Critical damping resistance measurement using ballastic galvanometer
For multiple aperture size laser beam projector
Abstract
A ballistic galvanometer will oscillate if it has not been properly damped. Galvanometers are damped by adding a
shunt resistor of just the right amount of resistance in parallel with them. The proper amount of resistance at which
the motion just ceases to be oscillatory is called the critical external damping resistance (CXDR). When shunted by
its CXDR, the galvanometer is said to be critically damped. With more resistance it is underdamped and with less
it is overdamped. When the galvanometer is critically damped, it will make one swing and return slowly to its zero
position.
Critical Damping
We can use these equations to discover when the energy dies out smoothly (over-damped) or rings (under-damped).
Look at the term under the square root sign, which can be simplified to: R2C2-4LC

When R2C2-4LC is positive, then α and β are
real numbers and the oscillator is over-damped.
The circuit does not show oscillation.

When R2C2-4LC is negative, then α and β are
imaginary numbers and the oscillations are
under-damped. The circuit responds with a sine
wave in an exponential decay envelope.

When R2C2-4LC is zero, then α and β are zero
and oscillations are critically damped. The
circuit response shows a narrow peak followed
by an exponential decay.
Purpose of the experiment:
Data table :
Observation no
Resistance Ω
Deflection Cm
cdrΩ
error
To observe damped oscillations in the RLC ci
rcuit and measure the amplitude, period, angular
frequency, damping constant and log decrement
of damped oscillatory signals. To find the
critical resistance for which the critical damping occurs.
2. What to learn?
Transfer of energy in LC circuit. The elec
trical-mechanical analogy. Differential equation
describing damped simple harmonic motion in
the RLC circuit.. Solution of this equation.
Angular frequency of the damped oscillator.
Damping constant. Angular frequency of the
undamped oscillator. Forced oscillations and re
sonance. Kirchhoff's rules. Log decrement of
damped oscillatory signals. Critical da
mping. How does the oscilloscope work?
Logarithmic decrement: The logarithmic decrement is defined a the ratio of any two successive peak amplitudes
natural logarithm as we can see in the damped simple harmonic motion if aA and A ‘ are two amplitude then
logarithmic decrement is log a/a’
The logarithmic decrement represents the rate at which the amplitude of a free damped vibration decreases. It is
Defined as the natural logarithm of the ratio of any two successive amplitudes. It is found from the time response of
under damped vibration (oscilloscope or real
time analyzer)
Figure 1Ballastic galvanometer
Electric oscillations can be excited in a circuit containing resistance R, inductance L and capacitance C. In terms of
topology, two types of circuits are often considered: series RLC-circuit (Figure 1) and parallel RLC-circuit (Figure
2).
Fig.1
Fig.2
We derive the differential equation describing the current change in a series RLC circuit.
The voltages VR, VC, VL, respectively, on the resistor R, capacitor C and inductor L are given by
It follows from the Kirchhoff's voltage law (KVL) that
where E(t) is the electromotive force (emf) of the power supply.
In the case of constant emf E, we obtain the following differential equation after substituting the expressions for VR,
VC, VL and differentiation:
If we denote
, the equation can be written as
This differential equation coincides with the equation describing the damped oscillations of a mass on a spring.
Hence, damped oscillations can also occur in series RLC-circuits with certain values of the parameters.
Now consider the parallel RLC-circuit and derive a similar differential equation for it.
By the Kirchhoff's current law (KCL), the total current is equal to the sum of currents through a resistor R, inductor
L and capacitor C (Figure 2):
Given that
For the case of constant total current I(t) = I0, we obtain the following differential equation of the second order with
respect to the variable V:
As one can see, we again have the equation describing the damped oscillations. Thus, the oscillatory mode can also
occur in parallel RLC-circuits.
Resonant Circuit. Thomson Formula
In the simplest case, when the ohmic resistance is zero (R = 0) and the source of emf is removed (E = 0), the
resonant circuit consists only of a capacitor C and inductor L, and is described by the differential equation
In this circuit there will be undamped electrical oscillations with a period
This formula is called the Thomson formula in honor of British physicist William Thomson (1824-1907), who
derived it theoretically in 1853.
Damped Oscillations in Series RLC-Circuit
The second order differential equation describing the damped oscillations in a series RLC-circuit we got above can
be written as
The corresponding characteristic equation has the form
Its roots are calculated by the formulas:
where the value of β = R/2L is called the damping coefficient, and ω0 is the resonant frequency of the resonant
circuit.
Depending on the values of R, L, C there may be three options.
Case 1. Overdamping: R2 > 4L/C
In this case, both roots of the characteristic equation λ1 and λ2 and real, distinct and negative. The general solution of
the differential equation is given by
In this mode, the current decreases monotonically, approaching zero (Figure 3).
Case 2. Critical Damping: R2 = 4L/C
This mode can be called boundary or critical. Here, both roots of the characteristic equation are equal, real and
negative. The general solution is expressed by the function
At the beginning of the process, the current may even increase, but then it quickly decreases exponentially.
Case 3. Underdamping: R2 < 4L/C
In this case, the roots of the characteristic equation are complex conjugate, which leads to damped oscillations in the
circuit. The change of current is given by
where the value of β = R/2L is, as above, the damping factor,
is the frequency of oscillation,
A, B are constants of integration, depending on initial conditions. Note that the frequency ω of damped oscillations
is less than the resonant frequency ω0 of the circuit. The typical shape of the curve I(t) in this mode is also shown in
Figure 3.
Fig.3
Forced Oscillations and Resonance
Fig.4
If the resonant circuit includes a generator with periodically varying emf, the forced oscillations arise in the system.
If the emf E of the source varies according to the law
then the differential equation of forced oscillations in series RLC-circuit can be written as
where q the charge of the capacitor,
.
This equation is analogous to the equation of forced oscillations of a spring pendulum, discussed on the page
Mechanical Oscillations. Its general solution is the sum of two components: the general solution of the associated
homogeneous equation and a particular solution of the nonhomogeneous equation. The first component describes the
decaying transient process, after which the behavior of the system depends only on the external driving force. The
forced oscillations will occur according to the law
where the phase φ is determined by the formula
Knowing the change of the charge q(t), it is easy to find the change of the current I(t):
where we have introduced the angle θ such that
. The angle
shift of the current oscillations I(t) with respect to oscillations in the supply voltage
The amplitude of the current I0 and the phase shift θ are given by
indicates the phase
.
The quantity
ohmic resistance R and a reactance
written as
is called the impedance, or impedance of the circuit. It consists of an
. Impedance of the resonant circuit in the complex form can be
We see from these formulas that the amplitude of steady-state oscillations of the current is maximum when
Resonance in the resonant circuit appears under this condition. The resonant frequency ω0 is equal to the frequency
of free oscillations in the circuit and does not depend on the resistance R.
We can transform the formula for the amplitude of the forced oscillations to get an explicit dependence on the
frequency ratio ω/ω0, where ω0 is the resonant frequency. As a result, we obtain
Dependencies of the current amplitude on the frequency ratio ω/ω0 for different values of R and C are shown in
Figures 5 and 6. These graphs are built at E = 100 V, L = 1 mH, Ñ = 10 µF (Figure 5), R = 10 ohms (Figure 6).
Fig.5
Fig.6
Resonance properties of a resonant circuit are characterized by the quality factor Q, which is numerically equal to
the ratio of the resonance frequency ω0 to the width Δω of the resonance curve at 1/√2 of the maximum value (see
Figure 4 above).
The Q factor in a series RLC circuit is given by
For a parallel RLC circuit, the Q factor is determined by the inverse expression:
Example 1
An electrical circuit consists of a series-connected resistor R = 100 ohms and a coil with inductance L = 50 H. At
time t = 0 a DC source with a voltage of V0 = 200 V is connected. Find:

