy loss). With real components, there is resistance and power (I R) is dissipated on each cycle. R di Q + ir+ = 0 (KV) dt d I di + R + I = 0 dt dt This differential equation has a solution like I = I 0 exp( γ t )sin( ωt+ φ ) where γ is the damping coefficient γ = R and ω = 4 Damping term R Note a reduction in the frequency 9 Damped oscillations Time constant of decay is τ = = γ R exp(-γt) The number of oscillations in one time constant of the decay is called the quality factor, or Q of the circuit: γ Q = π ω When R is small, this is given approximately by Q = R High Q means long ringing time and high voltage (high ) 0 5

6 Fourier series f 3f 5f The first few HARMONIS (multiples of the fundamental frequency) can be used to reconstruct any regular waveform. The accuracy of the reconstruction improves as you increase the number of harmonics. time Time domain This is important because it means that we can predict the behaviour of an electronic circuit with any complex waveform by studying the effect on pure sine waves of different frequencies Jean Baptiste Joseph Fourier The son of a tailor in Burgundy he developed much of the mathematical basis of heat transfer, which led to the Fourier expansion. This was heavily criticised by aplace and agrange. His life was much affected by turbulent French politics. He narrowly escaped the guillotine during the Terror, and under Napoleon he was a senior administrator in Egypt (where he wrote his Description of Egypt) and Prefect of Grenoble. 6

7 Frequency ranges Frequency Period Applications a.k.a. 0 Hz onstant voltages, battery circuits D 0 00 Hz 50 / 60 Hz 50 Hz - 0 khz 00 ms 0 ms 0 ms 50 μs Power generation / transmission; mains sockets; TV frame rate Audible frequencies (speech, music), RS- 3, phone modem, TV line rate F audio AF 60 khz 00 MHz 5 μs 0 ns AM SW- FM radio; computer data bus; Ethernet; D sampling rate Radio RF, VHF 00 MHz 00 GHz 0 ns 0 ps Mobile phones, TV channels, satellite links, radar, microwave ovens, P clocks UHF Microwave > 00 GHz < 0 ps??? medical imaging? communication? high speed super highways? Terahertz remember: time constants must be less than this 4 Heinrich Rudolf Hertz Born in Hamburg and became Professor of Physics in Bonn (via Berlin and Karlsruhe). In 885 he was the first person to demonstrate experimentally the electromagnetic waves that had been predicted theoretically by Maxwell in the previous year. (Marconi did not begin his work on radio until 894). 5 7

8 Some general properties of sine waves Average voltage over one cycle is ZERO Since power depends on V we get an effective value by taking the square root of the average of the SQUARE of the voltage or current over one cycle: T P T 0 V = V sin ( ωt+ φ) dt with T (the period)= π ω VP This works out to VRMS = = 0.707V φ P V P V = V sin( ωt+ φ) P time This is the RMS - root-mean-square value of the voltage 6 RMS values and power For resistors (current and voltage in phase), RMS values of voltage or current can be used to calculate power dissipation: V RMS P= = I R R RMS Example: Mains voltage has a peak voltage of V so: RMS voltage is x = 40 V If we connect this to a heater with a resistance of 57.6 Ω the total power disputed is P=40 x 40/57.6 = 000 W = kw We can t use this when the current and voltage are not in phase. 7 8