the current change I(t) in the circuit;

the voltage change across the resistor VR(t) and the inductor VL(t).
Solution.
The series RL-circuit is described by the differential equation
In accordance with the general theory, the solution of this equation is the sum of the general solution of the
homogeneous equation Ih and a particular solution of the nonhomogeneous equation In : I = Ih + In. The general
solution of the homogeneous equation
is expressed as
where A is the constant of integration.
The solution of the nonhomogeneous equation In corresponds to the steady state in which the current in the circuit is
determined only by the ohmic resistance R:
. Then the total current varies according to the law
The constant A is determined from the initial condition I(t = 0) = 0. Consequently,
So, after the circuit is closed, the current will vary according to the law
The graph I(t) is shown in Figure 7.
The voltages VR across the resistor and VL ascross the inductor are determined by the following formulas:
The graphs of the functions VR(t) and VL(t) are shown in Figure 8.
Fig.7
Fig.8
Example 2
An electrical circuit consists of a series-connected resistor R = 100 ohms and a capacitor Ñ = 0.01 µF. At the initial
moment a DC source with a voltage of V0 = 200 V is connected to the circuit. Find:

the current change I(t) in the circuit;

the voltage change across the resistor VR(t) and the capasitor VC(t).
Solution.
This problem is similar to the previous and differs from it only by the type of electrical circuit. In this problem we
consider an RC-circuit.
According to Kirchhoff's current law (KCL)
where the voltage across the resistor is given by
As a result, we obtain the following differential equation to describe the transition process in the RC-circuit:
The solution of this equation is the sum of the general solution Vh of the homogeneous equation and a particular
solution Vn of the nonhomogeneous equation. The homogeneous equation has the general solution Vh in the form
where A is the constant of integration, depending on initial conditions.
A particular solution of the nonhomogeneous equation corresponds to the steady state in which
. Then the
voltage across the resistor will be zero and all the voltage is applied to the capacitor, that is, VC = V0. Thus, the
voltage change on the capacitor is described by
Given the initial condition VC(t = 0) = 0, we find the constant A:
Consequently, the voltage change on the capacitor will look like this:
The voltage across the resistor is determined by the formula
The current in the RC-circuit will vary according to the law
Graphs of the voltages VC(t), VR(t) and current I(t) are shown in Figures 9 and 10.
Fig.9
Fig.10
Example 3
An electrical circuit consists of a series-connected resistor R = 1 ohm, a coil with inductance L = 0.25 H and a
capacitor Ñ = 1 µF. How many oscillations will it make before the amplitude of the current is reduced by a factor of
e?
Solution.
In this circuit, damped oscillations will occur with a frequency
The amplitude of the oscillations will decrease according to the law
Suppose that N complete oscillations occurred for time t:
If the amplitude decreased by e times, then one can write the following equation:
Hence we find the number of oscillations N:
Example 4
An AC source with amplitude E0 = 128 V and frequency ω = 250 Hz is connected to a series circuit consisting of a
resistance R = 100 ohms, a coil with inductance L = 0.4 H and a capacitor Ñ = 200 µF. Find:

the current amplitude in the circuit;

the voltage amplitude on the capacitor.
Solution.
The current oscillations in the steady state occur with the amplitude
The amplitude of the voltage oscillations on the capacitor will be equal to